ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES ANDREAS SØJMARK Abstract. We spresent a detailed exposition of new and old ideas in the regularity theory for quasilinear systems with quadratic right hand side. The results are used to prove existence of Nash equilibria for certain stochastic differential games, following Bensoussan & Frehse (2002). Contents 1. Introduction 2 1.1. Stochastic Differential Games 3 1.2. Solutions to the type of SDE we consider 4 2. Stochastic Differential Games 6 2.1. The set-up of the game 6 2.2. Motivation for how to find a Nash Equilibrium 6 2.3. Strategy for finding a Nash Equilibrium 7 2.4. Proving the existence of a Nash Equilibrium 8 3. General Regularity for Non-linear Elliptic Systems 13 3.1. Regularity Results 13 4. Existence and Regularity for Stochastic Games 16 4.1. Regularity Results 16 4.2. The Existence Proof 20 4.3. Existence of Nash Equilibria 25 5. The Classical Theory of Quasilinear Equations 27 5.1. Quasilinear elliptic systems 28 5.2. Hölder regularity for Quasi-linear equations 30 5.3. Diagonal Systems 37 6. Epilogue 50 7. Appendix A - Stochastic Analysis 51 7.1. An Aside on Brownian Motion 51 7.2. Sobolev version of Ito’s Formula 51 8. Appendix B - Regularity 51 References 52 Date: The present state of the note was completed on 14th April 2016. Acknowledgements: I would like to thank Prof. Jan Kristensen for many fruitful and entertaining discussions during the writing of this note. The note is a survey of existing ideas in the literature and contains no original results. 1
Transcript
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMSAND STOCHASTIC DIFFERENTIAL GAMES
ANDREAS SØJMARK
Abstract. We spresent a detailed exposition of new and old ideas in the regularity theory forquasilinear systems with quadratic right hand side. The results are used to prove existence ofNash equilibria for certain stochastic differential games, following Bensoussan & Frehse (2002).
Contents
1. Introduction 21.1. Stochastic Differential Games 31.2. Solutions to the type of SDE we consider 42. Stochastic Differential Games 62.1. The set-up of the game 62.2. Motivation for how to find a Nash Equilibrium 62.3. Strategy for finding a Nash Equilibrium 72.4. Proving the existence of a Nash Equilibrium 83. General Regularity for Non-linear Elliptic Systems 133.1. Regularity Results 134. Existence and Regularity for Stochastic Games 164.1. Regularity Results 164.2. The Existence Proof 204.3. Existence of Nash Equilibria 255. The Classical Theory of Quasilinear Equations 275.1. Quasilinear elliptic systems 285.2. Hölder regularity for Quasi-linear equations 305.3. Diagonal Systems 376. Epilogue 507. Appendix A - Stochastic Analysis 517.1. An Aside on Brownian Motion 517.2. Sobolev version of Ito’s Formula 518. Appendix B - Regularity 51References 52
Date: The present state of the note was completed on 14th April 2016.Acknowledgements: I would like to thank Prof. Jan Kristensen for many fruitful and entertaining discussions duringthe writing of this note. The note is a survey of existing ideas in the literature and contains no original results.
1
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 2
1. Introduction
In the present note, we discuss systems of quasilinear elliptic PDE’s with quadratic growthin right-hand-sides, motivated by the study of so-called Nash equilibria in stochastic differentialgames. An intuitive introduction to the latter is given in Section 1.1 below, while Section 2 providesthe rigorous formulation of the problem.
As we shall see in Section 2, the analysis of an N -player stochastic differential game sits naturallywithin the framework of stochastic control theory. Here the Dynamic Programming Principletranslates the problem into the study of a system of elliptic PDE’s with one equation for eachplayer in the game.
These PDE’s turn out to be nonlinear of the form∑k,`
Dk(θk`D`ui) = Hi(x, u,Du), κI ≤ θ ≤ KI, (1.1)
The key analytical issue is then to obtain a sufficiently regular solution under a suitably generalquadratic growth in the gradient Du. In this respect, the main result of the present report isTheorem 4.7 due to A. Bensoussan & J. Frehse ([1]). While this is a purely PDE related result, itis precisely what is needed in order to have existence of so-called Nash equilibria for the games westudy (see Example 2.1 in Section 2.4.1 and, more generally, Section 4.3).
We mention briefly that the standard method for dealing with irregularities in control theory isto consider so-called viscosity solutions. However, this approach does not generalize very easily tosystems as it rests upon comparison principles. Consequently, the methods we employ are basedin elliptic regularity theory. A nice review of old and new results within this area can be found inthe survey paper [6] by J. Kristensen & G. Mignione.
The modern history of the problem begins with the results of E. De Giorgi ([19]) and J. Nash([20]), who in 1957 and 1958 proved, independently, the Hölder regularity of solutions to scalarequations
−div(θ(x)Du) = 0
with bounded measurable coefficients. In the subsequent years, the questions of regularity forgeneral linear and non-linear elliptic equations were more or less fully resolved in the scalar caseby the works of C.B. Morrey and O. Ladyzhenskaya & N. Ural’tseva as well as G. Stampacchia.While much effort was invested in generalizing these results to systems, in 1968 it was provenindependently by De Giorgi and V. Maz’ya that this is, in general, not possible.
In the later sections, we shall touch upon two different directions of research, which grew outof the above. One is the notion of almost everywhere Hölder regularity for solutions to generalquasilinear systems (see Section 5.2), while the other concerns everywhere Hölder regularity for asimpler class of quasilinear systems in so-called diagonal form, where there is no coupling on theleft hand side (see Section 5.3).
Our main system of interest (1.1) is a prime example of a quasilinear system in diagonal form.When the right-hand-side H has quadratic growth in the gradient, it is a classic result due tothe combined efforts of Wiegner ([13]) and S. Hildebrandt & K.-O. Widman ([17]), from around1975 and 1976, that bounded solutions to (1.1) are Hölder continuous under the so-called smallnesscondition
a ‖u‖L∞ ≤ κ, where |H(x, u,Du)| ≤ a |Du|2 + b. (1.2)
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 3
This is obviously a very rigid condition on the structure of the quadratic growth and hence con-siderable efforts have been put into uncovering various structural conditions, which allow for moreflexibility. Nevertheless, (1.2) turns out be sharp (see Section 5.3.1) and consequently such flexi-bility always comes with a trade-off in terms of imposing extra structure on H.
Some of the first efforts in this direction are due to Ladyzhenskaya & Ural’tseva (see [L&U])and were further developed by Hildebrandt & Widman (see [17, 18]). In the ending remarks of[17], Hildebrandt & Widman proves Hölder continuity in the special case where H is of the form
Hi(x, u,Du) = Hi0(x, u,Du) +Q(x, u,Du)Dui (1.3)
and satisfies the growth conditions∣∣Hi0(x, u,Du)
∣∣ ≤ a |Du|2 + b, |Q(x, u,Du)| ≤ a′ |Du|+ b′
with a smallness conditiona ‖u‖L∞ ≤ κ.
Note that this smallness condition only involves H0. In particular, the extra structure (1.3) intro-duces some flexibility for the quadratic growth of Hi in Dui via Q.
In Section 5 of [18], Hildebrandt & Widman considered another important “generalization”.Specifically, they showed that if H satisfies the growth condition
∣∣Hi(x, u,Du)∣∣ ≤ N∑
j=1
aij∣∣Dui∣∣2 + bi
then they can get away with a smallness condition, which for each i = 1, . . . , N only involves thecoefficients
aij with j > i.
The result of Bensoussan & Frehse ([1]) is essentially a combination of the two above “generaliza-tions”. We give a short introduction to this in Section 2.4.1. The full details are treated in Sections3 and 4. In terms of flexibility, we also emphasize the simple yet crucial idea of transforming theequations under a linear automorphism to discover better structures (see Section 4.1.2).
1.1. Stochastic Differential Games.The notion of a game necessarily involves a number of players, say i = 1, . . . N , and - if it is tobe of any interest - some sort of cost (or profit), which the players aim to minimize (or maximize)over the duration of the game. Furthermore, for the game to be really interesting, we would likethe cost (or profit) at time t to depend on (i) all that is relevant about the world at time t asquantified by a vector Xt ∈ Rd, which we call the state of the game at time t, and (ii) the actionsof all the players at time t both directly as well as indirectly via an ability to influence the stateof the game.
Saying that a game is differential means that it evolves continuously in time over an interval[0, T ), where the dynamics of the state, i.e. the mapping t 7→ Xt, is governed by a differentialequation. What makes the game stochastic is then that we require each Xt to be a random variableand thus take the state dynamics to be given by a stochastic process X = (Xt)t≥0 arising as thesolution to some appropriate SDE (as opposed to an ODE in the deterministic setting).
The key analytical issue is to decide whether or not the game possesses a so-called Nash equilib-rium, which is simply a set of strategies for how the players will act satisfying that no player canimprove on his or her situation if holding fixed the strategies of the other players.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 4
1.2. Solutions to the type of SDE we consider.We use again and again the fact that we can find a probability measure and a filtered backgroundspace (Ω,F , (Ft)t) for which there exists a stochastic process X satisfying an SDE of the type
dXt = b(Xt, αt)dt+ σ(Xt)dWt, X0 = x,
where W is a Brownian motion under the postulated probability measure and αt is a boundedprocess adapted to (Ft)t. This is possible under our assumptions that
(1) σ and its inverse are Lipschitz and bounded(2) b is measurable and bounded
Let us briefly show how this can be achieved. We first of all need a Brownian motion, so let us fixthe probability space (Ω,F ,P) given by the Wiener space and Wiener measure. As argued above,we know that this probability space supports a Brownian motion.
Let B denote this standard Brownian motion on (Ω,F ,P) and let (Ft)t denote the filtrationgenerated by B. Now, since σ is assumed Lipschitz, the SDE
dXt = σ(Xt)dBt, X0 = x
has a so-called strong solution, meaning that there exists a process Xx = (Xxt )t≥0 on (Ω,F)
adapted to (Ft) with
Xxt = x+
∫ t
0
σ(Xxs )dBs,
where B is the Brownian motion under P that we pinned down above.Since σ−1 and b are both bounded, we can define a bounded, adapted process Y by
Yt := σ−1(Xxt )b(Xt, αt).
Then the stochastic integral (∫ t
0
YsdBs
)t≥0
is a well-defined continuous martingale. Now introduce a new probability measure Q on (Ω,F),equivalent to P, by defining its density as
dQdP
:= e∫ ∞0
YsdBs− 12
⟨∫ ∞0
YsdBs
⟩= e
∫ ∞0
YsdBs− 12
∫ ∞0
|Ys|2ds.
By the Cameron-Martin (Girsanov) Theorem, the process
Wt := Bt −⟨B,
∫YsdBs
⟩t
= Bt −∫ t
0
Ysds
is a Brownian motion under Q with respect to the original filtration. Rewriting things, we have
Bt =Wt +
∫ t
0
Ysds
and, noting that σ(Xxs )Ys = b(Xx
t , αt), we thus have
Xxt = x+
∫ t
0
σ(Xxs )dBt = x+
∫ t
0
σ(Xxs )d(Wt +
∫ t
0
Ysds)
= x+
∫ t
0
σ(Xxs )YsdBs + σ(Xx
s )dWs
= x+
∫ t
0
b(Xxt , αt) +
∫ t
0
σ(Xxs )dWs.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 5
Since W is a Brownian motion under Q, this means that Xx is indeed a solution to our SDE underQ with respect to the original filtration (Ft)t.
The fact that we needed to introduce a new probability measure to solve the SDE, makes Xx aso-called weak solution. (Had b been Lipschitz like σ, then we would have had a strong solution).
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 6
2. Stochastic Differential Games
Below we list the essential ingredients of our stochastic differential game. These constitute thecanonical control-theoretic formalization of the intuitive presentation given in Section 1.1 above.
2.1. The set-up of the game.
• The actions of each player is modeled by a stochastic process αi called a control. We writeα := (α1, . . . , αN ) for the control of all the players.
• We define the set of admissible controls A to be all bounded processes α valued in anappropriate control space A and adapted to the filtered probability space below.
• The state Xt of the game at time t is given by the SDE
dXt = b(Xt, α)dt+ σ(Xt)dWt, X0 = x, (2.1)
where W = (Wt)t≥0 is a standard d-dimensional Brownian motion on a filtered probabilityspace (Ω,F , (Ft)t≥0,P) satisfying the usual conditions1.
• Here σ : Rd → Rd×d and its inverse are assumed Lipschitz and bounded, while b : Rd×A→Rd is only assumed measurable and bounded (not necessarily Lipschitz!).
• The state space O, where Xt(ω) is allowed to live, is given by a bounded smooth domainin Rd. We denote by τx := inf t ≥ 0 : Xx
t /∈ O the first exit time of O. Note also thatE(τx) <∞.
• The (expected) costs are given by the so-called cost functionals
Ji(x, α) := E∫ τx
0
(li(Xxt , αt) + fi(X
xt )) e
−∫ t0ci(X
xs ,αs)dsdt
+ φi(Xxτx)e
−∫ τx0
ci(Xxs ,αs)ds
and the objective of each player is to minimize this cost functional.
• A Nash equilibrium for the game is defined as a control α = (α1, . . . , αN ) s.t. for eachi = 1, . . . , N we have
Ji(x, αi, α−i) ≤ Ji(x, α) ∀αi ∈ Ai,
where α−i denotes α with the i’th component αi removed. Here Ai denotes the set of αi
s.t. (αi, α−i) ∈ A.
2.2. Motivation for how to find a Nash Equilibrium.Suppose for a moment that, for any starting point x, we are given a Nash equilibrium
α(x) = (α(x),1, . . . , α(x),N )
.Then each player can be treated as a standard stochastic control problem with associated value
functionvi(x) := min
αi∈Ai
Ji(x, αi, α(x),−i) = Ji(x, α
(x)), x ∈ O.
Note that vi = φi on ∂O. Indeed, for x ∈ ∂O, we have τx = 0 (since Xx is already outside O attime t = 0) and hence Ji(x, α) = φi(x) for any control.
1The usual conditions: (1) the probability space (Ω,F ,P) is complete, (2) F0 contains all P-nullsets, and (3) thefiltration is right-continuous meaning that Ft = ∩s>tFs.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 7
Remark 2.1. Given the (stationary) Markovian nature of the problem, it is natural to consideronly so-called (stationary) Markovian controls of the form
α(y)t = ψ(Xy
t ),
where Xy is solution to the SDE with initial condition X0 = y. In particular, we then have
α(Xx
s )t (ω) = ψ(X
Xxs (ω)
t (ω)) = ψ(Xxt+s(ω)) = α
(x)t+s (ω) .
In light of the remark, we set a(y) := α(y)0 (this is deterministic by F0-measurability) and note
that a(Xxt ) = α
(x)t .
Proceeding formally, it can be derived from the Dynamic Programming Principle (and Ito’sformula) that vi satisfies the HJB equation− 1
2 tr(σσTD2vi) = Hi(x, vi, Dvi) + fi(x) on O
vi = φi on ∂O
where
Hi(x, λi, pi) := minai∈Ai
[−ci(x, ai, a−i(x))λi + b(x, ai, a−i(x)) · pi + li(x, a
i, a−i(x))].
Now, a standard verification theorem (for classical solutions) tells us that we should be able toback out α(x),i as α(x),i
and Xx is the Markov process with initial value Xx0 = x and generator
(Lϕ)(y) = b(y, β(y), a−i(y)) ·Dϕ(y) + 1
2tr(σ(y)σT (y)D2ϕ(y)).
The latter simply means that Xx is, as required, a solution to our SDE (2.1) with
αt = (β(Xxt ), a
−i(Xxt )) = (α
(x),it , α
(x),−it ) = α
(x)t .
Here we have used a−i(Xxt ) = α
(x),−it cf. the earlier remark.
2.3. Strategy for finding a Nash Equilibrium.Consider the functionals
Li(x, λi, pi, a) := −ci(x, a)λi + b(x, a) · pi + li(x, a), i = 1, . . . , N (2.2)
The arguments in the previous section suggest that, holding (x, λi, pi) fixed, we should solve thesystem (2.2) for
a = a(x, λ, p), a locally bounded (2.3)
satisfying the Nash type conditions
Li(x, λi, pi, ai, a−i) ≤ Li(x, λi, pi, a
i, a−i) ∀ai ∈ Ai, i = 1, . . . N.
This is simply a system of N minimization problems in ai, i = 1, . . . , N . We then define theHamiltonians by
Hi(x, λ, p) := Li(x, λi, pi, a(x, λ, p))
= minai∈Ai
[−ci(x, ai, a−i)λi + b(x, ai, a−i) · pi + li(x, a
i, a−i)]
(2.4)
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 8
and consider the associated system of HJB equations− 12 tr(σσTD2ui) = Hi(x, u,Du) + fi(x) on O
ui = φi on ∂O, i = 1, . . . , N.
Note that this is a system of nonlinear elliptic PDEs. Now, if a (weak) solution to this systemexists, we would then want to extract a Markovian control α, candidating to be a Nash equilibrium,by noting that (in the notation of the previous section)
β(y) = ai(y, u(y), Du(y))
and hence we takeαit := β(Xx
t ) = ai(Xxt , u(X
xt ), Du(X
xt )).
For this to make sense, we first of all need u and Du(x) bounded in order for α to be bounded.This can be achieved if the solution u is C1 on O (since a is assumed bounded on bounded sets),which may be obtained by proving the existence of a weak solution u ∈W 2,s(O) for all 2 ≤ s <∞.Note also that t 7→ αt is then in particular continuous (a.s.).
Secondly, the verification argument will rely on a generalized version of Ito’s formula, which alsorequires sufficient regularity in the form of u ∈W 2,d(O) ∩W 1,2d(O).
2.4. Proving the existence of a Nash Equilibrium.As already emphasized, there are two main ingredients: (i) proving existence and regularity forthe HJB system, and (ii) verifying that the extracted control is indeed a Nash equilibrium.
To simplify the presentation, we adopt the following two conventions:
(1) We only consider f i ∈ L∞ in which case we can simply incorporate each f i in the respectiveHamiltonian Hi. Hence f will not feature explicitly in what follows.
(2) We consider Dirichlet boundary conditions, meaning that we take φi = 0.
The above assumptions are generally harmless as regards the situations of interest in game theory.Moreover, there is really no significant loss of generality as emphasized in the remark below.
Remark 2.2. If fi ∈ Lq for some q > n (and q ≥ 2), then the only difference is that theW 2,p-regularity will only go as high as the integrability of f . Nothing is changed in the exis-tence proof.
2.4.1. Existence and Regularity for the HJB system. We introduce the simplified notation
θ :=1
2σσT , Θ := −
∑i,j
θij∂2
∂xi∂xj,
where θ is obviously symmetric and θij ∈W 1,∞(O) by the Lipschitz assumptions on σ. Assumingalso that we can solve for a (locally bounded!) in (2.2)-(2.3), we let
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 9
and our system of PDEs is
Θui = Hi(x, u,Du), i = 1, . . . , N.
Since θij is differentiable (they are in W 1,∞), we can rewrite this in divergence form (after incor-porating the resulting first order terms in Hi). Hence, we are left with the following system ofquasi-linear elliptic PDEs−
∑k,`
∂∂xk
(θk`∂
∂x`ui) = Hi(x, u,Du)
ui ∈W 2,s(O) ∩W 1,s0 (O) ∀s > d (s 6= ∞)
, (2.6)
under the conditionsθk` ∈W 1,∞(O), θk` = θ`k, ξθξT ≥ κ |ξ| . (2.7)
The contribution of A. Bensoussan & J. Frehse is that this can be solved under the followingconditions on the Hamiltonians in (2.5):
∣∣Hi0(x, s, p)
∣∣ ≤ ki +
N∑n=1
Kin |pn|
2, i = 1, . . . , N
Q(x, s, p) ≤ k +K |p| (2.8)
(s, p) 7→ H(x, s, p) continuous
where the Kin must also satisfy certain smallness conditions for n > i.
They key aspect here is that H(x, u,Du) is allowed to have quadratic growth in Du (exceptfrom the extra smallness conditions, which place some constraints on this). This is very importantbecause of the nature of the specific Hamiltonians that arise in stochastic game theory (for a simpleexample, see below).
Example 2.3. Consider a stochastic differential game with the SDE (2.1) given by
drift b(x, a) = g(x) +
N∑j=1
aj and diffusion matrix σ(x) ≡ I.
Let the cost functional Ji (as defined in section 2.1) be given by
li(x, a) =1
2
∣∣ai∣∣2 + πai · (∑j 6=i
aj), ci(x, a) ≡ c, f i ∈ L∞.
Then we have
Li(x, λi, pi, a) = −cλi +1
2
∣∣ai∣∣2 + πai · (∑j 6=i
aj) + pi ·(g(x) +
N∑j=1
aj), i = 1, . . . , N.
In order implement (2.2)-(2.5), we need to minimize this system in the ai’s. The first ordercondition for this amounts to
ai + π(∑j 6=i
aj) + pi = 0.
Provided thatπ 6= 1 and π 6= − 1
N − 1, (2.9)
this system has a unique solution p 7→ a(p) given by
ai(p) =π∑
j pj
(1− π)(1 + (N − 1)π)− pi
1− π. (2.10)
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 10
Using this (see Section 3.1.2 in Chapter 3 of [B&F]), the Hamiltonian can be computed as
Hi(x, λ, p) = −1
2
∣∣ai(p)∣∣2 − 1
πpi · (pi + ai(p)) + cλi + g(x) · pi. (2.11)
Since σ ≡ I, the differential operator is simply − 12∆ and hence the corresponding system of PDE’s
is−1
2(∆I)u = H(x, u,Du) + f, (2.12)
where H is given by (2.4)-(2.5) which has rather general quadratic growth in Du.which has rathergeneral quadratic growth in Du.
In Section 4.3, we will return to this example and show that the system of PDEs (2.12) has asufficiently regular solution. Hence the game has a Nash Equilibrium.
Once the existence and regularity theorems are in place, this boils down to verifying that,under a certain linear transformation of the problem (see Section 4.1.2), the Hamiltonians satisfythe structural conditions (2.8) as well as a crucial smallness condition (to be specified in Section4.1.1).
2.4.2. The Verification Argument. Let u ∈W 2,s denote the solution to our HJB system (2.6)and recall that our candidate Nash equilibrium control is
αt := a(Xxt , u(X
xt ), Du(X
xt )), (2.13)
where Xxt is the diffusion given by the SDE (2.1) from section 2.1.
As step 1, we show that ui(x) = Ji(x, α) and, as step 2, we then show that ui(x) ≤ Ji(x, αi, α−i)
for all admissible controls αi of player i.Step 1. Recall that (by definition) we have
Hi(y, u(y), Du(y))
= −ci(y, a(y, u(y), Du(y)))︸ ︷︷ ︸ui(y)=:ci(y)
+ b(y, a(y, u(y), Du(y)))︸ ︷︷ ︸=:b(y)
·Dui(y) + li(y, u(y), Du(y))︸ ︷︷ ︸=:li(y)
= −ci(y)ui(y) + b(y) ·Dui(y) + li(y)
Given the W 2,s regularity, our PDE is then the following a.e. equality of Ls-functions:
We shall need this in a second, but first we need to incorporate the diffusion Xxt into our
analysis. To this end, observe that, for a continuous semi-martingale Xt, stochastic integration byparts yields
d(ui(Xt)e
−∫ t0ci(Xs)ds
)= ui(Xt)d
(e−
∫ t0ci(Xs)ds
)+ e−
∫ t0ci(Xs)dsdui(Xt) (2.15)
= −ui(Xt)ci(Xt)e−
∫ t0ci(Xs)dsdt+ e−
∫ t0ci(Xs)dsdui(Xt).
Let X be the diffusion given as a solution to the SDE
dXt = b(Xt)dt+ σ(Xt)dWt, X0 = x
under some probability measure Px.2
2All equalities involving Xt are of course only Px-a.s. statements, but we suppress this from the notation.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 11
Assume for a moment, that u is in fact C2. Then Itô’s formula (and the SDE) implies that
dui(Xt) = Dui(Xt) · dXt +1
2D2ui(Xt) : d 〈X〉t
=(Dui(Xt) · b(Xt)dt+Dui(Xt) · σ(Xt)dWt
)+
1
2tr(σσT (Xt)D
2ui(Xt))dt.
Combined with (2.15), for the interval from 0 to τ = inf t ≥ 0 : Xt /∈ O, this reads as follows:
ui(x) = ui(Xτ )e−
∫ τ0
ci(Xs)ds − ui(X0)e−
∫ 00ci(Xs)ds =∫ τ
0
e−∫ r0ci(Xs)ds
[−ui(Xr)ci(Xr) +Dui(Xr) · b(Xr) +
1
2tr(σσT (Xr)D
2ui(Xr))
]dr (2.16)
+
∫ τ
0
e−∫ r0ci(Xs)dsDui(Xr) · σ(Xr)dWr.
For the first equality we have used that X0 = x and ui = 0 on ∂O, so ui(Xτ ) = 0 since Xτ ∈ ∂O.Now go back to u ∈W 2,s (O) and pretend (2.16) is still fine (we argue below that this is indeed
legitimate).Let Λ ⊆ Rd be a Lebesgue measurable set with Ld(Λc) = 0 s.t. the PDE (2.14) is a pointwise
equality for all y ∈ Λ. Note that, since X is a non-degenerate diffusion (i.e. σσT is strictly pos.def.) and Ld(Λc) = 0, we have L(t ∈ [0, τ) : Xt ∈ Λc) = 0 (Px-a.s., that is)3.
Hence (2.16) simplifies to
ui(x) =
∫ τ
0
e−∫ r0ci(Xs)ds li(Xr)dr −
∫ τ
0
e−∫ r0ci(Xs)dsDu(Xr) · σ(Xr)dWr.
With respect to Px, the final term is a martingale (stochastic integral of bounded integrand againsta Brownian motion) and hence its expectation vanishes. Consequently, taking expectations we get
ui(x) = EPx
[∫ τ
0
e−∫ r0ci(Xs)ds li(Xr)dr
]= EPx
[∫ τ
0
e−∫ r0ci(Xs,αs)dsli(Xr, αr)dr
]. (2.17)
Here we have used the fact that αs = a(Xs, u(Xs), Du(Xs) and hence
Since X satisfies our SDE (2.1) wrt. Px, this is precisely the statement that
ui(x) = Ji(x, α).
The only problem is that we used u ∈ C2 to get (2.16) when we only have Ls weak derivatives,but this can be shown to work. Indeed, by boundedness of the coefficients it can be shown, usingthe uniform ellipticity assumption ξθξT ≥ κ |ξ|, that the right-hand-side of (2.16) is well-definedfor u ∈ W 2,d ∩W 1,2d. A standard approximation argument using mollifiers then shows that theequality holds for u ∈W 2,d ∩W 1,2d. See p. 122 in [Krylov] for more details.
Remark 2.4. For classical solutions to the linear elliptic PDE (2.14), the representation of uiin (2.17) is often referred to as a Feynman-Kac representation. Our arguments show that thisrepresentation remains valid in a Sobolev setting.
Step 2. It remains to show that ui(x) ≤ Ji(x, αi, α−i) for all αi ∈ Ai. Recall that α was
defined via the map a : x 7→ a(x, u(x), Du(x)) in(2.13). By definition (see (2.3)-(2.4)), this map a
3The key point is that the law of each Xt has density wrt. the Lebesgue measure (e.g. for b = 0 and σ = I, X is aBrownian motion, so Bt has Gaussian density). Fubini then yields the conclusion:
E [L (t ≤ τ : Xt ∈ Λc)] = E[∫ τ
0 1Xt∈Λc(t)dt]=
∫ τ0 P [Xt ∈ Λc] dt =
∫ τ0
(∫1Λc (x)
dXt(P)dLd (x)dx
)dt = 0.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 12
The definitions of c, b, l should be obvious. Note that they satisfy
c(Xt, ·) = ci(Xt, ·, a(Xt)−i) = ci(Xt, ·, α−i), likewise for band l. (2.18)
With these definitions, the PDE for player i is just
Θui = minai∈Ai
[−ci(x, ai)ui + b(x, ai) ·Dui + li(x, a
i)]. (2.19)
Now, for any control αi ∈ Ai, introduce a diffusion Xt given by
dXt = b(Xt, αit)dt+ σ(Xt)dWt, X0 = x
(2.18)= b(Xt, α
it, α
−it )dt+ σ(Xt)dWt
under a probability measure Pxαi . Then define an objective functional (for player i alone) as
J(x, αi) := EPxαi
[∫ τ
0
e−∫ r0ci(Xs,α
is)dsli(Xr, α
ir)dr
](2.20)
(2.18)= EPx
αi
[∫ τ
0
e−∫ r0ci(Xs,α
is,α
−is )dsli(Xr, α
ir, α
−ir )dr
]= Ji(x, α
i, α−i),
where the last equality is just by its definition in Section 2.1.If ui was a classical solution of the PDE (2.19), then it is a standard verification theorem in
stochastic control that ui(x) is the value function corresponding to J in (2.20) and hence
ui(x) ≤ J(x, αi) ∀αi ∈ Ai.
Since J(x, αi) = Ji(x, αi, α−i), this is precisely what we needed to prove.
Of course, in our situation, ui only has weak derivatives. But the proof of the verificationtheorem is just an application of Itô’s formula in the same spirit as Step 1 above and hence all weneed is to note that the Itô formula still works in our particular Sobolev setting (as we argued inStep 1).
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 13
3. General Regularity for Non-linear Elliptic Systems
Let u denote an RN -valued vector function u = (u1, . . . , uN ) and suppose
u ∈(H1
0 (Ω))N ∩ (L∞(Ω))
N. (3.1)
We shall consider a uniformly elliptic operator Θ in divergence form, defined by
Θui = −∑k,`
∂
∂xk(θk`
∂
∂x`ui), (3.2)
where θ ∈ L(Rd,Rd×d) is a measurable map satisfying
κ |ξ|2 ≤ ξθ(x)ξT ≤ K |ξ|2 ∀ξ ∈ Rd. (3.3)
Remark 3.1. Compared to a general system, we emphasize that θ is the same for each ui andadditionally there is no coupling in the operator Θ. Hence the tensor of coefficients
(θµνk` )µ,ν=1,...,N
k,`=1,...,ds.t. Θuν = −
N∑µ=1
d∑k,`=1
∂
∂xk(θµνk`
∂
∂x`uµ)
is simply given by
θµνk` = δµνθk` =
θk` if µ = ν
0 if µ 6= ν.
Notice that (3.3) is then precisely the Legendre condition for (θµνk` ). Indeed, ∀ξ ∈ Rd×N ,N∑
µ,ν=1
d∑k,`=1
θµνk` ξkµξ`ν =
N∑i=1
d∑k,`=1
θk`ξkiξ`i =
N∑i=1
(θT coli(ξ)) · coli(ξ)
≥ κ
N∑i=1
|coli(ξ)|2 = κ |ξ|2 .
Finally, we assume Ω is a nice (bounded) domain with Lipschitz boundary, so in particular thefollowing sphere-condition holds:
∀x ∈ ∂Ω : |BR(x) ∩ Ωc| ≥ λRd, λ > 0. (3.4)
3.1. Regularity Results.Given a vector s ∈ RN , we use |·| to denote the standard euclidean norm, but we shall also need|·|∞ defined by
|s|∞ := max∣∣s1∣∣ , . . . , ∣∣sN ∣∣ .
For technical reasons (to be clear later), we introduce a function X0 : RN → R satisfying thefollowing properties:
X0(s) ≥ 0, X0(0) = 0, (3.5)∣∣∣∣∂X0
∂si(s)
∣∣∣∣ ≤ ∣∣si∣∣β(|s|∞), (3.6)
λ 7→ β(λ)is positive, increasing for λ ≥ 0. (3.7)
An important consequence of this is that X0 satisfies
X0(s) ≤1
2|s|2 β(|s|∞). (3.8)
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 14
This follows easily from (3.6) and (3.5) by writing
X0(s) =
∫ 1
0
DλX0(λs)dλ =
∫ 1
0
∑i∂iX0(λs)s
idλ ≤∫ 1
0
∑i
∣∣λsi∣∣β(|λs|∞)sidλ
≤ β(|s|∞)∑
i(si)2∫ 1
0
λdλ =1
2|s|2 β(|s|∞).
3.1.1. Gaining Integrability. Given u, we single out the vectors
We can now state the key condition that will allow us to get W 1,p estimates:
Assumption 1. ∃k0,K0 > 0 depending only on ‖u‖L∞ s.t. ∀c, ψ satisfying (3.9),(3.10)we have:
k0
∫Ω
|Du|2 ψdx+
∫Ω
θDX0(u− c) ·Dψdx ≤ K0
∫Ω
ψdx.
Here θ is the matrix of the elliptic operator Θ defined in (3.2),(3.3). Assumption 1 is the key toour first regularity result below, which in essence is a consequence of the famous Gehring’s Lemmaapplied to an estimate obtained from Assumption 1.
Theorem 3.2. Assume Ω is Lipschitz, θ is uniformly elliptic, and u ∈ H10 ∩ L∞ as listed in
(3.1)-(3.4). Let X0 be as defined in (3.6)-(3.7) and suppose Assumption 1 is satisfied. Then thereexists ε0 = ε0(κ,K,Ω, ‖u‖L∞) s.t.
u ∈ (W 1,p0 (Ω))N ∀p ∈ [2, 2 + ε0),
where ‖u‖W 1,p0
≤ Cp for Cp = C(κ,K,Ω, ‖u‖L∞ , p).
Proof. See Theorem 2.3, p. 68 in [B&F]. The proof is primarily about obtaining some fairly simpleestimates using the traditional tools of cut-off function and Hölder and Poincare inequalities. Thenone applies a corollary of Gehring’s Lemma.
3.1.2. Hölder Regularity. Consider the elements in (D′(Ω))N given by (f i0 − divf i)i, where
f i0 ∈ Lp(Ω), p > d2 , and f i = (f i1, . . . , f
iN ) ∈ (Lq(Ω))
N, q > d. (3.11)
We shall be considering the cases where there exist such functions (3.11), satisfying the followingrelationship with our differential operator Θ:
(i) Θui − f i0 + divf i ∈ L1(Ω)
(ii)∣∣Θui − f i0 + divf i
∣∣ ≤ mi +M i |Du|2 ,
for mi = mi(‖u‖L∞)and M i =M i(‖u‖L∞).
(3.12)
An obvious way for this to hold is, of course, if for some functions Hi(x, u,Du) with quadraticgrowth in Du, we have a weak solution u ∈ H1 ∩ L∞ of
Θui = Hi(x, u,Du) + f i0 − divf i.
Given the above, we now introduce a slightly altered version of Assumption 1, which will be thekey structural condition allowing us to obtain Hölder regularity.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 15
Assumption 2. ∃k0,K0 > 0 depending only on ‖u‖L∞ s.t. ∀c, ψ satisfying (3.9),(3.10)we have:
k0
∫Ω
|Du|2 ψdx +
∫Ω
θDX0(u− c) ·Dψdx
≤ K0
∫Ω
((1 + |f0|+ |f |2)ψ + |f | |Dψ|
)dx.
for some functions f0, f satisfying (3.11),(3.12).
Here X0 should of course satisfy its defining properties (3.5)-(3.7).
Theorem 3.3. Assume Ω is Lipschitz, θ is uniformly elliptic, and u ∈ H10 ∩ L∞ as listed in
(3.1)-(3.4). Assume also that there exists f0, f satisfying (3.11),(3.12). Finally, let X0 be asdefined in (3.6)-(3.7) and suppose Assumption 2 holds. Then there exists 0 < δ0 < 1 s.t.
u ∈ C0,δ(Ω) ∀δ ≤ δ0 where δ0 = δ0(κ,K, ‖f0‖Lp , ‖f‖Lq , ‖u‖L∞)
with an estimate of the form
‖u‖C0,δ ≤ C where C = C(κ,K, ‖f0‖Lp , ‖f‖Lq , ‖u‖L∞).
Proof. See Theorem 2.10, pp. 74-80 in [B&F]. This proof is quite technical. It relies heavily onproperties of the Green function and finishes off with Widman’s hole-filling technique.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 16
4. Existence and Regularity for Stochastic Games
As derived in Section 2, we consider the following system of quasilinear elliptic PDE’s(ΘI)u = H(x, u,Du) on Ω
u = 0 on δΩ, for a smooth bounded domain Ω, (4.1)
where the second order differential operator Θ is given in divergence form by
(iii)′ ρ 7→ k(ρ), ρ 7→ K(ρ) are positive, increasing.(4.4)
Furthermore, we introduce the quintessential smallness conditions alluded to at the end of Section2.4.1.
4.1.1. The Smallness Conditions. Whenever i < `, the coefficient Ki` is required to satisfy a
so-called “smallness” condition, which we are now ready to state in detail. Note that there is nocondition on the coefficients KN
1 , . . . ,KNN for HN , so the N ’th equation can have fully general
quadratic growth in Du.The need for special conditions on the K’s arises as we seek to gain regularity by establishing
estimates satisfying Assumptions 1 and 2 from Section 3. Specifically, we shall need the followinginequality for ρ = ‖u‖L∞ in order to achieve this (see the proof of Theorem 4.2):
The Smallness Property. ∃γi(ρ), λi(ρ) positive, increasing functions of ρ ≥ 0 s.t.∀i = 1, . . . , N it holds ∀ρ > 0 that
κγi(ρ)(λi(ρ))2 − γi(ρ)λi(ρ)Kii (ρ)−
K(ρ)
4κ
−∑6=i
γ`(ρ)λ`(ρ)K`i (ρ) exp
2ρλ`(ρ)
> 0
The obvious issue here is that, choosing the λi’s large enough might make the first line positive,but because of the exponential in the second line, there is no hope of making the full expressionpositive in general.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 17
Hence we need to introduce a very particular scheme for choosing the functions γi, λi that makesit clear what will be the appropriate smallness conditions on the K’s. Specifically, we define thefunctions recursively in the following backwards fashion:
and then, for i = N − 1, . . . , 1, we choose γi, λi recursively s.t.
λi(ρ)γi(ρ)(κλi(ρ)−Ki
i (ρ))−K(ρ)/4κ
−∑
n>iγn(ρ)λn(ρ)Kn
i (ρ) exp 2ρλn(ρ) > 0, κλi(ρ)−Ki
i (ρ) > 0 (4.6)
These choices are clearly possible by choosing the λi’s large enough. Indeed, (4.5) is a simplequadratic in λN and for (4.6) we note that the exponential only involves λ`’s for ` > i, which arejust fixed constants at “step i” of the recursion.
Observe that (4.5) and (4.6) only involve K`i for ` ≥ i, for which we are not imposing any
conditions. On the other hand, we can now restate the above Smallness Property in the followingway − in terms of assumptions solely on the K`
i ’s for ` < i:
The Smallness Condition. Let γi, λi be the functions defined above. Then, for eachi = 2, . . . , N , the K`
i ’s for ` < i should satisfy
∑`<i
γ`(ρ)λ`(ρ)K`i (ρ) exp
2ρλ`(ρ)
<
λi(ρ)γi(ρ)
(κλi(ρ)−Ki
i (ρ))− K(ρ)
4κ
−∑`>i
γ`(ρ)λ`(ρ)K`i (ρ) exp
2ρλ`(ρ)
Notice that all γ`, λ` are already fixed from (4.5),(4.6) in such a way that the RHS is strictlypositive, so this is purely a statement about the K`
i ’s for ` < i being sufficiently small. In particular,if K`
i = 0 whenever ` < i, then the Smallness Condition (and hence the Smallness Property) istrivially satisfied.
4.1.2. Change of Unknown Functions. In practice, a given system of PDEs may fail to satisfythe conditions in Section 4.1.1 straight out of the box, yet satisfy them after a linear change ofthe unknown functions. As long as the essential nature of the problem does not change, this isperfectly sufficient for our purposes.
To be specific, let Γ be a constant invertible N×N matrix and consider the “change of unknownfunctions” given by
Supposing that the transformed Hamiltonian H satisfies the growth assumptions (4.3),(4.4) andthe Smallness Property, we can then simply solve the system (4.7) for v and convert this to a
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 18
solution of the original system by means of u = Γ−1v. Consequently, we introduce the followingimportant convention.
Definition 4.1. The Hamiltonian H, or just the system as a whole, is said to satisfy a structure(a certain set of properties or conditions, say) if the transformed Hamiltonian H in (4.8) satisfiesthat particular structure for some invertible constant matrix Γ.
Despite its simplicity, it is worth emphasizing the significance of the above idea. First andforemost, it enables us to solve systems that á priori appear unsolvable by our machinery. Secondly,it can allow our system to have more general quadratic growth (e.g. violating the smallnesscondition) as long as there is some transformed Hamiltonian enjoying the appropriate properties.This latter observation will, of course, in general entail some kind of trade-off in terms of otherconditions placed on the original Hamiltonian in order to ensure the existence of a transformedHamiltonian satisfying the smallness condition and the right growth conditions (as set out inSection 2.1.1).
4.1.3. The Main Regularity Results. The game here is to define, in a clever way, a functionX0 satisfying the requirements in (3.5)-(3.7) and then test the distributional formulation of oursystem (4.1) with an equally clever test function. Combined with the Smallness Property (fromSection 4.1.1) this will then provide us with estimates in the form of Assumptions 1 and 2 (fromSections 3.1.1 and 3.1.2, respectively), which in turn will allow us to apply the general regularityresults of Section 3.
The proof of the next theorem presupposes the existence of a weak solution to ΘIu = H(x, u,Du)
in H10 (Ω) ∩ L∞(Ω). Since it is á priori only known that H(x, u,Du) ∈ (L1(Ω))N , via the growth
condition∣∣Hi∣∣ .u |Du|2 in (4.3), we emphasize that the solution should be understood in the sense∫
Ω
ΘIu · φ =
∫Ω
H(x, u,Du) · φ ∀φ ∈ (H10 (Ω) ∩ L∞(Ω))N .
Theorem 4.2. Suppose u ∈ H10 (Ω) ∩ L∞(Ω) is a solution to the system ΘIu = H(x, u,Du) as
given by (4.1),(4.2). Assume also that H satisfies the growth conditions in (4.3),(4.4) as well asthe Smallness Condition. Then
u ∈W 1,p0 (Ω) ∀p ∈ [2, 2 + ε0) and u ∈ C0,δ(Ω) ∀δ ∈ [0, δ0),
where the constants ε0, δ0 > 0 and the bounds on the norms only depend on the values
κ, k(‖u‖L∞), K(‖u‖L∞), ki(‖u‖L∞), Kin(‖u‖L∞).
Proof. For a detailed proof, see Theorem 3.7, pp.122-127 in [B&F]. As sketched above, the keyingredient is the Smallness Property, which yields the estimates needed to apply Theorem 3.2 (forthe gain in integrability) and Theorem 3.3 (for the Hölder regularity). Let us briefly see how thisworks.
We use the notation as introduced in the previous sections. On pp. 122-126 of B&F a cleverfunction X(s) is constructed s.t. X0(s) := X(s) − 2N satisfies the defining properties (3.5)-(3.7)and some technical estimates for the weak solution u is then performed.
For each i = 1, . . . , N , a slight variation Xi of X is defined, satisfying
2N ≤ X(s) ≤ Xi(s) ≤ X(s)e∣∣si∣∣ ≤ C(‖u‖L∞). (4.9)
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 19
Now, for tests functions ψ satisfying the defining properties (3.9)-(3.10), the final estimate of theform:∑
i
∫Ω
∣∣Dui∣∣2 Xi
[κγi(λi)2 − γiλiKi
i −K
4κ−∑6=i
γ`λ`K`i exp
2ρλ`
]dx
+
∫Ω
θDX ·Dψdx ≤∫Ω
[k
4κX +
∑i
γiλikiXi
]ψdx,
where all the functions from Section 4.1.1 (on the smallness condition) are taken to be evaluatedat ρ = ‖u‖L∞ .
The expression in brackets on the LHS is precisely the one from the Smallness Property. Writingε(i) for this expression, the Smallness Property says that ε(i) > 0 for each i = 1, . . . , N . From thisand (4.9) we deduce that∑
i
∣∣Dui∣∣2 Xiε(i)−k
4κX −
∑i
γiλikiXi ≥ 2N∑i
∣∣Dui∣∣2 ε(i)−( k
4κ+∑i
γiλiki
)C(‖u‖L∞)
≥ k0 |Du|2 −K0
for some constants k0,K0 > 0. This obviously depends crucially on the fact that ε(i) > 0 for all i.Plugging this back into the estimate, we obtain that
k0
∫Ω
|Du|2 ψdx+
∫Ω
θDX ·Dψdx ≤ K0
∫Ω
ψdx.
Since DX = DX0, this is precisely what we need in order to satisfy Assumptions 1 and 2 fromSection 3. Hence the claims follow by appealing to Theorem 3.2 and Theorem 3.3.
4.1.4. L∞-bounds. In this section, the goal is to obtain specific L∞ bounds for solutions u ∈H1
0 (Ω) ∩ L∞(Ω). In order to obtain this, we need some preliminary considerations.
Definition 4.3. Consider a constant invertible matrix Γ as in Section 4.1.2. Then we say that Γ
obeys a maximum principle if Γu ≥ 0 implies u ≥ 0.
By writing u = Γ−1Γu we see that it suffices for Γ to be inverse-positive (meaning that theinverse Γ−1 is a nonnegative matrix). A particular example of a family of inverse-positive matricesis the set of positive definite matrices with non-positive off-diagonal terms.
We are now in a position to state the final major structural assumption for our system, whichconcerns the nature of our bounds on the Hamiltonians from above and below.
The Boundedness Condition. We assume there exists a constant invertible matrixΛ, which satisfies the maximum principle. Given this, we assume there exists constantsλ, c, λi, ci, αi > 0 and positive increasing functions λ0(ρ), λi0(ρ) s.t.:Either
(i)
∑
`Λi`H`(x, s, p) ≤ λi + λi0(|s|∞)
∣∣∑`Λi`p
`∣∣2 − ci
∑`Λi`s
`∑`α`H
`(x, s, p) ≥ −λ− λ0(|s|∞)∣∣∑
`α`p`∣∣2 − ci
∑`α`s
`
or
(ii)
∑
`Λi`H`(x, s, p) ≥ −λi − λi0(|s|∞)
∣∣∑`Λi`p
`∣∣− ci
∑`Λi`s
`∑`α`H
`(x, s, p) ≤ λ+ λ0(|s|∞)∣∣∑
`α`p`∣∣2 − ci
∑`α`s
`
where the inequalities should hold for all i = 1, . . . , N .
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 20
Consider the case (i) with Λ = I (meaning that H itself satisfies the bounds). Then the firstinequality amounts to
Θui = Hi(x, u,Du) ≤ λi + λi0(‖u‖L∞)∣∣Dui∣∣− ciui
in the sense of distributions. Hence a weak maximum principle (see Theorem 7.1, Appendix B)implies that
ui ≤ λi/ci. (4.10)
On the other hand, if we introduce u(x) :=∑
` α`u`(x), then by linearity of Θ, the second inequality
becomesΘu =
∑`α`H
`(x, u,Du) ≥ −λ− λ0(‖u‖L∞) |Du|2 − ciu.
Thus another application of the weak maximum principle yields u ≥ −λ/c and consequently
ui =1
αi(u−
∑` 6=i α`u
`) ≥ − λ
cαi−∑` 6=i
α`
αiu` ≥ − λ
cαi−∑` 6=i
α`
αi
λ`
c`. (4.11)
The case (ii) yields similar estimates as (4.10) and (4.11), only in the opposite directions.In the case of a general linear transformation Λ, completely analogous arguments can be per-
formed for the transformed system Av = ΛH(x,Λ−1v,Λ−1Dv), leading to the following result.
Theorem 4.4. Let u ∈ H10 (Ω)∩L∞(Ω) be a solution of ΘIu = H(x, u,Du) as given by (4.1),(4.2).
Assume also that the Boundedness Condition is satisfied.If (i) holds, then we have
− λ
cαi−∑` 6=i
∑k
α`
αiΛ−1`k
λk
ck≤ ui ≤
∑`
Λ−1i`
λ`
c`,
while, if (ii) holds, we have
−∑`
Λ−1i`
λ`
c`≤ ui ≤ λ
cαi+∑` 6=i
∑k
α`
αiΛ−1`k
λk
ck.
Proof. As outlined above, the proof is a simple adaptation of the above argument for Λ = I to thecase of a general matrix Λ.
4.2. The Existence Proof.In addition to our current assumptions, we need a continuity condition on H in the sense that
(s, p) 7→ H(x, s, p) is continuous (4.12)
and a regularity condition on the coefficients (which we will actually be able to loose later) of theform
θij ∈W 1,∞(Ω). (4.13)
Before we embark on the existence theorem, we need the following lemma.
Lemma 4.5. Assume H satisfies the growth conditions (4.3),(4.4) and the Smallness Condition aswell as the Boundedness Condition with ci = c for all i = 1, . . . , N . Now define an approximationHε of H by
Hε(x, s, p) :=H(x, s, p) + cs
1 + ε |H(x, s, p) + cs|− cs.
Then Hε can be written in the same form as H and satisfies the same structures.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 21
Specifically, we claim that, ∀ε > 0, Hε satisfies the same growth conditions (4.3),(4.4) with thesame estimates by the functions
k(ρ),K(ρ),Ki`(ρ) and ki(ρ) := ki(ρ) + cρ.
In particular, the Smallness Condition remains true.Furthermore, the Boundedness Condition also remains true for Hε with the same parameters in
the bounds as for H, namelyαi, λ, λ
i, c, λ0(ρ), λi0(ρ).
Remark 4.6. Notice that the growth estimates and the bounds in the Boundedness Condition areall uniform in ε. This is absolutely crucial for the proof of Theorem 4.7 below.
Proof. Observe that Hε can be written in the same form as H:
Hiε =
Hi + csi
1 + ε |H + cs|− csi =
(Q · pi +Hi0) + csi
1 + ε |H + cs|− csi
=Q
1 + ε |H + cs|· pi + Hi
0 + csi
1 + ε |H + cs|− csi ≡ Qε · pi +Hi
ε,0.
Now, given a transformation H = Q · pi + Hi0 of H by some Γ inducing the desired structure on
H, we see that
ΓHε(x,Γ−1s,Γ−1p) = Γ
( H(x,Γ−1s,Γ−1p) + c(Γ−1s)
1 + ε |Hi(x,Γ−1s,Γ−1p) + c(Γ−1s)|− c(Γ−1s)
)=
ΓH(x,Γ−1s,Γ−1p) + cs
1 + ε |Hi(x,Γ−1s,Γ−1p) + c(Γ−1s)|− cs
=H(x, s, p) + cs
1 + ε |Γ−1|∣∣H(x, s, p) + cs
∣∣ − cs.
Letting Hε denote the transformation of Hε by Γ, this can be written as
Hε
i=
Q
1 + ε |Γ−1|∣∣H + cs
∣∣ + ( H0 + cs
1 + ε |Γ−1|∣∣H + cs
∣∣ − cs)≡ Qε · pi + (Hε)
i
0.
Hence the growth estimates for Q and H0 allows us to estimate∣∣Qε
∣∣ = Q
1 + ε |Γ−1|∣∣H + cs
∣∣ ≤ ∣∣Q∣∣2 ≤ k(|s|∞) +K(|s|∞) |p|2 , (4.14)
∣∣(Hε)i
0
∣∣ ≤∣∣Hi
0
∣∣1 + ε |Γ−1|
∣∣H + cs∣∣ + ∣∣csi∣∣ ε
∣∣H + cs∣∣
1 + ε |Γ−1|∣∣H + cs
∣∣ (4.15)
≤∣∣Hi
0
∣∣+ c |s|∞ ≤ (ki(|s|∞) +∑
`Ki`(|s|∞) |p|2) + c |s|∞
= ki(|s|∞) +∑`
Ki`(|s|∞) |p|2
It follows directly from (4.14) and (4.15) that the first claim is true. Since the Smallness Conditiononly involves the Ki
`’s and these are left unchanged, we conclude that the Smallness Condition isalso satisfied by Hε.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 22
It remains to observe that the Boundedness Condition is also satisfied. This follows easily from∑`Λi`Hε
`+ c∑
`Λi`s` =
∑`Λi`H
` + c∑
`Λi`s`
1 + ε |Γ−1|∣∣H + cs
∣∣≤
λi + λi0∣∣∑
`Λi`p`∣∣2
1 + ε |Γ−1|∣∣H + cs
∣∣ ≤ λi + λi0∣∣∑
`Λi`p`∣∣2
and similar computations for the remaining inequalities.
Now we can state main result of Bensoussan & Frehse, a variant of which was first obtained intheir 1984 article [1]. Here we follow the proof in [B&F].
Theorem 4.7. Consider the system ΘIu = H(x, u,Du) given by (4.1),(4.2) with the additionalassumptions (4.12),(4.13). Assume also that H satisfies the growth conditions (4.3)-(4.4) as wellas the Smallness Condition. Finally, assume the Boundedness Condition is satisfied with ci = c
for all i = 1, . . . , N .Then ΘIu = H(x, u,Du) has a weak solution u ∈ H1
0 (Ω)∩L∞(Ω) with the additional regularity
u ∈W 2,p ∀p ∈ [2,∞).
Proof. As is standard for nonlinear problems, we intend to apply a fixed point argument for anapproximating problem and then use compactness methods to pass to a solution in the limit. Tohis end, we consider the following approximating problem
ΘIuε = Hε(x, uε, Duε)
for Hε as introduced in Lemma 4.5.Notice that, for each ε > 0, the functions Hε(x, u,Du) + cu are bounded uniformly in x and u
since|Hε(x, s, p) + cs| =
∣∣∣∣ H(x, u,Du) + cs
1 + ε |H(x, u,Du) + cs|
∣∣∣∣ ≤ 1
ε∀x, s, p. (4.16)
In order to take advantage of this, we rewrite the problem as
ΘIuε + cuε = Hε(x, uε, Duε) + cuε.
Now consider the map Tε : (H10 (Ω))
N → (H10 (Ω))
N defined by
Tε(z) = v iff.⟨ΘIv + cv, φ
⟩=⟨Hε(x, z,Dz) + cz, φ
⟩∀φ ∈ (H1
0 (Ω))N . (4.17)
This is well-defined as Hε(x, z,Dz) + cz ∈ L∞(Ω). Since Ω is smooth and θ is Lipschitz by theassumption (4.13), the standard (global) regularity theory for linear elliptic systems tells us thatv ∈ (W 2,2(Ω))N and v ∈ (L∞(Ω))N with∥∥v∥∥
(W 2,2(Ω))N≤ C(
∥∥v∥∥(L2(Ω))N
+∥∥Hε(x, z,Dz) + cz
∥∥L∞).
. C∥∥Hε(x, z,Dz) + cz
∥∥L∞
Recalling from (4.16) that Hε(x, z,Dz) + cz is bounded uniformly in the functions z, we concludethat ∥∥Tε(z)∥∥(W 2,2(Ω))N
≤ Cε ∀z ∈ (H10 (Ω))
N .
It follows that the set
Eε :=z ∈ (W 2,2(Ω))N ∩ (H1
0 (Ω))N :
∥∥z∥∥(W 2,2(Ω))N
≤ Cε
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 23
is mapped into itself by Tε. Equally important, Eε is bounded in (W 2,2(Ω))N and hence compactin (H1
0 (Ω))N by Rellich-Kondrachov. Also, it is trivially convex. Finally, (s, p) 7→ Hε(x, s, p) is
continuous by the continuity assumption on H in (4.12) and hence Tε : Eε → Eε is continuous.To see this, fix (zj) ⊆ Eε s.t. zj → z in (H1
0 (Ω))N and let vj := Tε(zj). Then also (vj) ⊆ Eε
and by definition ⟨ΘIvj + cvj , φ
⟩=⟨Hε(x, zj , Dzj) + czj , φ
⟩∀φ ∈ (H1
0 (Ω))N . (4.18)
Now, for any subsequence, we can take a further subsequence (zjk(n)) s.t. zjk(n)
→ z and Dzjk(n)→
Dz a.e., from whence we get Hε(x, zjk(n), Dzjk(n)
) + czjk(n)→ Hε(x, z,Dz) + cz a.e. Since this
sequence is bounded in L∞, DCT implies that the RHS in (4.18) converges to⟨Hε(x, z,Dz)+cz, φ
⟩.
As this is true for any subsequence, we conclude that
limj→∞
⟨Hε(x, zj , Dzj) + czj , φ
⟩=⟨Hε(x, z,Dz) + cz, φ
⟩.
But then the LHS of (4.18) must also converge to this same limit. From this and the boundednessof (vj) in W 2,2, we deduce that vj converges strongly in H1
0 to some v satisfying⟨ΘIv + cv, φ
⟩= lim
j→∞
⟨ΘIvj + cvj , φ
⟩=⟨Hε(x, z,Dz) + cz, φ
⟩.
This shows that Tε(z) = v = limj Tε(zj), which proves the (strong) continuity of Tε on Eε.By the above we can apply Schauder’s Fixed Point Theorem to see that Tε has a fixed point
uε, which is in (H10 (Ω))
N ∩ L∞(Ω) and even (W 2,2(Ω))N . We deduce from (4.17) that uε solvesΘIuε = Hε(x, uε, Duε) in the weak H1
0 ∩ L∞ sense.In order to have any hope of passing to the limit, we now need estimates, which are uniform in
ε. This is precisely what the regularity results of Section 4.1 provides (when applied to uε).By Lemma 4.5, Hε satisfies the Boundedness Condition with parameters that are uniform in ε,
so Theorem 4.4 produces L∞-bounds that are uniform in ε, i.e.∥∥uε∥∥L∞ ≤ C ∀ε > 0. (4.19)
Similarly, Lemma 4.5 ensures that the estimates in the growth conditions (4.3),(4.4) for Hε areuniform in ε and hence Theorem 4.2 gives uniform bounds on the C0,δ and W 1,p
0 norms. That is,∥∥uε∥∥(W 1,p0 (Ω))N
≤ C,∥∥uε∥∥(C0,δ(Ω))N
≤ C ∀ε > 0 (4.20)
for all p ∈ [2, 2 + ε0) and δ ∈ (0, δ0].Now that the uniform bounds are in place, we can carry out the limiting argument. To this end,
observe that (uε) is bounded in (H10 (Ω))
N and (C0,δ(Ω))N by (4.20), so by Rellich-Kondrachovand the compact embedding of Hölder spaces we get uε u in (H1
0 (Ω))N ,∥∥uε − u
∥∥L2 → 0 and∥∥uε − u
∥∥∞ → 0 along a subsequence for some u ∈ (H1
0 (Ω))N ∩ (C0,δ(Ω))N . Using the uniform
ellipticity of θ, we further note that
κ
∫ ∣∣D(uiε − ui)∣∣2 ≤
∫θD(uiε − ui) ·D(uiε − ui)
=
∫θDuiε ·D(uiε − ui)−
∫θDui ·D(uiε − ui). (4.21)
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 24
Here the second term converges to 0 by uiε ui in H10 (Ω) and we claim the same is true for the
first term. To see this, note that testing the equation against uiε − ui yields∣∣∣∣∫ θDuiε ·D(uiε − ui)
∣∣∣∣ =
∣∣∣∣∫ Hiε(x, uε, Duε)(u
iε − ui)
∣∣∣∣ ≤ ∥∥Hiε(x, uε, Duε)
∥∥L1
∥∥uiε − ui∥∥∞
.∥∥|Duε|2∥∥L1
∥∥uiε − ui∥∥∞ .
∥∥uiε − ui∥∥∞ −→ 0. (4.22)
In the two last estimates we have used that (i)∣∣Hi
ε(x, uε, Duε)∣∣ . |Duε|2 since the bounding
functions in the growth conditions (4.3)(4.4) are the same for all ε (by Lemma 4.5) and they onlydepend on
∥∥uε∥∥∞ which is bounded uniformly in ε by (4.19), and (ii)∥∥Duε∥∥(L2)N
is boundeduniformly in ε by (4.20).
We conclude from (4.21) that κ∫ ∣∣D(uiε−ui)
∣∣2 → 0 and therefore∥∥Duε−Du∥∥(L2(Ω))
→ 0. Thisshows that in fact ∥∥uε − u
∥∥(H1
0 (Ω))N−→ 0.
Taking further subsequences, we thus have uε → u a.e. and Duε → Du a.e., which in turn impliesHε(x, uε, Duε) → H(x, u,Du) a.e. by the continuity of Hε.
Recalling that∣∣Hi
ε(x, uε, Duε)∣∣ . |Duε|2 and ‖Duε‖L2+ε ≤ C, we see that
Hi
ε(x, uε, Duε)
isbounded in Lq(Ω) for q = (2 + ε0)/2 > 1. Hence
Hi
ε(x, uε, Duε)
is uniformly integrable, whichcombined with the a.e. convergence yields Hi
ε(x, uε, Duε) → Hi(x, u,Du) strongly in L1(Ω) by theVitali Convergence Theorem.
It follows that for any φ ∈ (H10 (Ω) ∩ L∞(Ω))N , we have∣∣∣∣∫
Ω
ΘIuε · φ−∫Ω
H(x, u,Du)φ
∣∣∣∣ =
∣∣∣∣∫Ω
Hε(x, uε, Duε) · φ−∫Ω
H(x, u,Du)φ
∣∣∣∣≤
∥∥φ∥∥L∞
∥∥Hε(x, uε, Duε)−H(x, u,Du)∥∥L1 −→ 0.
On the other hand, Duε → Du in L2 implies∣∣∣∣∫Ω
ΘIuε · φ−∫Ω
ΘIuφ
∣∣∣∣ =
∣∣∣∣∫Ω
∑k,`θk`(D`uε −D`u) ·Dkφ
∣∣∣∣.
∥∥θ∥∥∞∥∥Duε −Du∥∥L2
∥∥Dφ∥∥L2 −→ 0.
By uniqueness of the limit, we must have∫Ω
ΘIuφ =
∫Ω
H(x, u,Du)φ ∀φ ∈ (H10 (Ω) ∩ L∞(Ω))N .
This shows that u is indeed a weak solution to our problem.It only remains to argue that u ∈ (W 2,p(Ω))N for all p ∈ [2,∞).4 From the above, we know that u
is in fact in (W 1,2+ε0 (Ω))N since (uε) started out as a bounded sequence in W 1,2+ε
0 . Alternatively,this also follows immediately from another application of Theorem 4.2 to our newly obtainedsolution u.
Recalling that |H(x, u,Du)| . |Du|2 for u ∈ L∞(Ω) we get H(x, u,Du) ∈ Lq(Ω) with q :=
(2+ε)/2 > 1. Since⟨Θui, ϕ
⟩=⟨H(x, u,Du), ϕ
⟩for all ϕ ∈ C∞
c (Ω) we conclude that Θui ∈ Lq(Ω).Using the fact that the coefficients are Lipschitz and the boundary is smooth, the linear theorytells us thatΘ is an isomorphism from W 2,q(Ω) to Lq(Ω), and so it follows that ui ∈ W 2,q(Ω).Since we also have C0,δ regularity, an interpolation argument (combined with bootstrapping) gives
4Here we rely on interpolation. Another approach is to apply, to each equation separately, the program outlinedafter Theorem 5.3 in Section 5 (i.e. obtain a linearized system for the gradient and apply the linear Schaudertheory). Note, however, that the presentation in Section 5 requires differentiability of H. See Theorem 3 and 4 inSection 3 of Tomi (1972) for a version of this approach that applies when H is only measurable.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 25
ui ∈ W 2,p(Ω) for all p ∈ [2,∞). See the Miranda-Nirenberg interpolation inequality in Theorem7.2 of Appendix B. For details on how to carry the argument, see the proof of Theorem 2.23 on p.97 of [B&F].
Remark 4.8. In terms of existence, we emphasize that the above limiting argument can be carriedout without any Hölder regularity. Indeed (uε) is bounded in L∞, so we can take our subsequences.t. uε
∗ u in L∞ (instead of strong convergence in the sup-norm, when Hölder estimates are
available). This ensures that the limit is in H10 ∩ L∞. Now, since Hε(x, uε, Duε) is bounded in
Lq(Ω) for q = (2 + ε)/2 > 1 all we need is to observe that uε → u strongly in Ls(Ω) for alls ∈ [1,∞), which follows immediately from interpolation using the strong L2-convergence. Hencewe can simply change (4.22) to∣∣∣∣∫ θDuiε ·D(uiε − ui)
∣∣∣∣ = ∣∣∣∣∫ Hiε · (uiε − ui)
∣∣∣∣ ≤ ∥∥Hiε(x, uε, Duε)
∥∥Lq
∥∥uiε − ui∥∥Lq′ −→ 0.
In this way, the proof that uε → u strongly in H10 still works and the limit remains in L∞.
Remark 4.9. The strongH10 -convergence can also be obtained without use of the uniform L∞-bounds.
Recall (Θuε) is uniformly bounded in L1(Ω), so it is a bounded sequence in M(Ω). Using the com-pact embedding M(Ω) → W−1,q for each q < d
d−1 , we thus have that (Θuε) is precompact inW−1,q0 for any fixed q0 <
dd−1 . Since Θ is an isomorphism between W 1,q0
0 and W−1,q0 it followsthat (uε) is precompact in W 1,q0
0 . But we also know that (uε) is bounded in W 1,2+ε0 , so by Sobolev
interpolation we obtain that (uε) is precompact in H10 (Ω).
4.3. Existence of Nash Equilibria.Recall Example 2.3 from Section 2. In order to prove existence of a Nash Equilibrium, it fol-lows from Theorem 4.7 that it suffices to verify the structural growth conditions (4.3)-(4.4), theSmallness Condition and the Boundedness Condition.
Here we shall only provide a brief summary of the results covered in Sections 3.6 and 3.7 of[B&F]. To this end, we fix the notation from Section 2 and, for notational convenience, we define
a :=∑j
aj , a−i :=∑j 6=i
aj , p−i :=∑j 6=i
pj .
Consider a game given by an SDE with
drift b(x, a) = g(x) + a and diffusion matrix σ(x) ≡ I,
where g ∈ L∞. Let the cost functionals Ji be given by
li(x, a) =1
2(ai)TΣai + (ai)TΠa−i, ci(x, a) ≡ c, f i ∈ L∞.
Then Example 2.3 corresponds to the case Σ = I and Π = πI.More generally, we can take Σ and Π to be symmetric matrices with Σ > 0 and
Σ−Π, Σ+ (N − 1)Π invertible.
It can then be shown that the Nash controls are
ai(p) = (Σ − Π)−1Π(Σ + (N − 1)Π
)−1p−i −(Σ + (N − 2)Π
)(Σ + (N − 1)Π
)−1pi
and the Hamiltonian can be computed as
Hi(x, u, p) = −1
2¯a(p)TΣ¯a(p) + ¯a−i(p)T
(Σ2−Π
)¯a−i(p) + cλi + g(x) · pi + f i(x).
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 26
It turns out that these Hamiltonian do not satisfy neither the structural growth conditions(4.3)-(4.4) nor the Smallness Condition from Section 4.1.1.
However, there exists a linear transformation Γ such that the transformed Hamiltonians H doindeed satisfy all of these conditions. This is proved in Proposition 3.12 in [B&F].
Moreover, some extra assumptions on Σ and Π ensure that the Hamiltonians also satisfy theBoundedness Condition from Section 4.1.4. For details of this, the reader is referred to Propositions3.14, 3.15 and 3.17 in [B&F].
In the case of Example 2.3, where Σ = I and Π = πI, the assumptions are simply
π ≤ 1
2, π 6= − 1
N − 1; or π > 1.
The case of two players, i.e. N = 2, is particular instructive (with less technicalities) and is treatedin Section 3.7 of [B&F].
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 27
5. The Classical Theory of Quasilinear Equations
In the present exposition, we are mostly interested in systems the form
−N∑
µ=1
d∑k,`=1
Dk (θµνk` (x, u,Du)D`u
µ) = Hν(x, u,Du), ν = 1, . . . , N, (5.1)
where the leading coefficients θµνk` are bounded and the right-hand-side H is allowed to have qua-dratic growth in the gradient.
For notational convenience, we note that (5.1) can be written in matrix form as
Θu = H(x, u,Du), where Θνµu := −d∑
k,`=1
Dk (θµνk` (x, u,Du)D`u
µ) .
It is important for our presentation that the above can also be studied from the point of viewof Euler-Lagrange type equations
These are typically referred to as quasilinear equations with principal part in divergence form.Observe that, when p 7→ θνk(x, z, p) is differentiable (as we will always assume), the fundamental
Consequently, we are back in the general class of quasilinear systems represented by (5.1).When considering equations (5.2), we will impose conditions on the nonlinearity in Du of the
type|θνk(x, z, p)| ≤ c(1 + |p|), |Dpθ
νk(x, z, p)| ≤M.
This implies in particular that∣∣∣θµνk` (x, z, p)∣∣∣ ≤M and
∣∣∣hνk(x, z)∣∣∣ ≤ c in the rewritten equation (5.3).That is, the leading coefficients θµνk` (x, z, p) are bounded and the h-term is unimportant from atechnical point of view.
Finally, we single out the special case
−N∑
µ=1
d∑k,`=1
Dk (θµνk` (x, u)D`u
µ) = Hν(x, u,Du), ν = 1, . . . , N, (5.4)
where there is no non-linear dependence on the gradient in the principal part. In relation to (5.2),this corresponds to
θνk(x, u,Du) =
N∑µ=1
d∑`=1
θµνk` (x, u)D`uµ.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 28
5.1. Quasilinear elliptic systems.It turns out that the De Giorgi-Nash-Moser theory is quintessential for a good theory of quasilinearequations. To motivate this, we discuss below its rôle in the questions of existence and regularity.In this way, the present section also serves to introduce (in a very informal way) the most basicaspects of the theory of quasi-linear elliptic systems.
5.1.1. De Giorgi-Nash and higher regularity. We begin with some basic considerations con-cerning the main obstacle to regularity for the systems in (5.2). For the sake of clarity alone, weconsider the case where θνk(x, u,Du) = θνk(Du) and ϑν ≡ 0, leaving us with
−d∑
k=1
Dk (θνk(Du)) = 0. (5.5)
We remark that this is the system satisfied by minimizers of the functional
I[u] :=
∫Ω
L(Du(x))dx with θνk(p) := DpνkL(p).
Now suppose u ∈ (H1(Ω))N satisfies (5.5) in the weak H10 -sense, where we impose the following
conditions on the coefficients:
|θνk(p)| ≤ c(1 + |p|), θνk ∈ Cj,δ(RN×d) for some j ∈ N,∣∣Dpν
kθνk∣∣ ≤M.
Using the standard difference quotient technique, we easily obtain that Du ∈ (W 1,2loc (Ω))
N×d and,passing to the limit in the resulting weak formulation, we further deduce (as suggested by formaldifferentiation of (5.5)) that for each i = 1, . . . , d
N∑µ=1
d∑k,`=1
∫Ω
Dpµ`θνk(Du)D`(Diu
µ)Dkφν = 0 ∀φ ∈ (H1
0 (Ω))N .
Setting θµνk` (x) := Dpµ`θνk(Du(x)), we thus have that vi = Diu is a weak solution of a linear system
in divergence formN∑
µ=1
d∑k,`=1
Dk
(θµνk`D`v
µi
)= 0, ν = 1, . . . , N. (5.6)
In the hope of applying elliptic regularity theory, we may impose an ellipticity condition5 on thederivatives of the original coefficients θνk , i.e.
N∑µ,ν=1
d∑k,`=1
Dpµ`θνk(p)ξkµξ`ν ≥ κ |ξ|2 ∀ξ ∈ RN×d.
In this way, our linearized system (5.6) satisfies the Legendre ellipticity condition. Unfortunately,no matter how much regularity we impose on the original coefficients θνk , the new coefficients θµνk`will in general only be in L∞ unless we have some continuity on the gradient Du.
Suppose, for a moment, that mercy has shined upon us and Du is in fact continuous. Then thecoefficients θµνk` in (5.6) are also continuous and hence the linear Schauder theory tells us that thesolution v = Du is Hölder continuous.6 Since Dpµ
`θνk ∈ Cj,δ for some j ≥ 1, we then have that
the coefficients θµνk` are Hölder continuous7 and hence the Schauder theory implies Du ∈ C1,δ. But
5This condition is very natural and simply corresponds to ellipticity of the operator in non-divergence form. More-over, it is equivalent to strict convexity of the Lagrangian L in the variational integral.6This is essentially the continuous coefficient version of De Giorgi-Nash, obtained as part of the standard Schaudertheory by the method of frozen coefficients.7N.B.: δ is just a placeholder for the appropriate Hölder exponents. In general, on bounded domains, f ∈ C0,δ andg ∈ C0,γ implies f g ∈ C0,γδ , while f ∈ Cj,δ and g ∈ Cj,δ implies f g ∈ Cj,δ .
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 29
then the coefficients θµνk` are in C1,δ, so we get Du ∈ C2,δ by another application of the lineartheory. By bootstrapping this argument we obtain u ∈ Cj+1,δ.
In particular, if solutions to linear elliptic systems of the form (5.6) are continuous under theminimal assumptions of bounded measurable coefficients, then solutions to the quasi-linear system(5.5) are as smooth as the corresponding coefficients.
That this is true in the scalar case, is precisely the content of the celebrated De Giorgi-Nashtheorem, which says that bounded coefficients are sufficient for the Hölder continuity of weaksolutions to linear elliptic equations.
While this theory is quite flexible in terms of handling non-linearities, sadly, the conclusion failsto hold for systems. We discuss this in detail in Section 5.2 below.
5.1.2. Existence by Leray-Schauder Fixed Point Method. We mention briefly that the DeGiorgi-Nash-Moser theory also plays a crucial role in the existence theory. To get a flavor of this,consider the problem of finding a classical solution u ∈ C2,δ(Ω) to
d∑k,`=1
θk`(x, u,Du)Dk`u+ ϑ(x, u,Du) = 0, u|∂Ω = ϕ, (5.7)
where θ is elliptic, θk`, ϑ ∈ C0,γ(Ω,R,Rd) and ϕ ∈ C2,γ(Ω) on a C2,γ-domain Ω.Now define a family of mappings
Tσ : C1,δ(Ω) → C1,δ(Ω), σ ∈ [0, 1],
by letting uσ := Tσ(v) be the solution to the linear equationd∑
k,`=1
θk`(x, v,Dv)Dk`u+ σϑ(x, v,Dv) = 0, u|∂Ω = σϕ.
This is well-defined by the linear theory and since θk`(x, v,Dv), ϑ(x, v,Dv) are in C0,γδ(Ω) theglobal Schauder theory implies uσ ∈ C2,γδ(Ω). In particular, Tσ is compact by the compactembedding of C2,γδ(Ω) into C1,δ(Ω) and continuity is easily verified.
What we want is then a fixed point of T1. Noticing that Tσ(v) = σT1(v), the Leray-Schaudertheorem says that this is achieved if only we can find a constant M > 0 s.t. for all σ ∈ [0, 1] wehave
‖u‖C1,δ(Ω) ≤M
whenever u = Tσ(u).In short, existence boils down to the ability to establish uniform á priori bounds on the Hölder
norms. We refer to Chapter 11 of [G&T] for proofs of the above statements. In the following, webriefly sketch a typical method for obtaining the á priori bounds. To this end, recall that
‖u‖C1,δ ≤ ‖u‖ L∞ + ‖Du‖L∞ + [Du]C0,δ .
The first task is thus to find a bound on ‖u‖ L∞ , which can be obtained from De Giorgi-Mosertype iteration (or quasilinear maximum principles as in Chapter 10 of [G&T]). Secondly, one needsto estimate ‖Du‖L∞ , which can be done by estimating it in terms of sup∂Ω |Du| and ‖u‖ L∞ , andthen in turn estimating sup∂Ω |Du| by ‖u‖ L∞ . These two steps are covered respectively in Chapter14 and 15 of [G&T].
Finally, the crucial ingredient is estimates for the Hölder coefficient [Du]C0,δ of the gradient.This is precisely the domain of the De Giorgi-Nash-Moser techniques. Recall, for example, theequation from the previous subsection, where we showed that Du satisfied a linearized equation
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 30
with bounded measurable coefficients and then applied De Giorgi-Nash to get Hölder estimates forDu. This idea is pursued in detail for general quasilinear equations in Chapter 13 of [G&T].
In the case of quasi-linear systems, a similar approach can be implemented, where De Giorgi-Nash-Moser plays much the same role. A classic account of this can be found in Chapter 8 of [L&U].It is important to emphasize, however, that the results for systems are far less general (since rathernon-trivial structural conditions are required in order to obtain the a priori estimates).
5.2. Hölder regularity for Quasi-linear equations.In the following we consider the question of (interior) Hölder regularity for solutions to ellipticequations. Note that by Hölder continuity on a domain Ω, we really mean local Hölder continuityin the sense that any point in Ω has a neighborhood on which the function is Hölder continuous.
5.2.1. The case of scalar equations. We begin by stating the elliptic De Giorgi-Nash theoremfor linear (2nd order) elliptic equations.
Theorem 5.1. Let L be a linear differential operator
Lu = −div(A(x)Du+ b(x)u) + c(x) ·Du+ h(x)u
with bounded measurable coefficients satisfying the ellipticity condition
κ |ξ|2 ≤ ξTA(x)ξ ≤ K2 |ξ|2 ∀ξ ∈ Rd
and the bound8
κ−2∑i(|bi(x)|
2+ |ci(x)|2) + κ−1 |h(x)| ≤M2.
Suppose also that f ∈ (Lq(Ω))d and f0 ∈ Lq/2(Ω) for some q > d. If u ∈ W 1,2(Ω) satisfiesLu = f0 + divf in Ω, then u ∈ C0,δ(Ω) and for any Ω0 b Ω we have the estimate
where δ = δ(d,K/κ,M, ε), C = C(d,K/κ,M, ε, q) and ε = dist(Ω0, ∂Ω).
Proof. See Theorem 8.24, p. 202 in [G&T].
The true power of this theorem lies in the fact that the coefficients need only be measurable.As already emphasized in Section 5.1.1, the result is much more easily obtained in the case ofcontinuous coefficients and here one gets Hölder regularity for any exponent δ < 1. This is muchbetter than the measurable version above, which only holds for some δ ∈ (0, 1) and this δ can evenbe shown to satisfy δ → 0 as K/κ→ ∞.
Remark 5.2. We note that, assuming u is bounded, the result can be obtained from Theorem3.3 in Section 3 by choosing X0(s) := s2/2 and then verifying Assumption 2 by testing the weakformulation with φ = (u− cR)ψ.
In terms of nonlinearities, the De Giorgi-Nash theory is surprisingly flexible. The richest theoryis for quasi-linear equations in divergence form of the type (5.2), i.e.
d∑k=1
Dk(θk(x, u,Du)) + ϑ(x, u,Du) = 0. (5.8)
Note that if u ∈ W 2,2 (and θ is taken to be symmetric), then rewriting 5.8 in non-divergenceform yields 1
2 (Dpiθj(x, u,Du) + Dpjθi(x, u,Du)) as the principal coefficients. Hence the natural
8As is usually the case, the conditions on b, c and h can be replaced by b, c ∈ (Lq(Ω))d and h ∈ Lq/2(Ω).
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 31
where υ(·), ω(·) are positive functions, then we obtain the following extension of De Giorgi-Nash.9
Theorem 5.3. Let u ∈ H1(Ω) ∩ L∞(Ω) be a weak solution of (5.8) and assume the conditions(5.9)-(5.10)10 are satisfied. Then u ∈ C0,δ(Ω) and for any Ω0 b Ω we have
‖u‖C0,δ(Ω0)≤ C,
where C = C(‖u‖L∞ , ω(‖u‖L∞), υ(‖u‖L∞), ε), δ = δ(‖u‖L∞ , ω(‖u‖L∞), υ(‖u‖L∞)) and ε = dist(Ω0, ∂Ω).
Proof. See Theorem 1.1, p. 251 in Chapter 4 of [L&U].
We stress that the above Hölder continuity is the crucial part of the regularity theory. Once wehave this, further regularity can be obtained by implementing the same ideas as in the motivatingexample from Section 5.1.1. (although with much more technical á priori estimates).
Here we briefly outline the approach in [L&U]. For this, we shall need differentiability of(x, z, p) 7→ ϑ(x, z, p), θ(x, z, p) with the following growth conditions on the derivatives: ∑d
i,k=1 |Dxiθk| ≤ ω(|z|)(1 + |p|)∑d
k=1 |Dzθk| ≤ ω(|z|)(1 + |p|),
∑di=1 |Dpi
ϑ| ≤ ω(|z|)(1 + |p|)|Dzϑ|+
∑di=1 |Dxi
ϑ| ≤ ω(|z|)(1 + |p|)2(5.11)
The first step is to use the C0,δ-estimate from Theorem 5.3, together with the above assumptions, toshow that the gradient Du is bounded. Based on this, one can derive a priori estimates for the weaksecond order derivative D2u and deduce its existence in L2 from the difference quotient technique.This then leads to the gradient Du satisfying a linear equation with bounded coefficients, whichshows that Du ∈ C0,δ by De Giorgi-Nash. If the coefficients θ and ϑ are more regular, we can nowgain higher regularity by precisely the same bootstrap argument as in Section 5.1.1 (based on thelinear Schauder theory). The results can be summarized as follows.
Theorem 5.4. Let u ∈ H1(Ω) ∩ L∞(Ω) be a weak solution of (5.8). Assume that conditions(5.9),(5.10) and (5.11) are satisfied. Suppose θ ∈ Cj−2,δ (C0,1 if j = 1) and ϑ ∈ Cj−1,δ (C0,1 ifj = 1, 2). Then u ∈ Cj,δ(Ω0) for any Ω0 b Ω.
Proof. See Theorem 6.3 in Chapter 4 of [L&U].
Remark 5.5. On pp. 98-111 in [B&F], a proof of a very similar result is based on Theorem 3.2 andTheorem 3.3 from Section 3 followed by a series of technical estimates using the Green function.In particular, we emphasize that Theorem 3.3 essentially includes both the linear and quasilinearversions of De Giorgi-Nash presented above.
5.2.2. Boundedness in the definition of weak solution. As we have seen in the previoussection, the convention that weak solutions of quasilinear equations should belong to L∞ leads toa very rich regularity theory.
9One can also consider mth-order growth for ϑ, in which case Theorems 5.3-5.4 apply to solutions in W 1,m.10Instead of ellipticity, it suffices to assume
∑dk=1θk(x, z, p)pk ≥ υ(|z|) |p|2 − ω(|z|). Note that this is implied by
(5.9)-(5.10) as can e.g. be seen from the mean value theorem.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 32
Another important reason for only considering solutions in L∞ concerns the question of localuniqueness. In this respect, we emphasize that, unlike the contraction mapping principle, theLeray-Schauder theorem does not entail uniqueness of the fixed point.
We do, however, have the following local uniqueness theorem for bounded solutions:
Theorem 5.6. Let u, v ∈ H10 (Ω) ∩ L∞(Ω) be two weak solutions of (5.8). If they agree on the
surface of a ball Bρ ⊆ Ω0 b Ω, then they also agree inside Bρ, provided that the radius ρ is smallerthan some number ρ0 determined by the L∞-norms of u and v, the ellipticity constants, and thedistance from Ω0 to Ω.
Proof. See Theorem 2.1 in Chapter 4 of [L&U].
If, on the other hand, we drop the assumption of boundedness, then there are counterexamplesshowing that the local uniqueness fails.
5.2.3. The case of quasilinear systems. Let us now consider weak solutions u ∈ H10 ∩ L∞ to
elliptic systems of the form (5.4) where θµνk` = θµνk` (x, u). That is,
−N∑
µ=1
d∑k,`=1
Dk (θµνk` (x, u)D`u
µ) = Hν(x, u,Du), ν = 1, . . . , N. (5.12)
This system is studied extensively in Chapter 8 of [L&U] and Chapters 4-6 of [Giaquinta].We shall always assume the operator satisfies the Legendre ellipticity condition
N∑µ,ν=1
d∑k,`=1
θµνk` ξkµξ`ν ≥ κ |ξ|2 ∀ξ ∈ RN×d. (5.13)
Supposing that we have an estimate for ‖u‖C0,δ(Ω), then the L∞-norm of the gradient can beestimated in essentially the same way as in the scalar case and the a priori estimates for furtherregularity also follow by similar methods (utilizing the linear Schauder theory). The decisive factorin the regularity theory is thus to establish the Hölder continuity.
Unfortunately, the question of C0,δ-regularity is very involved for systems. In general, the bestwe can do is C0,δ-regularity almost everywhere outside a set of small Hausdorff dimension, referredto as partial regularity.11 This is the case even when H ≡ 0. When the right-hand-side H(x, u,Du)
has quadratic growth in the gradient, we furthermore need to impose a smallness condition on thegrowth in order to obtain such a partial regularity result.
In a nutshell, the partial regularity is obtained by requiring continuity of the leading coefficients(x, z) 7→ θµνk` (x, z) and then freezing these coefficients around Lebesgue points for u. Using thecontinuity of θµνk` and the properties of Lebesgue points we can then estimate (the mean value of) thedifference |θµνk` (x0, (u)x0,r)− θµνk` (x, u)| and thus transfer the estimates for the (frozen) linearizedequation to the full equation in a small neighborhood of the Lebesgue point.
The key technical tools are a Caccioppoli type inequality and a higher integrability result (provedby Caccioppoli, Poincaré and Gehring’s Lemma). Based on this one establishes a decay estimatefor the gradient and then appeals to the Morrey-Campanato theory.12 When H(x, u,Du) growsquadratically in Du, the role of the aforementioned smallness condition is precisely to make aCaccioppoli inequality possible (see the discussion after Theorem 5.9 in the next subsection).
11The case Ω ⊆ R2 is special and here there are results on full regularity, even for systems, due to Morrey (longbefore De Giorgi-Nash).12Here I am referring to the so-called direct approach to obtaining the decay estimate. There is also an indirectapproach, where instead of using higher integrability one argues by contradiction using a certain blow-up technique.A more recent method is that of A-harmonic approximation introduced in Duzaar & Grotowski [2].
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 33
Theorem 5.7. Let u ∈ H10 (Ω)∩L∞(Ω) be a weak solution of the system Θu = H(x, u,Du) given
by (5.12)-(5.13). Assume (x, z) 7→ θµνk` (x, z) is continuous and |θµνk` | ≤ K(‖u‖L∞). Finally, supposethat H(x, u,Du) has either subquadratic growth
|H(x, u,Du)| ≤ ω(‖u‖L∞) |Du|q + C, 1 +2
d< q < 2
or quadratic growth with a smallness condition|H(x, u,Du)| ≤ ω(‖u‖L∞) |Du|2 + C
2ω(‖u‖L∞) ‖u‖L∞ ≤ κ.
Then there is an open set Ω0 ⊆ Ω s.t. u ∈ C0,δ(Ω0) for any δ ∈ (0, 1). Moreover, dimH(Ω \Ω0) <
d− 2.
Proof. See Theorem 1.3 in Chapter 6 of [Giaquinta].
Once we have established the partial Hölder continuity, we can readily gain higher partialregularity. Indeed, if in Theorem 5.6 we further assume that the leading coefficients (x, z) 7→θµνk` (x, z) are in C0.α, then we can exploit the Hölder regularity of u on the regular set Ω0 toget Du ∈ C0,α(Ω0) by methods completely analogous to those underlying Theorem 5.6. Now,given Du ∈ C0,α(Ω0), the coefficients θµνk` (x) := θµνk` (x, u(x)) and Hν(x) := Hν(x, u(x), Du(x)) areHölder continuous and hence bootstrapping arguments based on the linear Schauder theory yieldshigher regularity for more regular coefficients.
While everywhere regularity is in general not within reach, there are certain special cases whereit can be achieved. One such example, concerning continuity, is presented below. Another moreinvolved example concerns systems in diagonal form, which is the topic of Section 5.3.
Proposition 5.8. Consider the same assumptions as in Theorem 5.7, but now assume that thecoefficients are functions of x only and x 7→ θµνk` (x) is continuous. Then u ∈ C0,δ(Ω) for anyδ ∈ (0, 1).
Proof. In essence, the continuity of x 7→ θµνk` (x) means that we can freeze coefficients around anypoint and hence apply the methods of Theorem 5.7 everywhere in Ω.
Note that continuity of (x, z) 7→ θµνk` (x, z) does in no way ensure continuity of
θµνk` (x) := θµνk` (x, u(x)),
so the setting is very different from Theorem 5.7. It does, however, show that if we considercontinuous solutions, then we have Hölder regularity everywhere.
5.2.4. Dimension of the singular set. In the case of partial regularity, there is naturally aconsiderable interest in estimating as precisely as possible the “size” of the singular set Ω\Ω0. Forexample, in the setting of Theorem 5.7 we claimed that it is not only of full Lebesgue measure butalso has Hausdorff dimension strictly less than d − 2. The tools for proving this are presented inSection 2 of Chapter 4 in [Giaquinta].
It is, however, beyond the scope of this report to explore these issues any further. Section 2 ofthe survey paper [6] by Kristensen & Mignione provides a very brief overview of the key results.Details of these results can be found in the original papers by Mignione ([4, 5]) and Kristensen &Mignione ([7]).
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 34
5.2.5. Nonlinear dependence on the gradient in the principal part. In this subsection, webriefly discuss the partial regularity results for systems (5.2), i.e.
where the principal terms θνk = θνk(x, u,Du) now depend nonlinearly on the gradient Du.While the business of obtaining decay estimates for the gradient becomes much more technical,
the central ideas remain essentially the same. As such, the guiding principle is to linearize aroundtriples
(x0, (u)x0,r, (Du)x0,r/4)
and then transfer the linear estimates to the full equation in sufficiently small neighborhoods ofLebesgue points. Leaving all technicalities aside, let us try to convey in some detail how to actuallyimplement this strategy.
We begin by recalling from the very beginning of Section 5 that we can transform systems (5.14)into systems of the form (5.1). In line with this, we define
Assuming still that θµνk` is elliptic, we also know from the linear theory that∫Bρ(x0)
|Dv − (Dv)x0,ρ|2dx . (
ρ
r)d+2
∫B r
4(x0)
∣∣Dv − (Dv)x0,r4
∣∣2 dx ∀ρ < r/4
and hence∫Bρ(x0)
|Du− (Du)x0,ρ|2dx . (
ρ
r)d+2
∫B r
4(x0)
∣∣Du− (Du)x0,r4
∣∣2 dx ∀ρ < r/4
+
∫B r
4(x0)
|Dw|2 dx.
What we need is then to appropriately estimate the integral of |Dw|2 by estimating in turn thethree integrals bounding it in (5.15). Now, by imposing sufficient continuity properties on θµνk` (inthe p-variable) and on θνk (in the x, z-variables) it should at least seem plausible from (5.15) thatsomething interesting can be achieved by comparing, around Lebesgue points, Du with (Du)x0,
r4
in the first integral and u with (u)x0,r in the second integral.Basically, the goal is the following. By pursuing the above in combination with a higher inte-
grability result for the gradient (so Hölder can be applied) and some technical lemmas, we aim tofind a small radius R > 0 s.t.∫
Bρ(y)
|Du− (Du)y,ρ|2 dx ≤ C(ρ
R)d+ε ∀ρ < R
for each y in a small enough neighborhood of a Lebesgue point x0. From the Morrey-Campanatotheory we then get that Du is Hölder continuous in this neighborhood.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 36
Theorem 5.9. Let u ∈ H10 ∩ L∞ be a weak solution to (5.14) with
|Hν(x, z, p)| ≤ a |p|2 + b.
Assume the following structural conditions are satisfied:
(1) |θνk(x, z, p)| ≤ c(1 + |p|)(2) (x, z) 7→ (1 + |p|)−1θνk(x, z, p) is in C0,α(Ω× Rn) uniformly in p.(3)
∣∣∣Dpµ`θνk(x, z, p)
∣∣∣ ≤M
(4)∑
k,`
∑µ,ν Dpµ
`θνk(x, z, p)ξ
νkξ
µ` ≥ κ |ξ|2
Suppose also that the following smallness condition holds:
(5) 2a ‖u‖ L∞ < κ
Then u ∈ C1,δ(Ω0), where Ω0 is an open set of full Lebesgue measure.
Proof. See Theorem 2.1 in Chapter 5 of [Giaquinta] for the case H ≡ 0. For the adjustmentsneeded to handle the general case, see Theorem 2 in [10].
Remark 5.10. The conditions (1)-(5) are in particular satisfied by
θνk(x, z, p) :=
N∑µ=1
d∑`=1
θµνk` (x, z)pµ`
for θµνk` (x, z) as in Theorem in 5.7 with the additional assumption θµνk` ∈ C0,α(Ω× Rn).
We end this section with a quick discussion of the conditions (1)-(5). Note first that (3) providesLipschitz continuity for θµνk` in the p-variable and together with (2) these assumptions are thus thecontinuity conditions that we asked for immediately before the statement of the theorem. Moreover,(4) ensures ellipticity of θµνk` , which we also asked for in the discussion preceding the theorem.
Finally, (1) and (4) yields a coercivity estimate which together with the smallness condition (5)gives us a Caccioppoli type inequality. Specifically, (1) and (4) implies∑
ν,k
θνk(x, z, p)pνk =
∑ν,k
(∑µ,`
(∫ 1
0
Dpµ`θνk(x, z, tp)dt
)pµ` + θνk(x, z, 0)
)pνk
≥ κ |p|2 − c |p| ≥ (κ− ε) |p|2 −M(ε).
To see how we obtain the Caccioppoli inequality, test the equation (5.14) with
φ := (u− (u)x0,2r)χ2,
where χ is a cut-off function supported in B2r(x0) with χ ≡ 1 on Br(x0) and |Dχ| . 1/r. Thenthe above coercivity estimate implies
(κ− ε)
∫B2r
|Du|2 χ2 ≤ M(ε)
∫B2r
χ2 − 2
∫B2r
θνk [u− (u)2r]χDχ+
∫B2r
Hν [u− (u)2r]χ2
Recalling the assumptions |θνk | ≤ c(1 + |Du|) and |Hν | ≤ a |Du|2 + b this becomes
(κ− ε)
∫B2r
|Du|2 χ2 ≤ M(ε)
∫B2r
χ2 + 2c
∫B2r
(1 + |Du|) |u− (u)2r| |χ| |Dχ|
+ b
∫B2r
|u− (u)2r|χ2 + a
∫B2r
|Du|2 |u− (u)2r|χ2
Now the key point is that, because of the smallness condition (5), we can take ε > 0 so small that2a ‖u‖L∞ < κ− ε, and thus we can absorb the last term (on the RHS) into the LHS.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 37
The remaining terms can then be estimated by the usual tricks (Cauchy-Schwarz, Young’sinequality with ε, Poincaré inequality, and absorbing like terms into the LHS), leaving us with∫
Br
|Du|2 dx ≤ C
r2
∫B2r
(1 + |u− (u)2r|2
)dx.
This is the desired Caccioppoli inequality. Applying the Poincaré inequality on the RHS followedby Gehring’s Lemma, we get a crucial higher integrability result for the gradient, which allows usto finish the program outlined before the statement of Theorem 5.9.
5.3. Diagonal Systems.In this section we prove the original result on Hölder regularity for diagonal systems, which led toFrehse & Bensoussan’s proof of Theorem 4.7.
The result arose out of a series of papers in the years 1971-1976 by Widman ([8]), Wiegner([12, 13, 14]) and Hildebrandt & Widman ([17, 18]). Later, two alternative proofs were providedby Caffarelli ([9]) and Tolksdorf ([15]).
In the present section, we shall follow the approach of Tolksdorf. This is similar in spirit to theapproach of Cafarelli ([9]) in the sense that both are of a more succinct geometric nature than theoriginal arguments by Wiegner and Hildebrandt & Widman, which rely heavily on properties ofthe Green function and Widman’s hole-filling technique.
While Cafarelli’s proof relies on Moser’s Harnack inequality, Tolksdorf’s arguments are basedon a generalization of the Strong Maximal Principle proved by Tolksdorf in [16]. In either case,the legacy of the De Giorgi-Nash-Moser theory is evident.
In line with the rest of the paper, we focus strictly on the elliptic case. The generalization toparabolic systems was first achieved in 1981 by M. Struwe (see [11]).
In the previous section, we considered partial Hölder regularity for general quasilinear systemsof the form
Θ11 · · · Θ1N
.... . .
...ΘN1 · · · ΘNN
u1
...uN
=
H1
...HN
,where
Θµνuν =
∑i,j
Di
(θµνij (x, u,Du)Dju
ν)
and H = H(x, u,Du).
We now restrict attention to systems in diagonal formΘ 0
. . .0 Θ
u1
...uN
=
H1
...HN
,where θ = θ(x, u,Du) is assumed to be bounded for bounded values of u.
Unlike the partial regularity results by Giaquinta & Modica, we make no assumptions of conti-nuity and, in particular, we shall not use the dependence of θ on u and Du. Hence, from the pointof view of regularity theory, we might as well set
θ(x) := θ(x, u(x), Du(x))
and consider θ as a function of x only. Of course, our results will then still apply to a generalθ = θ(x, u,Du) as long as it is bounded for bounded values of u.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 38
Given a weak solution u ∈ (H1(Ω) ∩ L∞(Ω))N , we write the weak formulation concisely as∫Ω
∑i,j
θij(x)Dju ·Diφdx =
∫Ω
(H(x, u,Du) + g(x)
)· φdx ∀φ ∈ (H1
0 ∩ L∞)N . (5.16)
As usual, we require ellipticity of θ, i.e.∑i,j
θij(x)ξiξj ≥ κ |ξ|2 . (5.17)
In addition, we assume that θ and g are measurable and bounded with∑i,j
|θij(x)| ≤ b, |g(x)| ≤ b (5.18)
Finally, we impose the following two structural conditions
|H(x, u(x), Du(x))| ≤ a∑i,j
θij(x)Dju(x) ·Diu(x) + b (5.19)
H(x, u(x), Du(x)) · u(x) ≤ a∗∑i,j
θij(x)Dju(x) ·Diu(x) + b∗ (5.20)
along with a smallness condition
aM + a∗ < 2 where ‖u‖L∞ ≤M. (5.21)
Observe that, on its own, (5.19) implies
H(x, u(x), Du(x)) · u(x) ≤ aM∑i,j
θij(x)Dju(x) ·Diu(x) + bM,
so we can always assume that a∗ ≤ aM and b∗ ≤ bM . Hence the related condition
aM < 1 (5.22)
implies thataM + a∗ ≤ aM + aM ≤ 2
and thus our smallness condition (5.21) is a weaker assumption than (5.22).In particular, all our results remain true if we assume (5.22) instead of (5.21) and then we can
even drop (5.20).It is clear that the combination of the one-sided growth condition (5.20) and the smallness
condition (5.21) is only significant if a∗ is assumed strictly less than aM (otherwise we get just asgood a constant from (5.22) and (5.19)). We shall not pursue this here, but having a∗ < aM doesindeed turn out to give some improved properties. This can e.g. be seen from Theorem 3.2 in [18].
Remark 5.11. The smallness conditions (5.21) and (5.22) are optimal in the sense that thereare examples of discontinuous bounded solutions even if they are only just violated. The classicexample is the Harmonic mapping u : R3 → S2 given by u(x) = x/ |x|, which is a weak solution of
−4 u = u |Du|2 .
Here a∗ = a = 1 and b∗ = b = 0 while M = 1. This counterexample is due to Hildebrandt &Widman ([17]).
Remark 5.12. Let us briefly connect the assumptions of the present section with those of Section5.2. To this end, suppose that instead of (5.18)-(5.19) and (5.21) we are given
|H(, u,Du)| ≤ a |Du|2 + b and a ‖u‖ L∞ < κ.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 39
Then the ellipticity (5.17) implies that
|H(x, u,Du| ≤ a
κ
∑i,j
θijDju ·Diu+ b
and hence alsoH(x, u,Du) · u ≤ a ‖u‖ L∞
κ
∑i,j
θijDju ·Diu+ b ‖u‖ L∞
witha
κM +
a ‖u‖ L∞
κ< 1 + 1 = 2 for M := ‖u‖ L∞ .
Consequently, we are back in the setting of (5.19) and (5.21), so the setup from Section 2 is indeedincluded in our analysis.13
5.3.1. Proof of everywhere Hölder regularity. We shall prove (local) Hölder continuity firstin the interior and then up to the boundary under some additional assumptions. To this end, wefix a point x0 ∈ Ω and a ball BR0
(x0) with radius R0 ∈ (0, 1].For the interior regularity, we assume that
B2R0(x0) ⊆ Ω. (5.23)
For the boundary regularity, we need the following two assumptions on the regularity of u|∂Ωand ∂Ω. First of all, the trace u|∂Ω must be Hölder continuous on the boundary with
|u(x)− u(y)| ≤ c′ |x− y|γ′
∀x, y ∈ ∂Ω ∩B2R0(x0) (5.24)
for some constant c′ > 0 and Hölder exponent γ′ ∈ (0, 1). Secondly, the boundary ∂Ω must satisfya sphere condition of the form
where γ′ > 0 is the Hölder exponent from (5.24).We begin our efforts by proving an auxiliary result on Hölder continuity, which we shall employ
later.
Lemma 5.13. Fix a ball BR0(x0) and a smaller radius R1 ≤ R0. Suppose there are constants
σ, λ ∈ (0, 1) such that, for any y ∈ Ω ∩BR0(x0), we have
oscBσiR1
(y)∩Ωu ≤ 2λiM ∀i ∈ N,
where supΩ u ≤M . Then
supx,y∈BR∩Ω, x 6=y
|u(x)− u(y)||x− y|γ
≤ 1
λ
2M
Rγ1
, γ :=log λ
log σ.
Proof. Fix x, y ∈ BR(x0) with x 6= y and set ρ := |x− y|. Obviously,
|u(x)− u(y)||x− y|γ
≤ 1
ργosc
Bρ(y)∩Ωu.
Suppose first that R1 < ρ ≤ R. Then we have1
ργosc
Bρ(y)∩Ωu ≤ 1
Rγ1
oscBR(y)∩Ω
u ≤ 1
Rγ1
2M ≤ 1
λ
1
Rγ1
2M.
Now suppose 0 < ρ ≤ R1 and take i ∈ N s.t.
σiR1 < ρ ≤ σi−1R1.
13In fact, we have even improved the smallness condition by a factor of 2, from 2a ‖u‖L∞ < κ to a ‖u‖L∞ < κ.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 40
In particular,log(σi) + log(R1) = log(σiR1) < log(ρ),
Noting that γ = log λi/ log σi, we thus have
1
ργ= exp
(log λi
log σi
(− log ρ
))≤ exp
(log λi
log σi
(− log σi − logR1
))= exp
(− log λi
)exp
(log λi
log σi
(− logR1
))=
1
λi1
Rγ1
Since we also have Bρ(y) ⊆ Bσi−1R1(y), it follows that
1
ργosc
Bρ(y)∩Ωu ≤ 1
ργosc
Bσi−1R1(y)∩Ω
u ≤ 1
ργλi−12M =
1
λ
1
Rγ1
2M.
This proves the claim.
Before moving on, we shall need some notation. Given a fixed y ∈ Ω ∩BR0(x0), we define
M(R) := supBR(y)∩Ω
|u|
for any R ≤ R0.The first step towards Hölder regularity is a Caccioppoli-like estimate for the nonnegative func-
tionv := |u− u1|2 ,
where u1 belongs to a particular class of vectors in RN . To this end, we shall need a class ofsuperlevel sets for v of the form
Ak,r :=x ∈ Br(y) ∩ Ω : |u− u1|2 > k
,
for arbitrary r ≥ 0 and k ≥ 0.
Lemma 5.14. Fix y ∈ Ω ∩ BR0(x0) and a ball BR(y) of arbitrary radius R ≤ R0. Assume
u ∈ H1 ∩ L∞ satisfies conditions (5.16)-(5.20) and assume also that there is a δ > 0 s.t.
aM(R) + a∗ + 2δaM ≤ 2. (5.26)
Fix r, % > 0 s.t. 0 < r < % < R. Let k ≥ 0 and u1 ∈ RN be given. Suppose the following twoconditions are satisfied
|u1| ≤ δM(R) (5.27)
|u− u1|2 ≤ k on ∂Ω ∩BR(y) (5.28)
Then there is a positive constant
C = C(d, κ, b, δ, bounds on b∗,M(R))
such that ∫Ak,r
∣∣∣D |u− u1|2∣∣∣2 dx ≤ C
(%− r)2
∫Ak,%
(|u− u1|2 − k
)2dx+ C |Ak,%| . (5.29)
Proof. Take a cut-off function χ ∈ C∞c (B%(y)) with φ ≡ 1 in Br(y) and
|Dχ| ≤ c
(%− r), c = c(d). (5.30)
Consider the test function φ ∈ (H10 (Ω) ∩ L∞(Ω))N given by
φ := (u− u1)max|u− u1|2 − k, 0
χ2.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 41
Clearly φ ∈W 1,2 by the Sobolev chain rule for Lipschitz functions and, moreover, it has trace zeroon ∂Ω by 5.28 and the fact suppχ ⊆ B%(y). Note that Diφ = 0 on (Ak,%)
Since (u− u1) ·Dju = 12Dj |u− u1|2 we can rewrite this as
1
2
∫Ak,%
θijDj |u− u1|2Di |u− u1|2 χ2dx = −∫Ak,%
θijDju ·Diu(|u− u1|2 − k)χ2dx
−∫Ak,%
θijDj |u− u1|2 (|u− u1|2 − k)χDiχdx +
∫Ak,%
(H + g) · (u− u1)(|u− u1|2 − k)χ2dx.
From the ellipticity (5.17), we then obtain thatκ
2
∫Ak,%
∣∣∣D |u− u1|2∣∣∣2 χ2dx ≤ −
∫Ak,%
θij(Dj |u− u1|2 χ
)((|u− u1|2 − k)Diχ
)dx [I1]
+∫Ak,%
g · (u− u1)(|u− u1|2 − k)χ2dx [I2]
+∫Ak,%
H · (u− u1)(|u− u1|2 − k)χ2dx [I3]
−∫Ak,%
θijDju ·Diu(|u− u1|2 − k)χ2dx [I4]
The first integral is estimated by
I1 ≤ b
ε
∫Ak,%
∣∣∣Dj |u− u1|2∣∣∣2 χ2dx+
c(ε, d)
(%− r)2
∫Ak,%
(|u− u1|2 − k
)2dx
. (5.31)
This follows easily from the bounds (5.18),(5.30) on |θ| and |Dχ|, using Cauchy-Schwarz and thenYoung’s inequality with ε.
The second integral is estimated by:
I2 ≤ b(M(R) + δM(R))
εc(d)
∫Ak,%
∣∣∣Dj |u− u1|2∣∣∣2 χ2dx
+c(ε, d)
(%− r)2
∫Ak,%
(|u− u1|2 − k
)2dx+
∫Ak,%
χ2dx
. (5.32)
This follows from |u− u1| ≤M(R)+ δM(R) on Ak,% and |g| ≤ b cf. (5.27),(5.18) by first applyingCachy-Schwarz, then Poincaré, and finally Young’s inequality with ε (using also the bound (5.30)on |Dχ|).
For the third and fourth integrals, the estimation relies crucially on the smallness condition.First of all, we notice that because of (5.27), we can write
H · (u− u1) =1
2
(H · u+H · (u− 2u1)
)≤ 1
2H · u+
1
2|H| (M + 2δM(R))
14Here summation of repeated indices is understood
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 42
Hence the two structural conditions (5.19) and (5.20) on H, imply that
I3 ≤ 1
2
(a∗ + aM(R) + 2δaM(R)
) ∫Ak,%
θijDju ·Diu(|u− u1|2 − k)χ2dx
+1
2
(b∗ + bM(R) + 2δbM(R)
) ∫Ak,%
(|u− u1|2 − k)χ2dx
Since a∗ + aM(R) + 2δaM(R) ≤ 2 by (5.26), the former of these two integrals cancels with I4.The latter integral can be estimated using Cauchy Schwarz, then Poincaré and finally Young’sinequality with ε (using also the bound (5.30) on |Dχ|). Hence we obtain
I3 + I4 ≤ 1
2
(b∗ + bM(R) + 2δbM(R)
)εc(d)
∫Ak,%
∣∣∣Dj |u− u1|2∣∣∣2 χ2dx
+c(ε, d)
(%− r)2
∫Ak,%
(|u− u1|2 − k
)2dx+
∫Ak,%
χ2dx
. (5.33)
Now, combining the estimates (5.31),(5.32) and (5.33), and absorbing the gradient terms into theLHS, we conclude that∫
Ak,%
∣∣∣D |u− u1|2∣∣∣2 χ2dx ≤ C
(%− r)2
∫Ak,%
(|u− u1|2 − k
)2dx+ C
∫Ak,%
χ2dx,
where C = C(d, κ, b, δ, bounds on b∗,M(R)). From the definition of χ, this gives us the desiredinequality.
The previous lemma allows us to state the following Proposition, which is the workhorse ofTolksdorfs approach.
Proposition 5.15. Fix y ∈ Ω ∩BR0(x0) and a ball BR(y) of arbitrary radius R ≤ R0. Let ε > 0
be given. Assume u ∈ H1 ∩ L∞ satisfies the estimate (5.29) of Lemma 5.12. Assume also thateither (5.23) holds or (5.24)-(5.25) holds (depending on whether we are interested in interior orboundary regularity).
Then there exists a constant σ > 0 and an exponent γ′′ ∈ (0, 1) with
σ = σ(d,N, κ, b, c′, γ′, δ, ε, a bound for b∗ and M(R)), γ′′ = γ′′(d,N, γ′),
Finally, if we are in the interior case (5.23), then γ′′ = 1/2 and σ is independent of c′ and γ′.
Proof. The key to the proof is that the estimate (5.29) in Lemma 5.12 allows us to apply somegeneralizations of the Strong Maximum Principle developed by Tolksdorf in [16]. We give a sketchof the proof in subsection 5.3.2 after Theorem 5.17 below.
Proposition 5.16. Fix an arbitrary y ∈ Ω ∩BR0(x0) and let u ∈ H1 ∩ L∞ be a weak solution of(5.16) satisfying the conditions (5.17)-(5.21). Assume also that either (5.23) or (5.24)-(5.25) aresatisfied.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 43
Then there exists δ, ε > 0, σ ∈ (0, 1), and R1 ∈ (0, R0] s.t.
oscBσiR1
(y)∩Ωu ≤ 2
(max (1− σ), (1− δ + 2ε
)iM,
where the dependence of σ is as in Proposition 5.14 and
δ = δ(a, a∗,M), ε = ε(δ), R1 = R1(σ, ε,M, γ′′).
Proof. First of all, we emphasize that the role of the assumptions is to allow us to apply Proposition5.14. For reasons to be apparent later, we fix a δ = δ(a, a∗,M) > 0 s.t.
aM + a∗ + 3δaM ≤ 2 (5.37)
and then we fix, for now, ε = ε(δ) > 0 such that 2ε < δ. Note that (5.37) is possible because ofthe smallness condition (5.21).
Given the above, we fix σ and γ′′ from Proposition 5.14 (as applied to our weak solution u).For notational convenience, we define M0 := M . Then we can choose an R1 ∈ (0, R0] sufficientlysmall so that
σ−1Rγ′′
1 ≤M0 min σ, ε . (5.38)
Without loss of generality we may assume that our σ > 0 satisfies
σγ′′≤ 1/2 ≤ min (1− δ), (1− σ) . (5.39)
Indeed, we can always choose σ smaller in Proposition 5.13.15
In the following, we initiate an inductive construction of a sequence (Mi), which will allow usto prove the theorem. We split this procedure into seven steps.
Step 1. Suppose we are in Case (1) of Proposition 5.13, meaning that (5.34) holds for R = R1.Then we set
M1 := (1− σ)M0, U1 := u.
Observe thatM(σR1) ≤M1. (5.40)
Indeed, it follows from (5.38) and M(R1) ≤M0 that
M(σR1) ≤ (1− 2σ)M(R1) + σ−1Rγ′′
1 ≤ (1− 2σ)M(R1) + σM0 ≤ (1− σ)M0 =M1.
Now suppose instead that we are in Case (2) of Proposition 5.13, meaning that (5.35)-(5.36)holds for R = R1. Then we set
M1 := (1− δ − 2ε)M0, U1 := u− u0,
where u0 is the vector provided by (2) of Proposition 5.13. Notice that
15Simply note that the RHS’s of (5.34) and (5.36) are decreasing in σ while the LHS of (5.34) is increasing in σ.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 44
Step 2. We now want to define Mi inductively by the following scheme: Let
Ri := σi−1R1
and defineMi := (1− σ)Mi−1, Ui := Ui−1 if Ri satisfies (1) of Prop. 5.13
Mi := (1− δ + 2ε)Mi−1, Ui := Ui−1 − u0,i−1 if Ri satisfies (2) of Prop. 5.13
where u0,i−1 is given by (2) of Proposition 5.13 applied to Ui−1. Note that, since we are applyingProposition 5.13 to Ui−1 and Ri, the statements in (1) and (2) are about MUi−1
(Ri) in place ofM(R).
For this to work, we of course need to ensure the estimate (5.29) of Lemma 5.12 holds for Ui−1,so that Proposition 5.13 can in fact be applied to Ui−1. This follows from Lemma 5.12 if wecan show that Ui−1 satisfies the assumptions (5.16)-(5.20) and (5.26). In addition, we need to becareful that we can choose the same σ in each application (we summarize why this can be done inStep 6).
Step 3. Consider first i = 1. If we are in Case (1), then there is nothing to check (sinceU1 = u). So assume we are in Case (2) where U1(x) = u(x)− u0.
First we show that U1 satisfies all the assumptions (5.16)-(5.20) with H(x) := H(x, u(x), Du(x))
thought of as a function of x only.16 Noting that DiU1 = Diu, only (5.20) requires explanation.However,
H · U1 ≤ H · u+H |u0| ≤ a∗∑i,j
θijDjU1DiU1 + b∗ + δM(R1)(a∑i,j
θijDjU1DiU1 + b)
≤ (a∗ + δaM0)∑i,j
θijDjU1DiU1 + (b∗ + δbM0),
so the condition is indeed satisfied by setting
a∗1 := a∗ + δaM0, b∗1 := b∗ + δbM0.
It remains to verify the final condition (5.26). From (5.41) we find that
by simply choosing ε small enough so that 2εaM0 ≤ δaM0.Note finally that σ can be taken to be the same for U1 as for U0 := u. Indeed, σ only depends
on the particular function Ui via ε, δ and a bound for MUi(Ri+1) and b∗i . But δ is fixed and ε only
depends δ, so this is fine. Moreover, MUi(Ri+1) ≤ M1 ≤ M0 by (5.41), so we can indeed use the
same bound for both U0 and U1. Similarly, we can use the same bound for b∗1 and b∗.Step 4. Now fix an arbitrary time j and assume the conclusion holds for all i < j. Let us begin
by observing that, for all i < j, we then have
MUi(Ri+1) ≤Mi. (5.42)
16We emphasize that U1 is of course not going to be a solution of the original equation, but this is unimportantfrom the point of view of obtaining the estimate in Lemma 5.12.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 45
To see this, first observe that it follows from Ri+1 = σiRi, the definition of Mi, and (5.39) and(5.39) that
σ−1Rγ′′
i+1 = σ−1Rγ′′
1 σiγ′′≤ min (1− σ), (1− δ + 2ε)iM0 min σ, ε ≤Mi min σ, ε .
Using this, the claim follows from either (5.34) or (5.36) depending on whether we are in case 1 or2. In particular, we have
MUi(Ri+1) ≤Mi ≤ max (1− σ), (1− δ + 2ε)iM0 ≤M0,
so we can choose the same bound for all MUi(Ri+1) uniformly in i.Step 5. We are now in a position to check the conditions for Lemma 5.12. If we are in Case
(1), then Uj = Uj−1 so the conclusion holds trivially with a∗j := a∗j−1 and b∗j := b∗j−1 .Now consider Case (2), where Uj = Uj−1 + u0,i−1. To verify (5.20), we proceed as in Step 3,
using now (5.42), to see that
H · Uj ≤ H · Uj−1 + |H| |u0,j−1| ≤ a∗j−1
∑k,`
θk`D`Uj−1DkUj−1 + b∗j−1
+ δMUj−1(Rj)
(a∑k,`
θk`D`Uj−1DkUj−1 + b)
≤ (a∗j−1 + δaMj−1)∑k,`
θk`D`Uj−1DkUj−1 + (b∗j−1 + δbMj−1).
Thus we simply need to define
a∗j := a∗j−1 + δaMj−1, b∗j := b∗j−1 + δbMj−1.
For the smallness-type condition (5.37), we use again (5.42) and obtain
If we suppose that Mj is defined inductively for all j ∈ N, then the sum of the “ε-corrections”only keeps growing if Mn = (1 − δ + 2ε)Mn−1 occurs infinitely often. However, even in thisworst-case-scenario, we obviously have
Nj∑i=1
2εaMn(i)−1 ≤∞∑i=1
2εaMn(i)−1 ∀j ∈ N,
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 46
where we claim that the infinite sum is convergent (and more importantly, dominated by a constantnot depending on σ). Indeed, by definition of n(i), we have
so we can choose ε = ε(δ) sufficiently small such thatNj∑i=1
2εaMn(i)−1 ≤ δaM0 ∀j ∈ N. (5.44)
It follows from the above, i.e. (5.43) and (5.44), together with the definition of δ in (5.37) that,for any j ∈ N,
aMUj(Rj+1) + a∗j + 2aδM0 ≤ aM0 + a∗ +
Nj∑i=1
2εaMn(i)−1 + 2aδM0
≤ aM0 + a∗ + 3aδM0 ≤ 2.
Hence the smallness-type condition (5.26) is satisfied for any Uj .Step 6. By the same arguments as for the choice of ε above (and using the same notation), we
see that
b∗j = b∗ +
Nj∑i=1
δbMn(i)−1 ≤ b∗ + δb
∞∑i=0
Mn(i)−1
≤ b∗ + δb∞∑i=0
(1− δ + 2ε)iM0 = b∗ + δbM0
δ − 2ε.
Consequently, b∗j is bounded uniformly in j. Likewise, we saw in (5.42) that
MUj(Rj+1) ≤M0
and hence MUj(Rj+1) is also bounded uniformly in j. Finally, from Step 5 we know that the
smallness-type condition (5.26) can be satisfied for all j by a choice of ε only depending on δ.It follows that, for this choice of ε, we can define Ui inductively for all i ∈ N by repeated
applications of Proposition 5.14. Since the bounds for b∗i and MUi(Ri+1) are uniform in i, wecan choose the same σ and γ′′ in each application, and hence the induction definition of Mi isconsistent.
Step 7. Now, going back to the beginning of the proof (before Step 1 in the construction ofMi) and choosing ε in accordance with the above, we thus get a σ > 0 such that our inductivedefinition of Mi can be carried out for all i ∈ N.
By construction, the Mi’s satisfy
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 47
Mi ≤ max (1− σ), (1− δ + 2ε)iM0.
Moreover, by definition of Ui we have
oscBσiR1
(y)∩Ωu = sup
BσiR1(y)∩Ω
u− infBσiR1
(y)∩Ωu
= supBσiR1
(y)∩Ω
Ui − infBσiR1
(y)∩ΩUi ≤ 2MUi(σ
iR1).
Recalling that σiR1 = Ri+1 and MUi(Ri+1) ≤Mi, it finally follows that
oscBσiR1
(y)∩Ωu ≤ 2Mi ≤ max (1− σ), (1− δ + 2ε)i 2M0.
This finishes the proof of the proposition.
Based on Proposition 5.15, we deduce the Hölder continuity of our weak solution.
Theorem 5.17. Let u ∈ H1 ∩ L∞ be a weak solution of (5.16) satisfying (5.17)-(5.21). Supposealso that the condition (5.23) for interior regularity is satisfied, i.e.
B2R0(x0) ⊆ Ω.
Then there is positive constant C = C(d,N, a, a∗, b, b∗,M,R0) s.t.
supx,y∈BR0
(x0),x 6=y
|u(x)− u(y)||x− y|γ
≤ C,
for some exponent γ ∈ (0, 1) with γ = γ(d,N, a, a∗, b, b∗,M,R0).
Proof. This follows from Proposition 5.15 and Lemma 5.1 with R1 and
λ = max (1− σ), (1− δ + 2ε)
as provided by Proposition 5.14. The dependences follow by keeping track of the dependencesfor the parameters δ, ε, R1 and σ as given in the statements of Proposition 5.14 and Proposition5.15.
Theorem 5.18. Let u ∈ H1 ∩ L∞ be a weak solution of (5.16) satisfying (5.17)-(5.21). Supposealso that the conditions (5.24)-(5.25) for boundary regularity are satisfied. Then there is positiveconstant C = C(d,N, a, a∗, b, b∗,M,R0, c
′, γ′) s.t.
supx,y∈BR0
(x0),x 6=y
|u(x)− u(y)||x− y|γ
≤ C,
for some exponent γ ∈ (0, 1) with γ = γ(d,N, a, a∗, b, b∗,M,R0, c′, γ′).
Proof. This follows again from Proposition 5.15 and Lemma 5.1. The extra dependence on c′ andγ′ arises because σ depends on these in the boundary case cf. Proposition 5.14.
It remains to see that Proposition 5.14 is true. This is the topic of the next subsection.
5.3.2. Proof of Proposition 5.14. Given any R ≤ R0 we fix a ball BR(y) and define
Ak,r := x ∈ Br(y) ∩ Ω : v > k .
Now fix a non-negative function v ∈ H1(BR(y)∩L∞(BR(y)) and suppose there is a constant C > 0
s.t. ∫Ak,r
|Dv|2 dx ≤ C
(%− r)2
∫Ak,%
(v − k
)2dx+ C |Ak,%| (5.45)
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 48
for all k ≥ 0 and all 0 < r < % ≤ R. Recall that Lemma 5.12 is precisely the statement that thisis satisfied for v := |u− u1|2.
For a non-negative function v satisfying the above, we get the following two lemmas fromProposition 4 and 5 in Tolksdorfs paper [16]. They are generalizations of Lemmas 6.1 and 6.2 inChapter 2 of [L&U].
Lemma 5.19. Fix ε > 0 and let v be as in the above. Then there is a ρ ∈ (0, 1) s.t.∣∣A0,R
∣∣ ≤ ρ∣∣BR(y)
∣∣ ⇒ supBR/2(y)
v ≤ ε supBR(y)
v +R.
Moreover, ρ = ρ(d,C, ε), where C is the constant from (5.45).
Proof. If supBRv ≤ R, then we have
supBR/2
v ≤ supBR
v ≤ R ≤ ε supBR
v +R,
so the statement is trivial. Now assume supBRv ≥ R. Then the claim follows from Proposition 4
in [16].To see this, let v := v − R and fix attention to k ≤ ε supBR
v. Then k ≤ supBRv, so |v − k| ≤
supBRv and hence
(v − k
)2 ≤ (supBRv)2. Thus, it follows from (5.45) that∫
Ak,r
|Dv|2 dx =
∫Ak+R,r
|Dv|2 dx ≤ C
(%− r)2
∫Ak+R,%
(v − k
)2dx+ C |Ak+R,%|
≤ C
(%− r)2(supBR
v)2∣∣Ak,%
∣∣+ C∣∣Ak,%
∣∣.Recalling that supBR
v ≥ R, we have
1 ≤(supBR
v)2
R2≤
(supBRv)2
(%− r)2.
and hence ∫Ak,r
|Dv|2 dx ≤2C(supBR
v)2
(%− r)2∣∣Ak,%
∣∣.Noting also that |Ak,%| ≤ |BR| = |B1|Rd , it follows that∫
Ak,r
|Dv|2 dx ≤2C |B1|α (supBR
v)2
(%− r)2Rdα
∣∣Ak,%
∣∣1−α
for an arbitrary α ∈ (0, 2/d). Setting
M := 21/2C1/2 |B1|α/2 supBR
v, δ :=ε
21/2C1/2 |B1|α/2
we have shown that∫Ak,r
|Dv|2 dx ≤ M2
(%− r)2Rdα
∣∣Ak,%
∣∣1−α ∀k ≤ δM = ε supBR
v.
Consequently, Proposition 4 of [16] yields a ρ = ρ(d, δ) = ρ(d,C, ε) > 0 such that the inequality∣∣A0,R
∣∣ ≤ ρ∣∣BR(y)
∣∣implies
supBR/2
v −R = supBR/2
v ≤ δM = ε supBR
v.
This proves the claim.
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 49
Lemma 5.20. Let v be as in the above and suppose there is a ρ ∈ (0, 1) s.t.∣∣A0,R/2
∣∣ ≤ ρ∣∣BR/2(y)
∣∣ .Then there is a δ = δ(d,C, ρ) > 0 s.t.
supBR/4(y)
v ≤ (1− δ) supBR(y)
v + δ−1R.
Proof. Let M := supBR(y) v. Then (5.45) implies that∫Ak,r
|Dv|2 dx ≤ C |Ak,%|(%− r)−2
(M − k
)2dx+ 1
.
≤ C |Ak,%|(%− r)−2
(M − k
)2dx+ (%− r)−1/2R1/2
.
Here we have simply used that 1 ≤ (% − r)−1/2R1/2 since (% − r) ≤ R. Taking N := R1/2 andµ = ρ in Proposition 5 of [16] gives the result.
The proof of Proposition 5.14 boils down to a clever application of these two lemmas. For thedetails, we refer to Proposition 1 in [15].
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 50
6. Epilogue
We end the report by discussing some wider perspectives and commenting on more recent results.First of all, we stress that most of the theory carry over to the parabolic case. As already
mentioned, Struwe [11] proved an analogous result to Theorem 5.17 for parabolic systems. For in-depth coverage of parabolic regularity theory in the case of quadratic growth (and, more generally,polynomial), we refer to the monograph [21] by Duzaar, Mingione & Steffen.
Another interesting issue is to consider degenerate systems. For a basic treatment of this, werefer to the monograph [22] by Bögelein, Duzaar & Mignione.
Moreover, in view of the game theoretic applications, it is interesting to note that the workof Bensoussan & Frehse has given rise to whole industry of related results. See for example[23, 24, 25, 26, 27, 28] as well as [29, 30, 31]. In particular, the parabolic analogue of the results inthe present report were fully resolved by Bensoussan & Frehse in [23].
Recently, their work has shifted to study the dynamic programming approach to stochasticMean Field Games. The book [32] by Bensoussan, Frehse & Yam provides a nice survey of theauthors’ contributions. See also their recent paper [33]. In the mean field theory, the result in thepresent report are still of significant interest. First of all, because they provide a means of studyinghow well the essentially 1-player mean field game approximates the limit of an N -player game asN goes to infinity. Secondly, many of the analytical techniques remain very much the same.
The final thing we would like to discuss is the more recent results in elliptic regularity theory atlarge. For diagonal systems and the relation to stochastic games, we refer to the 2015 paper [34]by Beck, Frehse & Bulíček. For a detailed review of general elliptic regularity results, we refer tothe survey [35] by Mignione. For a more recent and much more compact survey, we refer to [6] byKristensen & Mignione.
Finally, we would like to direct attention to an interesting article [36] by Mooney & Savin,where they construct a singular minimizer for a smooth uniformly convex integral functional.Several classical counterexamples to regularity can by found in Chapter 2 of [Giaquinta].
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 51
7. Appendix A - Stochastic Analysis
7.1. An Aside on Brownian Motion.In general, we take our measurable space (Ω,F) to be given by the classical Wiener space
C0([0,∞);Rd) =ω ∈ C([0,∞);Rd) : ω(0) = 0
together with the Borel σ-algebra under the uniform topology.
We then consider the unique Wiener measure P on (Ω,F) under which the identity map Id :
Ω → Ω, which we shall denote by W , has the law of a standard Brownian motion.As for notation, we think of C0([0,∞);Rd) as a subspace of (Rd)[0,∞) =
∏t∈[0,∞) Rd, i.e. we
think of ω : [0,∞) → Rd as (ω(t))t≥0 and hence we write W = (Wt)t≥0 where Wt = πt|C0is the
restriction to C0([0,∞);Rd) of t’th projection on (Rd)[0,∞), i.e. ω 7→Wt(ω) = ω(t).It is a standard fact that the Borel σ-algebra on C0([0,∞);Rd) coincides with the one induced
by the product σ−algebra on (Rd)[0,∞), which is just the σ-algebra generated by the restrictedprojections πt|C0
=Wt, i.e.F = σ(Wt : t ∈ [0,∞)).
This makes it clear that each Wt : (Ω,F) →(Rn,B(Rd)
)is a measurable random variable and the
marginal laws of the Wt’s of course satisfy the defining properties of a Brownian motion.The canonical nature of Ω = C0([0,∞);Rd) is apparent when we talk about the continuous
path of W = (Wt)t≥0 by which we mean the mapping t 7→ Wt(ω) = ω(t) for fixed ω ∈ Ω. Whatis perhaps less canonical about the Wiener space, however, is that it fails to be Banach. Yet,this is easily remedied by noting that ω(t)/t → 0 as t → ∞ (by the Law of large numbers forBrownian motion). Hence we may restrict attention to the subset of continuous functions forwhich this growth condition is satisfied, thus leaving us with a separable Banach space under thenorm ‖ω‖ := sup |ω(t)/(t+ 1)|.
7.2. Sobolev version of Ito’s Formula.The required estimates are developed in Section 2 of Chapter 2 in [Krylov]. See in particularTheorem 4 on p. 54. The generalization of Ito’s formula is proved in Section 10 of Chapter 2.
8. Appendix B - Regularity
Theorem 8.1. Consider an elliptic operator Θ of the form (3.2) satisfying the ellipticity condition(3.3). Suppose that a scalar function u ∈ H1
0 (Ω) ∩ L∞(Ω) satisfies
Θu ≤ λ+ λ0(ρ) |Du|2 − cu in H−1(Ω).
for some λ, c ≥ 0 with λ0(·) a positive increasing function evaluated at ρ = ‖u|L∞ . Then
u ≤ λ/c.
Proof. See Theorem 2.16, p. 87 of [B&F].
Theorem 8.2. Let u ∈W 2,r(Ω) ∩ C0,δ(Ω). Then u ∈W 1,q(Ω) with
q =rd
dσ + r(1− 2σ − δ(1− σ))∀σ ∈
[1− δ
2− δ, 1
].
Moreover, the following interpolation inequality holds
‖Du‖Lq ≤ C∥∥D2u
∥∥σLr [u]
1−σC0,δ + C [u]C0,δ .
Proof. See the references on p. 9 of [B&F].
ELLIPTIC REGULARITY FOR QUASILINEAR SYSTEMS AND STOCHASTIC DIFFERENTIAL GAMES 52
References
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[Giaquinta] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Prince-ton University Press, Princeton, New Jersey, 1983.
[G&T] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Grundlehrender Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-Heidelberg-New York, 1977.
[L&U] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and quasilinear elliptic equations, Mathematics inScience and Engineering. 46. New York-London: Academic Press., 1968.
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