On the boundary value problems for fully nonlinear elliptic equations of second order M.V. SAFONOV 127 Vincent Hall, University of Minnesota, Minneapolis, MN, 55455 Abstract Fully nonlinear second-order, elliptic equations F (x, u, Du, D 2 u) = 0 are considered in a bounded domain Ω ⊂ R n ,n ≥ 2. The class of equations includes the Bellman equations sup m (L m u + f m ) = 0, where the functions f m and the coefficients of the linear operators L m are bounded in the H¨older space C α ( Ω), 0 <α< 1. We prove the interior C 2,α -smoothness of solutions in Ω with some small α> 0. Under the Dirichlet boundary condition u = φ on ∂ Ω with φ ∈ C 2,α ( Ω) and ∂ Ω ∈ C 2,α , the solutions u ∈ C 2,α ( Ω). Under the oblique derivative condition b 0 u + b · Du = φ on ∂ Ω, where b =(b 1 , ··· ,b n ) is not tangent to ∂ Ω, the solutions u ∈ C 2,α ( Ω) if b i ,φ ∈ C 1,α ( Ω), and also ∂ Ω ∈ C 1,α . Contents 1. Introduction 2 2. The H¨older spaces 4 3. Formulation of main existence results 9 4. Interior C 2,α − estimates: the simplest nonlinear equations 12 5. Interior C 2,α − estimates: general equations 16 6. Some boundary estimates for solutions of linear elliptic equa- tions 20 7. Boundary C 2,α − estimates: the Dirichlet problem 22 8. Boundary C 2,α − estimates: the oblique derivative problem 25 1
Transcript
On the boundary value problemsfor fully nonlinear elliptic equations
of second order
M.V. SAFONOV127 Vincent Hall, University of Minnesota, Minneapolis, MN, 55455
Abstract
Fully nonlinear second-order, elliptic equations F (x, u,Du,D2u) = 0 are considered
in a bounded domain Ω ⊂ Rn, n ≥ 2. The class of equations includes the Bellman
equations supm(Lmu+ fm) = 0, where the functions fm and the coefficients of the
linear operators Lm are bounded in the Holder space Cα(Ω), 0 < α < 1. We prove
the interior C2,α-smoothness of solutions in Ω with some small α > 0. Under the
Dirichlet boundary condition u = φ on ∂Ω with φ ∈ C2,α(Ω) and ∂Ω ∈ C2,α, the
solutions u ∈ C2,α(Ω). Under the oblique derivative condition b0u + b · Du = φ
on ∂Ω, where b = (b1, · · · , bn) is not tangent to ∂Ω, the solutions u ∈ C2,α(Ω) if
bi, φ ∈ C1,α(Ω), and also ∂Ω ∈ C1,α.
Contents
1. Introduction 2
2. The Holder spaces 4
3. Formulation of main existence results 9
4. Interior C2,α− estimates: the simplest nonlinear equations 12
5. Interior C2,α− estimates: general equations 16
6. Some boundary estimates for solutions of linear elliptic equa-tions 20
7. Boundary C2,α− estimates: the Dirichlet problem 22
8. Boundary C2,α− estimates: the oblique derivative problem 25
1
1. Introduction
In this paper we consider general nonlinear elliptic equations including theBellman equations
(1.1) supm
(Lmu+ fm) = supm
(amijDiju+ bmi Diu+ cmu+ fm) = 0
(the summation over repeated indices is everywhere understood). Such equa-tions are also important from the viewpoint of the applications to the theoryof controlled diffusion processes (see [10]).
We investigate the Dirichlet and the oblique derivative problems in a boun-ded domain Ω ⊂ Rn, n ≥ 2, for nonlinear elliptic equations in the Holder spaceC2,α(Ω), 0 < α < 1. Leaving aside the simpler one- and two- dimensionalcases, we note that the interior C2,α-smoothness of solutions to the equation(1.1) was first proved in 1977 by Brezis and Evans [2] in the case when massumes only two values. In 1981, Krylov [11], [12] has established the C2,α-smoothness of the solutions of the Bellman elliptic and parabolic equationsin the higher-dimensional case, both in the interior of the domain and nearits boundary, under appropriate smoothness of the boundary and the bound-ary values of the solutions. At about the same time, Evans [7] (see also [9])independently proved the C2,α-smoothness of solutions of elliptic equations(1.1) in the interior of the domain. Under the oblique derivative condition,the C2,α-smoothness of solutions near the boundary was proved in [16], [17].In all those papers, and also in [4], [5], [9], [13], as they apply to (1.1), itis assumed that the functions amij , b
mi , c
m, fm are uniformly bounded, togetherwith all their first and second derivatives.
The C2,α-estimates of solutions fo the Bellman equation (1.1) with coef-ficients amij , b
mi , c
m, fm in Cα for some small α ∈ (0, 1), were first obtainedin [24], including the estimates near the boundary in the Dirichlet case. Forthe oblique derivative problem, the corresponding result was proved in [1].Some other extensions of the results in [24], both for the Dirichlet and for theoblique derivative problem, were derived by Trudinger [28]. There are alsosome close results in the papers of Caffarelli [3] and Wang [29], where theytreat both the C2,α-estimates and the estimates in the Sobolev spaces W 2,p
when fm ∈ Lp, 1 < p <∞.Here we give an enlarged exposition of results in [24], [26], [1]. We intro-
duce a class of nonlinear equations including the Bellman equaitons (1.1) withcoefficients in Cα for some small α ∈ (0, 1), and we show that the Dirichletproblem and the oblique derivative problem are solvable in C2,α. Under min-imal assumptions on the boundary and the boundary data, we receive alsothe C2,α-estimates near the boundary for solutions of these problems. Theseresults are formulated in Section 3 (Theorems 3.1–3.3). Notice that in the caseof linear elliptic equations, they turn into the classical Schauder-type results(see [9], Ch.6; [19], Ch.3), even with an improvement: Theorem 3.3 states thatthe solution of the oblique derivative problem in Ω still belongs to C2,α(Ω) if∂Ω ∈ C1,α. For linear elliptic equations such an improvement was proved byLieberman [15].
2
The most essential part in the proof of Theorems 3.1–3.3 consists in theappropriate a priori C2,α-estimates of the solutions. The basic idea of derivingsuch estimates is the “local” decomposition of the solution into “smooth” and“small” terms. The “smooth” term is the solution of an auxiliary problem forthe simplest nonlinear equation corresponding to the case when in (1.1) wehave amij , f
m = const, and bmi = cm = 0. For the technical realization of thisidea, it is convenient to use some equivalent seminorms in C2,α introduced byCampanato [6]. In Section 2, we expose a simple approach to the Campanatotype of seminorms which are equivalent to the usual seminorms in “weighted”Holder spaces (Theorem 2.1).
For the completeness of the presentation, in Section 4 we prove the interiorC2,α-estimates of Krylov [11] and Evans [7], in a particular case of the simplestnonlinear equaitons. Our approach is new in some details, while it relies, aswell as [11], [7], on the results of [14], [22]. In Section 5, we extend the interiorC2,α-estimates to the solutions of the general nonlinear equaitons. The nextSection 6 is devoted to the boundary behaviour of solutions to the linear ellipticequations with measurable coefficients. These auxiliary results help us to getthe C2,α-estimates near the boundary for soluitons of nonlinear equations; theDirichlet and the oblique derivative conditions are treated in Sections 7 and 8correspondingly.
BASIC NOTATIONS. Rn is Euclidean space of dimension n, with stan-dard basis e1, · · · , en, and points x = (x1, · · · , xn) written in coordinatesrelative to this basis; (x, y) = xiyi is the inner product of x, y ∈ Rn; |x| =(x, x)1/2 = (
∑x2i )
1/2; Rn+ = x ∈ Rn : xn > 0, Rn
0 = x ∈ Rn : xn = 0, Sn
denotes the n(n+1)/2- dimensional space of all real symmetric n×n matrices.We will identify Rn
0 and Rn−1.∂Ω is the boundary of the set Ω ⊂ Rn, Ω = Ω ∪ ∂Ω; Bρ(x) = y ∈ Rn :
|y − x| < ρ is the ball of radius ρ > 0 centered at x ∈ Rn,
B+ρ (x) = Rn
+ ∩Bρ(x), B0ρ(x) = Rn
0 ∩Bρ(x), Ωρ(x) = Ω ∩Bρ(x).
l = (l1, · · · , ln) is a multi-index, i.e. li = integer ≥ 0, with |l| =∑li.
We define xl = xl11 xl22 · · · xlnn , l! = l1! l2! · · · ln!. For functions u = u(x),
we set Diu = ∂u/∂xi, Diju = ∂2u/∂xi∂xj; Du = (D1u, · · · , Dnu) is thegradient of u, D2u = [Diju] is the Hessian matrix. Moreover, we definethe first and the second derivatives of u in the direction λ ∈ Rn as follows:Dλu = ∂u/∂λ = λiDiu, Dλλu = ∂2u/∂λ2 = λiλjDiju. We will also usethe multi-index notation Dlu = ∂|l|u/∂xl11 · · · ∂xlnn , with the understandingD0u = u. For integer k ≥ 0, Pk denotes the collection of all polynomials ofdegree at most k. In particular, the Taylor polynomial of degree k for thefunction u at the point y ∈ Rn is
(1.2) Ty,k u(x) =∑|l|≤k
Dlu(y) · (x− y)l/l! ∈ Pk.
Throughout this paper, N will denote various positive constants. In theintermediate calculations, we will usually omit the dependence of N on theoriginal quantities.
3
Acknowledgements
This work was completed while the author was visiting the Australian NationalUniversity during Spring 1994. A considerable part of the research was carriedout ealier, during the visit of the Dipartimento di Matematica Applicata dell’Universita di Firenze in 1990.
2. The Holder spaces
Let Ω be a domain in Rn, n ≥ 1. For k = 0, 1, 2, · · · , we denote Ck(Ω) the setof functions u = u(x) whose derivatives Dlu for |l| ≤ k are continuous in Ω.We set
We will also use the similar notations for closed domains Ω and moregenerally, for Ω∪Γ, where Γ ⊂ ∂Ω. Obviously, for bounded domain Ω we haveCk,0(Ω) = Ck(Ω). For simplicity we will write C0,α = Cα, if 0 < α < 1. Fromthe elementary inequality
with a constant N(ε) = N(ε, n, k, α, β). The similar inequalities are also truefor B+
r = B+r (x0), x0 ∈ Rn
+ = x ∈ Rn : xn ≥ 0.
Further, let a subset Γ ⊂ ∂Ω be given, Γ = ∂Ω (the case Γ = ∅ is notexcluded). For k = 0, 1, 2, · · · , 0 ≤ α ≤ 1, γ ∈ R1, and u ∈ Ck(Ω∪Γ), we set
(2.8) [u](γ)k,α = [u]
(γ)k,α;Ω∪Γ = sup
x∈Ω∪Γdk+α+γ(x) · [u]k,α;Ω(x),
where
(2.9) d(x) =1
2dist(x, ∂Ω \ Γ), Ω(x) = Ωd(x)(x) = Ω ∩Bd(x)(x).
Definition 2.3. For Γ ⊂ ∂Ω, k = 0, 1, 2, · · · , and γ ∈ R1, Ck;γ(Ω ∪ Γ) =Ck,0;γ(Ω ∪ Γ) is the Banach space of functions u ∈ Ck(Ω ∪ Γ) with the finitenorm
(2.10) ∥u∥(γ)k,0 = ∥u∥(γ)k,0;Ω∪Γ =
k∑j=0
[u](γ)j,0;Ω∪Γ.
Definition 2.4. For Γ ⊂ ∂Ω, k = 0, 1, 2, · · · , 0 < α ≤ 1, and γ ∈ R1,the weighted Holder space Ck,α;γ(Ω ∪ Γ) is the Banach space of functions u ∈Ck(Ω ∪ Γ) with the finite norm
Multiplying both sides of this onequality by dγ(x), and then taking the supover x ∈ Ω∪Γ, we arrive at the following interpolation inequalities for weightedHolder spaces.
Corollary 2.1. Suppose j + β < k + α, and let u ∈ Ck,α;γ(Ω ∪ Γ), γ ∈ R1.Then for and ε > 0 we have
(2.12) [u](γ)j,β;Ω∪Γ ≤ ε [u]
(γ)k,α;Ω∪Γ +N(ε, n, k, α, β) · |u|(γ)0,0;Ω∪Γ.
The following lemma is related to the approximation of a function u bymeans of its Taylor polynomial Ty,ku defined in (1.2).
5
Lemma 2.2. Let u ∈ Ck,α(Ω), 0 < α ≤ 1. Then for any x, y ∈ Ω such thatthe segment [x, y] ⊂ Ω, we have
Corollary 2.2. Let u ∈ Ck,α(Ωρ), where Ωρ = Bρ(x0), x0 ∈ Rn, or Ωρ =B+
ρ (x0), x0 ∈ Rn+. Then
(2.14) Ek[u; Ωρ] = infp∈Pk
supΩρ
|u− p| ≤ N(n)[u]k,α ρk+α.
Lemma 2.3. Let k = 0, 1, 2, · · · , 0 < α ≤ 1, and u ∈ Ck,α(Bρ), Bρ = Bρ(x0).Then for any ε > 0 we have
(2.15) ρ−α max|l|=k
oscBρ
Dlu ≤ ε[u]k,α;Bρ +N(ε, n, k, α) · ρ−k−αEk[u;Bρ],
where osc f = sup f − inf f . The similar inequalities are also true for B+ρ =
B+ρ (x0), x0 ∈ Rn
+.
Proof: Using the elementary inequality osc f ≤ 2 sup |f | and (2.7) withr = ρ, j = k, β = 0, we have
1
2ρ−α max
|l|=koscBρ
Dlu ≤ ρ−α[u]k,0;Bρ ≤ ε[u]k,α;Bρ +N(ε)ρ−k−α supBρ
|u|.
For arbitrary p ∈ Pk, the left-hand side of this inequality and [u]k,α remain thesame if we replace u by u− p. After the replacement, we take the infimum ofthe right-hand side over p ∈ Pk. On redefining ε, this will give us the desiredestimate.
The next theorem is similar to Theorem 2.1 in [26] (see also [6]).
6
Theorem 2.1. Let k = 0, 1, 2, · · · , 0 < α ≤ 1, γ ∈ R1, and u ∈ Ck(Ω ∪ Γ)
has a finite seminorm [u](γ)k,α in (2.8). Set
(2.16) ωk(x, ρ) = max|l|=k
oscΩρ(x)
Dlu, Ωρ(x) = Ω ∩Bρ(x),
(2.17) M(γ)k,α = M
(γ)k,α[u; Ω ∪ Γ] = sup
x∈Ω∪Γdk+α+γ(x) sup
ρ∈(0,d(x)]ρ−αωk(x, ρ),
(2.18)
M(γ)k,α =M
(γ)k,α[u; Ω ∪ Γ] = sup
x∈Ω∪Γdk+α+γ(x) sup
ρ∈(0,d(x)]ρ−k−αEk[u; Ωρ(x)],
where d(x) = 12dist(x, ∂Ω\Γ), Ek is defined in (2.14). Then all the seminorms
[u](γ)k,α, M
(γ)k,α, and M
(γ)k,α are equivalent :
(2.19) N−11 [u]
(γ)k,α ≤ M
(γ)k,α ≤ N2[u]
(γ)k,α,
(2.20) N−13 [u]
(γ)k,α ≤M
(γ)k,α ≤ N4[u]
(γ)k,α,
where the constants N depend only on n, k, α, γ.
Proof: To prove the first inequality in (2.19), we fix x0 ∈ Ω∪Γ, d = d(x0) =12dist(x0, ∂Ω \ Γ), |l| = k, and x, y ∈ Ωd(x0), such that
We consider separately the cases (a) ρ = |x − y| < d/2 and (b) ρ ≥ d/2.In the case (a), we have x, y ∈ Ωd/2(y) ⊂ Ω3d/2(x0). Since d/2 ≤ d(y) ≤ 3d/2,from (2.21) it follows
(2.22) [u](γ)k,α ≤ N dk+α+γ(z) · ρ−α osc
Ωρ(z)Dlu
with N = N(k, γ), z = y. Obviously, this inequality is also true in the case(b) for z = x0, ρ = d, and N = 21+α. In any case, we have (2.22), where0 < ρ ≤ d(z), that yields the first inequality in (2.19) with N1 = N1(k, γ).The second inquality is trivial with N2 = 2α.
Further, from Lemma 2.3 and the definition of [u](γ)k,α, M
(γ)k,α, and M
(γ)k,α, we
get the inequailty
M(γ)k,α ≤ ε[u]
(γ)k,α +N(ε, n, k, α) ·M (γ)
k,α.
Setting ε = (2N1)−1 and using (2.19), we obtain the first inequality in (2.20).
Finally, the last inequality follows immediately from Corollary 2.2 with N4 =N4(n).
We will investigate the boundary value problems in a bounded domain Ωunder some natural restrictions on the boundary ∂Ω. The following assump-tions are usually called the Lipschitz condition on ∂Ω.
7
Assumptions 2.1. There exist positive constants r0 and K0, such that foreach x0 ∈ ∂Ω we have(2.23)
This estimate with arbitrary x1, x2 ∈ Ω implies (2.25).
Remark 2.1. In the standard approach to the Schauder interior estimates(see [9], Ch.6) the notation [u]
(γ)k,α is used for
(2.28) A = max|l|=k
supx,y∈Ω
dk+α+γx,y
|Dlu(x)−Dlu(y)||x− y|α
= supδ>0
δk+α+γ[u]k,α;Ωδ,
8
where 0 < α ≤ 1, k + α + γ ≥ 0, dx,y = dist (x, y, ∂Ω) , and
(2.29) Ωδ = x ∈ Ω : dist(x, ∂Ω) > δ.
In particular, in the case γ = −k−α we have A = [u]k,α;Ω . One can show
that for Lipschitz domains Ω seminorms [u](γ)k,α in (2.8) and A in (2.28) are
equivalent, if k + α + γ ≥ 0.However, in the case k + α + γ < 0 we have A <∞ only for polynomials
of degree at most k (and then A = 0), while [u](γ)k,α <∞ for more general class
of functions. For example, if k + α + γ < 0 ≤ k + 1 + γ and u ∈ Ck+1(B1),then by the mean value theorem we obtain
[u](γ)k,α ≤ N1[u]
(γ)k+1,0 ≤ N2[u]k+1 <∞.
3. Formulation of main existence results
Let Ω be a bounded domain in Rn and constants K,K1 ≥ 0, ν ∈ (0, 1],α ∈ (0, 1) be fixed. We will consider nonlinear elliptic equations
(3.1) F [u] = F (x, u,Du,D2u) = 0 in Ω,
where F (x, u, ui, uij) is defined on Ω×R1×Rn×Sn and satisfies the followingconditions:
Assumptions 3.1. (F0) The function F (x, u, ui, uij) is lower convex withrespect to [uij] ∈ Sn; (F1) (the ellipticity condition)
ν|ξ|2 ≤ F (x, u, ui, uij + ξiξj)− F (x, u, ui, uij) ≤ ν−1|ξ|2 for all ξ ∈ Rn;
(F2) F (x, u, ui, uij) is nonincreasing with respect to u, and
|F (x, u, ui, uij)− F (x, u, ui, uij)| ≤ K · (|u− u|+∑i
|ui − ui|)
for all x, u, u, ui, ui, uij; (F3) |F (x, 0, 0, 0)| ≤ K1 for all x ∈ Ω;(F4) for any fixed (u, ui, uij) ∈ R1 ×Rn × Sn, the seminorm in Cα,
[F (·, u, ui, uij)]α;Ω ≤ K · (|u|+∑i
|ui|+∑ij
|uij|) +K1.
Remark 3.1. It is easy to see that if F satisfies the additional condition(F∗) F (x, u, ui, uij) is infinitely differentiable with respect to (u, ui, uij) ∈R1 ×Rn × Sn, then conditions (F0)–(F2) can be rewritten as follows:(F0∗) ∂2F/∂uij∂upq · ηijηpq ≥ 0 for all [ηij] ∈ Sn; (F1∗ ) the functionsaij = ∂F/∂uij satisfy
(3.2) aij = aji, ν|ξ|2 ≤ aijξiξj ≤ ν−1|ξ|2 for all ξ ∈ Rn;
(F2∗ ) the functions bi = ∂F/∂ui, c = ∂F/∂u satisfy
(3.3) |bi| ≤ K, −K ≤ c ≤ 0.
9
We notice that our equations (3.1) include the Bellman equations (1.1), ifaij = amij satisfy (3.2), cm ≤ 0 for all m, and the norms in Cα(Ω),
|amij , bmi cm|α;Ω ≤ K, |fm|α;Ω ≤ K1.
Consequently, the following Theorems 3.1–3.3 can be viewed as generalizationsof known Schauder-type results for linear equations (see [9], Theorems 6.13,6.14, 6.31; [19], Ch.3).
Theorem 3.1. Let Ω be a bounded Lipschitz (satisfying Assumptions 2.1)domain in Rn, d0 = diamΩ ≤ R0 = const < ∞, φ ∈ C(Ω), and letF (x, u, ui, uij) satisfy Assumptions 3.1. Then the Dirichlet problem
(3.4) F [u] = 0 in Ω, u = φ on ∂Ω
has a unique solution u ∈ C2,α;0(Ω) ∩ C(Ω), provided 0 < α < α for someconstant α = α(n, ν) ∈ (0, 1). Moreover, we have
We will use the following definition for the classification of the boundaries∂Ω having higher then the Lipschitz smoothness.
Definition 3.1. The boundary ∂Ω of a bounded domain Ω ⊂ Rn belongsto the class Ck,α for k = 1, 2, · · · , 0 < α ≤ 1, if there exists a functionΨ(x) ∈ Ck,α(Rn) such that
Ω = x ∈ Rn : Ψ(x) > 0 and |DΨ| ≥ 1 on ∂Ω.
Theorem 3.2. Let Ω be a bounded domain in Rn with the boundary ∂Ω ∈C2,α, φ ∈ C2,α(Ω), and let F (x, u, ui, uij) satisfy Assumptions 3.1. Then theDirichlet problem (3.4) has a unique solution u ∈ C2,α(Ω), provided 0 < α < αfor some constant α = α(n, ν) ∈ (0, 1). Moreover, we have (3.5) and
Theorem 3.3. Let Ω be a bounded domain in Rn with the boundary ∂Ω ∈C1,α, the functions b0, b1, · · · , bn, g ∈ C1,α(Ω), and let F (x, u, ui, uij) satisfyAssumptions 3.1. Suppose that for some constant ν0 > 0,
(3.8) b0 ≥ ν0, b · N =n∑i
biNi ≥ ν0|b| > 0 on ∂Ω,
where N = −|DΨ|−1DΨ is the outward unit normal on ∂Ω. Then the obliquederivative problem
(3.9) F [u] = 0 in Ω, Bu = b0u+ b ·Du = g on ∂Ω
10
has a unique solution u ∈ C2,α(Ω), provided 0 < α < α for some constantα = α(n, ν, ν0) ∈ (0, 1). Moreover, we have
(3.10) |u|2,α;Ω ≤ N · (U0 +K1 + |g|1,α;Ω),
where the constant N depends only on n, ν,K, ν0, domain Ω (with ∂Ω ∈ C1,α),and on the norms of the functions b0, b1, · · · , bn in C1,α(Ω).
Theorems 3.1, 3.2 are similar to Theorems 1.1, 1.2 from [26], Theorem 3.3for Bellman equations (1.1) in Ω with ∂Ω ∈ C2,α was proved in [1]. Analogousresults are also true in the parabolic case, with the same modifications as inthe theory of linear equations ([18], Theorems 5.1–5.3 in Ch.4).
Below, in Sections 4–8, we describe the technique of deriving C2,α-estimatesfor solutions of the problems (3.4), (3.9) by given constants n,K,K1, α, · · · .On the grounds of the C2,α-estimates, one can prove Theorems 3.1–3.3 by thestandard continuity method (see [9], Ch.17; [13], Sec. 1.3). Therefore, we willonly supplement these sections with some remarks concerning the continuitymethod applied to our problems.
Our approach to C2,α-estimates in general case uses analogous estimatesin the case of simplest nonlinear equations and the comparison principle fornonlinear equations. The simplest nonlinear equations are defined as follows.
Definition 3.2. The simplest nonlinear elliptic equation has the form F0[u] =F0(D
2u) = 0, where the function F0(uij) satisfies (F0), (F1) on Sn with some
constant ν ∈ (0, 1], and F0(0) = 0. We denote F(ν) the class of all suchfunctions F0(uij).
The comparison principle is based on the next simple lemma.
Lemma 3.1. Let F (x, u, ui, uij) satisfies (F0)–(F2) with some constantsK ≥ 0, ν ∈ (0, 1], and let u, v ∈ C2(Ω). Then F [u] − F [v] = L(u − v) in Ω,where the linear elliptic operator L = aijDij + biDi + c has the coefficientssatisfying (3.2), (3.3) on Ω. Furthermore, in F0(uij) ∈ F(ν), then
where aij = aij(x) satisfy (3.2) on Ω. In particular (v = 0),
(3.12) F0[u] = L0u = aijDiju in Ω.
The coefficients aij, bi, c can be constructed directly (see [26], Lemma 1.1)or through approximation of F by smooth functions. For example, (3.11)follows from the equality
(3.13) F0(uij)− F0(vij) = aij · (uij − vij) for all [uij], [vij] ∈ Sn,
where aij (depending on uij, vij) satisfy (3.2). If F0(uij) ∈ F(ν) is smooth,then we can take in (3.13)
aij =
∫ 1
0
aij(θuij + (1− θ)vij) dθ, where aij(uij) = ∂F0(uij)/∂uij,
11
and (3.11) holds by virtue of (F1∗ ).From this lemma and the classical maximum principle (see [9], Theorem
3.7; [19], Ch.3, §1), we get
Corollary 3.1. Let F satisfy (F0)–(F2), and let u, v ∈ C2(Ω) ∩ C(Ω). Then
supΩ|u− v| ≤ sup
∂Ω|u− v|+N(n, ν,K, diamΩ) · sup
Ω|F [u]− F [v]|.
In particular, under the assumptions of Theorems 3.1 and 3.2, there exists atmost one solution of the problem (3.4).
Taking in this corollary v = 0, we arrive at the following:
Corollary 3.2. Under the assumptions of Theorems 3.1 and 3.2, the estimate(3.5) holds.
Corollary 3.3. If u ∈ C2(Ω) is a solution of the simplest nonlinear equaitonF0[u] = 0 in Ω, then supΩ |u| = sup∂Ω |u|.
4. Interior C2,α− estimates: the simplest nonlinear equations
In this section we will obtain the interior C2,α-estimates for solutions of sim-plest nonlinear equations F0[u] = F0(D
2u) = 0. In the next section, theseestimates will be applied to the proof of similar estimates for solutions ofgeneral nonlinear equations.
Theorem 4.1. Let ν ∈ (0, 1], x0 ∈ Rn, r > 0, Br = Br(x0), φ ∈ C(Br) , andthe function F0(uij) ∈ F(ν). Then the problem
(4.1) F0[v] = F0(Dijv) = 0 in Br, v = φ on ∂Br
has a unique solution v ∈ C2,α;0(Br) ∩ C(Br) , and
(4.2) ∥v∥(0)2,α;Br≤ N · sup
∂Br
|φ|,
where the constants α ∈ (0, 1] and N > 0 depend only on n, ν.
This theorem (for more general equations) was proved independently byN.V. Krylov [11] and L.C. Evans [7] (see also [9], Section 17.4). We give hereanother proof which we precede with three lemmas. We will use the followinglemma from [14], [22] (see also [13], [9], Sec. 9.8).
Lemma 4.1. Let ν ∈ (0, 1], ρ > 0, µ ∈ (0, 1], V ∈ C2(B2ρ), and suppose thatV ≥ 0, aijDijV ≤ 0 in B2ρ, where the functions aij = aij(x) satisfy
(4.3) aij = aji, ν|ξ|2 ≤ aijξiξj ≤ ν−1|ξ|2 for all ξ ∈ Rn.
Moreover, let the Lebesgue measure |x ∈ Bρ : V (x) ≥ 1| ≥ µ · |Bρ|. Then
infBρ
V ≥ β = β(n, ν, µ) > 0.
12
The next lemma is standard in the study of Holder spaces.
Lemma 4.2. Let constants q > 1, α > 0, ρ0 > 0 be given, and let ω(ρ) be apositive non-decreasing function on (0, ρ0] satisfying the inequality
(4.4) qαω(ρ) ≤ ω(qρ) for all ρ ∈ (0, ρ0/q).
Then
(4.5) ρ−αω(ρ) ≤ qαρ−α0 ω(ρ0) for all ρ ∈ (0, ρ0].
Proof. By virtue of monotony of ω(ρ), (4.5) is evident for ρ ∈ (ρ0/q, ρ0]. Ifρ < ρ0/q, then q
kρ ∈ (ρ0/q, ρ0] for some natural k, and using (4.4), we obtain:
where the surface integral over Λ = ∂B1(0) is considered. Then
(4.7) N−1ω ≤ ω∗ ≤ N ω, where N = N(n).
Proof. Consider the function Q(λ, x) = Dλλv(x) = λiλjDijv(x) on theset Rn ×Bρ. Notice that for all i, j, λ, x,
2Dijv(x) = Q(λ+ ei + ej, x)−Q(λ+ ei, x)−Q(λ+ ej, x) +Q(λ, x).
Integrating this inequality over λ ∈ B1(0), we obtain:
2 |B1| ·Dijv(x) =
∫B1(ei+ej)
−∫B1(ei)
−∫B1(ej)
+
∫B1(0)
Q(λ, x) dλ.
In the right-hand side we have four integrals over unit balls with centers atei + ej, ei, ej, 0 ∈ Rn. All these balls are contained in B3(0), therefore,
(4.8) 2 |B1| · ω ≤ 4
∫B3(0)
oscx∈Bρ
Q(λ.x) dλ.
Further, Q(rλ, x) ≡ r2Q(λ, x) for all r > 0. Hence by passing to thespherical coordinates, the integral in (4.8) can be rewritten as follows:∫ 3
0
dr
∫∂Br(0)
oscx∈Bρ
Q(λ, x) dsλ =
∫ 3
0
rn+1dr
∫Λ
oscx∈Bρ
Dλλv · dsλ =3n+2
n+ 2ω∗.
13
This relation together with (4.8) give us the first inequality in (4.7). Thesecond inequality is obvious because for all λ ∈ Λ,
oscx∈Bρ
Dλλv ≤ ω ·∑i,j
|λiλj| ≤ω
2
∑i,j
(λ2i + λ2j) = nω.
Proof of Theorem 4.1. Now we will prove only the estimate (4.2) assumingthat the problem (4.1) is solvable in C2,α;0(Br)∩C(Br). This assumption willbe substantiated below, in Remark 5.1.
Step 1. Notice that each function F0(uij) ∈ F(ν) can be easy approximatedby smooth functions F δ
0 (uij) ∈ F(ν), δ > 0, such that
(4.9) |F0(uij)− F δ0 (uij)| ≤ δ for all [uij] ∈ Sn, δ > 0.
If vδ is the solution of the problem
F δ0 [v
δ] = 0 in Br, vδ = φ on ∂Br,
then by Corollary 3.1 and (4.9) we obtain
supBr
|vδ − v| ≤ N · supBr
|F0[vδ]− F0[v]| ≤ N · sup
Br
|F0[vδ]− F δ
0 [vδ]| ≤ Nδ.
Therefore, if vδ satisfy the estimate (4.2) for all δ > 0, then this estimateremains valid for v, the solution of initial problem (4.1). Thus we can as-sume without loss of generality that F0(uij) is smooth. Then automaticallyv ∈ C∞(Br) (see [13], Lemma I.3.2; [9], Lemma 17.16). Next, replacing r byr − ε and then letting ε→ 0+, we can assume v ∈ C4(Br).
Step 2. Let us fix(4.10)
z ∈ Br, 0 < ρ ≤ d(z) =1
2dist(z, ∂Br), ω = ω(ρ) = max
i,joscBρ(z)
Dijv.
Relying on the smoothness of F0(uij) and v(x), we differentiate the equalityF0[u] = 0 twice in the direction λ ∈ Λ. Using (F0∗), (F1∗) in Remark 3.1, wehave
(4.11) aijDijDλλv = −∂2F0/∂uij∂upq ·DijDλv ·DpqDλv ≤ 0 in Br,
where aij = ∂F0/∂uij. Hence the functions
V λ(x) = Dλλv(x)− infB2ρ(z)
Dλλv
satisfy the relations
(4.12) V λ ≥ 0, aijDijVλ ≤ 0 in B2ρ(z).
Step 3. Next, we fix x ∈ Bρ(z). From the definition of ω in (4.10) it follows
that there exists a point y ∈ Bρ(z) such that
ω = ω(ρ) ≥ maxi,j|Diju(x)−Diju(y)| ≥ ω/2.
14
By virtue of (3.13), we have aij ·(Dijv(x)−Dijv(y)) = 0, where aij (dependingon x, y) satisfy (4.3). Therefore, if µ1 ≤ µ2 ≤ · · · ≤ µn are eigenvalues of thematrix M = [Mij] = [Dijv(x) − Dijv(y)], then µn ≥ 2µω for some constantµ = µ(n, ν) > 0. Moreover, we can assume µ = µ(n, ν) > 0 to be suitablechosen so that the Lebesgue measure on Λ = ∂B1(0),
|λ ∈ Λ :Mijλiλj ≥ µω| ≥ 2µ · |Λ|.
Since x ∈ Bρ(z) can be taken arbitrarily, and
Mijλiλj = Dλλu(x)−Dλλu(y) ≤ V λ(x),
we arrive at the estimate
(4.13) |λ ∈ Λ : V λ(x) ≥ µω| ≥ 2µ · |Λ| for all x ∈ Bρ(z).
Step 4. Now we set
Γ = (λ, x) ∈ Λ×Bρ(z) : Vλ(x) ≥ µω ⊂ Λ×Bρ(z),
Γλ = x ∈ Bρ(z) : Vλ(x) ≥ µω for λ ∈ Λ,
Γx = λ ∈ Λ : V λ(x) ≥ µω for x ∈ Bρ(z),
(4.14) Λ0 = λ ∈ Λ : |Γλ| ≥ µ · |Bρ|.
By (4.13) and Fubini’s theorem, the product-measure on Λ×Bρ(z),
|Γ| =∫Bρ(z)
|Γx| dx ≥ 2µ · |Λ| · |Bρ|.
On the other hand, by definition of Λ0,∫Λ\Λ0
|Γλ| dsλ ≤ |Λ \ Λ0| · µ|Bρ| ≤ µ|Λ| · |Bρ|,
therefore,
|Λ0| · |Bρ| ≥∫Λ0
|Γλ| dsλ = |Γ| −∫Λ\Λ0
|Γλ| dsλ ≥ µ · |Λ| · |Bρ|,
so we obtain |Λ0| ≥ µ · |Λ|.Step 5. The relations (4.12) allow us to apply Lemma 4.1 to the functions
V (x) = (µω)−1V λ(x), λ ∈ Λ0. This gives us
oscB2ρ(z)
Dλλv − oscBρ(z)
Dλλv ≥ infBρ(z)
Dλλv − infB2ρ(z)
Dλλv = infBρ(z)
V λ ≥ βµ · ω(ρ)
for all λ ∈ Λ0. Taking into account (4.14) and using Lemma 4.3, we obtain:
ω∗(2ρ)−ω∗(ρ) =
∫Λ
[osc
B2ρ(z)Dλλv − osc
Bρ(z)Dλλv
]dsλ ≥ |Λ0|·βµ·ω(ρ) ≥ µ1·ω∗(ρ)
15
for some constant µ1 = µ1(n, ν) > 0.
Step 6. Taking α = α(n, ν) = log2(1 + µ1) > 0 and ρ0 = d(z), we see that
2αω∗(ρ) = (1 + µ1)ω∗(ρ) ≤ ω∗(2ρ) for all ρ ∈ (0, ρ0/2].
Using Lemma 4.2 with q = 2 and then again Lemma 4.3, we get
(4.15) ραω(ρ) ≤ N(n, ν)ρ−α0 ω(ρ0) for all ρ ∈ (0, ρ0].
This estimate implies
ρ2+α0 ρ−αω(ρ) ≤ Nρ20ω(ρ0) ≤ N · [v](0)2,0;Br
.
Taking the sup over z ∈ Br and ρ ∈ (0, ρ0] = (0, d(z)], and applying Theorem2.1, we get
[v](0)2,α ≤ N M
(0)2,α ≤ N [v]
(0)2,0.
Finally, from the interpolation inequalities (2.12) and Corollary 3.3 it follows
∥v∥(0)2,α ≤ N supΩ|v| ≤ N sup
∂Ω|φ|.
So we have proved the desired estimate (4.2).
Remark 4.1. Theorem 4.1 is the only point where the convexity condition(F0) is used. It is an open problem whether it remains valid without (F0). Inthe case n = 2, this is so and follows from well-known result of Nirenberg [21]on C1,α-smoothness of solutions of linear elliptic equations with measurablecoefficients. Therefore, in the case n = 2 all our results, including Theorems3.1–3.3, remain in effect without (F0). It was shown in [27] that the result ofNirenberg fails for n = 3, and hence the convexity condition (F0) is essentialin our considerations when n ≥ 3.
5. Interior C2,α− estimates: general equations
We will use the interior estimate (4.2) in C2,α;0 for solutions of simplest non-linear equations in the proof of estimates in C2,α;0, 0 < α < α for solutions ofgeneral nonlinear equations. In the case of linear equations, we can considerlinear equations L0v =
∑aijDijv = 0 with constant coefficients in place of
simplest nonlinear equations F0[v] = 0, so (4.2) is true with α = 1, and hencefor general linear equations with coefficients in Cα, we can take arbitraryα ∈ (0, 1).
Theorem 5.1. Let Ω be a bounded domain in Rn with d0 = diamΩ ≤ R0 =const <∞, and let F (x, u, ui, uij) satisfy Assumptions 3.1 with some constantsK,K1 ≥ 0, ν ∈ (0, 1], α ∈ (0, α), where α = α(n, ν) is the constant in Theorem4.1. Suppose u ∈ C2,α;0(Ω) ∩ C(Ω) is a solution of the equation F [u] = 0 inΩ. Then the estimate (3.6) holds.
16
Proof. Throughout the proof we will denote by N different constants de-pending only on n, ν,K, α,R0. We will also use the brief notations
U2,α = [u](0)2,α, Uk = [u]
(0)k,0.
Step 1. Let us fix
y ∈ Ω, d = d(y) =1
2dist(y, ∂Ω), ρ ∈ (0, d], and ε ∈ (0, 1/2].
We set r = ρ/ε and consider separately the cases (a) r ≤ d and (b) r > d.In the case (a), we take
We have considered the case (a) r = ρ/ε ≤ d. In the case (b) r = ρ/ε > d,we have d2+αρ−2−α < ε−2−α, and E2[u;Bρ] ≤ supBd
|u| ≤ U0, so the left handside of (5.5) does not exceed ε−2−αU0. Since y ∈ Ω and 0 < ρ ≤ d = d(y) arechosen in an arbitrary manner, we get the following estimate for the seminormin Theorem 2.1:
for all ε > 0. By this theorem, (5.6) remains valid with U2,α in place of M(0)2,α.
Choosing then ε = ε(n, ν,K, α,R0) > 0 such that the coefficient of U2,α wouldbe less then 1/2, we get
(5.7) U2,α ≤ N · (U2 + U1 + U0 + d2+α0 K1).
Finally, from (5.7) and the interpolation inequalities (2.12),
U2 + U1 ≤ εU2,α +N(ε)U0, ε > 0,
it follows
∥u∥(0)2,α = U2,α + U2 + U1 + U0 ≤ N · (U0 + d2+α0 K1),
completing the proof of theorem.
18
Remark 5.1. We relied on Theorem 4.1, though its proof given in Section4 was not quite complete because we assumed the solvability of the problem(4.1). This gap can be removed with the help of some variant of the methodof continuation with respect to the parameter ν. By virtue of (3.12), in thecase ν = 1 the class F(ν) = F(1) consists of the only function tr[uij] =
∑uii
corresponding to the Laplace operator ∆, so the standard results of the lineartheory of elliptic equations yield all the statements of Theorem 4.1 with α = 1.
Setting out from ν ′ = 1, F ′(uij) = tr[uij], and using (3.12), we notice thatthe functions F0(uij) ∈ F(ν), 0 < ν < 1, satisfy the estimate
(5.8) |F ′0(uij)− F0(uij)| ≤ γ ·max
i,j|uij|,
where the constant γ = γ(n, ν) → 0 as ν → 0. In the proof of Theorem 5.1with ν close to ν ′ = 1, one can consider the solution v of the problem
F ′0[v] = F0(Dijv) = 0 in Br = Br(y), v = φ on ∂Br
instead of the problem (5.1). Then the estimate (5.2) remains valid. By virtueof (5.8) we have
supBr
|F ′0[φ]− F0[φ]| ≤ γ[φ]2,0;Br ≤ γrα[φ]2,α;Br ,
A′ = r−α supBr
|F ′0[φ]| ≤ γ[φ]2,α;Br + A.
The last estimate together with (5.3) imply
d2+αA′ ≤ γU2,α +N · (U2 + U1 + U0 + d2+α0 K1).
Hence in the right hand sides of (5.4)–(5.6) we will have an additional termNε−2−αγU2,α. If ν is close enough to 1, then γ is small and the coefficient ofU2,α in (5.6) still can be made less then 1/2.
Thus, starting from ν ′ = 1, we see that Theorem 5.1 remains true for someν = ν1 < 1. As was pointed out in Section 3, Theorem 3.1 can be obtained onthe grounds of the estimates provided by Corollary 3.2 and Theorem 5.1. Inturn, Theorem 4.1 is a special case of Theorem 3.1 with Ω = Br and F = F0.So, all Theorems 3.1, 4.1, and 5.1 are true for ν = ν1.
Further moving past ν ′ = ν1, we can approximate F0(uij) ∈ F(ν) by thefunctions
F ′0(uij) = θδij + (1− θ)F0(uij), 0 < θ < 1.
Moreover, if ν < ν1 is close to ν1, then for small θ > 0 we have F ′0(uij) ∈
F(ν1) and the constant γ in (5.8) will also be small. Hence all the previousconsiderations are valid for some ν = ν2 < ν1. Continuing this procedure, wecan embrace the arbitrary ν ∈ (0, 1].
19
6. Some boundary estimates for solutions of linear elliptic equations
We will essentially use the following Lemma 6.1 announced in [23] (see also[25]). For applications to nonlinear equations, the same results can be ob-tained by Krylov’s method (see [12] and [13], comments to §1 of Ch.5), whichimplies the consideration of auxiliary degenerate elliptic or parabolic equationfor V (x)/xn.
Lemma 6.1. Let ν ∈ (0, 1], x0 ∈ Rn0 , r > 0, and B+
r = Rn+∩Br(x0). Suppose
that V ∈ C2(B+r ) ∩ C(B+
r ) be a solution of the equation
(6.1) aijDijV = 0 in B+r ,
where aij = aij(x) satisfy the conditions (4.3), and moreover,
(6.2) V = 0 on Γ = Rn0 ∩Br(x0).
Then the function ω(ρ) = oscB+
ρ
V (x)/xn satisfies the estimate
(6.3) ρ−αω(ρ) ≤ Nr−αω(r) for all ρ ∈ (0, r],
where the constants α ∈ (0, 1], N > 0 depend only on n, ν.
Proof. By Lemma 4.2, for the proof of (6.3) it is sufficient to obtain, forexample, the estimate
(6.4) 10αω(ρ) ≤ ω(10ρ) for ρ ∈ (0, r/10].
Using the transformation x −→ ρ−1 · (x− x0), we can consider only the casex0 = 0, ρ = 1. In addition, replacing, if necessary, V (x) by one of the functionsλxn ± V (x), λ = const, we assume that
(6.5) 0 ≤ x−1n · V (x) ≤ ω = ω(10) in B+
10,
and moreover,
(6.6) |x ∈ B2(x∗) : x−1
n · V (x) ≥ ω/2| ≥ |B2|/2,
where x∗ = (0, · · · , 0, 4). Since V ≥ 0 and aijDijV = 0 in B4(x∗) ⊂ B+
10,and by virtue of (6.6),
|x ∈ B2(x∗) : V (x) ≥ ω| ≥ |B2|/2,
from Lemma 4.1 it follows that V (x) ≥ βω on B2(x∗), where β = β(n, ν) > 0.
where the constant γ = γ(n, ν) > 0 is so large that aijDijw(x) ≥ 0 for
all x = y∗. Moreover, we have V (x) ≥ 0 = w(x) on ∂B4(y∗) ⊂ B+
10, and
20
V (x) ≥ βω ≥ w(x) on ∂B1(y∗) ⊂ B2(x
∗). Consequently, by the classicalmaximum principle, V (x) ≥ w(x) on B4(y
∗) \B1(y∗). In particular,
V (y) ≥ w(y) = ((4− yn)−γ − 4−γ)βω ≥ β1ω · yn,
where β1 = β1(n, ν) > 0. Since y ∈ B1 can be selected in an arbitrary manner,we get V (x)/xn ≥ β1ω for all x ∈ B+
1 . This estimate together with (6.5)yield ω(1) ≤ (1 − β1) · ω(10). Taking α = α(n, ν) = − log10(1 − β1) > 0, weobtain the desired inequality (6.3).
Remark 6.1. By standard barrier technique, one can show that if (6.1), (6.2)are valid for B+
2r in place of B+r , then
supB+
r
|V (x)/xn| ≤ N(n, ν) · r−1 supB+
2r
|V |.
Therefore, in this case we have
ρ−αω(ρ) ≤ Nr−1−α supB+
2r
|V | for all ρ ∈ (0, r].
We observe that since V (x0) = 0, the estimate (6.3) yields the existence ofthe derivative
DnV (x0) = limρ→0+
ρ−1 · V (x0 + ρen).
Lemma 6.1 can be applied to each point y0 ∈ Γ in place of x0, hence thereexists DnV on Γ and moreover, the following assertions hold.
Corollary 6.1. Under the assumptions of Lemma 6.1, we have
(6.7) [DnV ]α;B0r/2≤ Nr−αω(r),
where B0r/2 = Br/2(x0) ∩Rn
0 . In addition, if DnV ∈ C(B+r ), then
(6.8) [DnV ]α;B0r/2≤ Nr−αosc
B+r
DnV.
Proof. Applying (6.3) to different half-balls B+ρ (y0)), we get (6.7). Further,
if DnV ∈ C(B+r ), then
V (x)/xn = x−1n
∫ xn
0
DnV (x1, · · · , xn−1, t) dt,
Therefore,
(6.9) ω(r) = oscB+
r
V (x)/xn ≤ oscB+
r
DnV,
that yields (6.8).
Corollary 6.2. Let the assumptions of Lemma 6.1 be satisfied and supposethat DnV ∈ C(B+
r ). Then
ρ−1−αE1[V ;B+ρ ] ≤ Nr−αosc
B+r
DnV for all ρ ∈ (0, r].
21
Proof. Since ρ/xn > 1 in B+ρ , from Lemma 6.1 applied to the functions
V (x)− λxn, λ ∈ R1, we obtain:
ρ−1−αE1[V ;B+ρ ] ≤ ρ−1−α inf
λ∈R1supB+
ρ
|V (x)− λxn|
≤ ρ−α infλ∈R1
supB+
ρ
|V (x)/xn − λ| =1
2ρ−αω(ρ) ≤ Nr−αω(r).
By virtue of (6.9), the desired estimate holds.
7. Boundary C2,α− estimates: the Dirichlet problem
In order to obtain C2,α-estimates of solutions near the boundary, with certainboundary conditions, we need appropriate extensions of Theorem 4.1. Thefollowing result of N.V. Krylov [12] can be treated as such an extension in thecase of the Dirichlet boundary condition.
Theorem 7.1. Let ν ∈ (0, 1], x0 ∈ Rn0 , r > 0, B+
r = B+r (x0), φ ∈ C(B+
r ),φ = 0 on Γ = Rn
0 ∩ Br(x0) , and the function F0(uij) ∈ F(ν). Then theproblem
(7.1) F0[v] = F0(Dijv) = 0 in B+r , v = φ on ∂B+
r
has a unique solution v ∈ C2,α;0(B+r ∪ Γ) ∩ C(B+
r ) , and
(7.2) ∥v∥(0)2,α;B+
r ∪Γ ≤ N · sup∂B+
r
|φ|,
where the constants α ∈ (0, 1], N > 0 depend only on n, ν.
Our proof of this theorem is different from [12] and is based on Lemma 6.1.We precede it with the following auxiliary statement.
Lemma 7.1. In addition to the assumption of Theorem 7.1, let the functionsF0 = F0(uij) and v = v(x) in (7.1) be smooth. Then for any ε > 0 and0 < ρ ≤ r, we have
(7.3) ρ−αω(x0, ρ) ≤ ε[v]2,α;B+r+N(ε, n, ν)r−αω(x0, r),
where
(7.4) ω(x, ρ) = maxi,j
oscB+
ρ (x)Dijv,
B+r = B+
r (x0), and α = α(n, ν) ∈ (0, 1] is the constant in Lemma 6.1.
Proof. For k = 1, 2, · · · , n−1, set V k = Dkv. Since v = φ = 0 on Γ ⊂ Rn0 ,
we have also V k = 0 on Γ. Further, differentiating the equality F0[v] = 0with respect to xk, we obtain: aijDijV
k = 0 in B+r , where aij = ∂F0/∂uij.
22
Applying to V k first Lemma 2.3 with k = 1, and then Corollary 6.2, we findthat
ρ−αoscB+
ρ
Dikv = ρ−αoscB+
ρ
DiVk
≤ ε[V k]1,α;B+ρ+N(ε)ρ−1−αE1[V
k;B+ρ ]
≤ ε[v]2,α;B+r+N(ε)r−αω(x0, r)
(7.5)
for all ε > 0, 0 < ρ ≤ r, and i = 1, 2, · · · , n.Since k ≤ n− 1, on the left hand side of (7.5) there can appear any second
derivative of v except Dnnv. Further, by virtue of (3.11), for each x, y ∈ B+ρ
we haveaij · (Dijv(x)−Dijv(y)) = 0
with aij satisfying (3.2). Since ann ≥ ν > 0, we get
oscB+
ρ
Dnnv ≤ N(n, ν)∑
i+k≤2n−1
oscB+
ρ
Dikv.
From the last relation and (7.5), after redefining ε, the desired estimate follows.
Proof of Theorem 7.1. Let α = α(n, ν) ∈ (0, 1] be the smaller of theconstants α in Theorem 4.1 and Lemma 6.1. As in the proof of Theorem 4.1, wewill assume without loss of generality that F0(uij) is smooth and the problem(7.1) has a solution v ∈ C2,α;0(B+
r ∪ Γ) ∩ C(B+r ). The last assumption can
be substantiated by the method of continuation with respect to the parameterν, which is outlined in Remark 5.1. Notice that from the smoothness of F0 itfollows v ∈ C3(B+
r ∪ Γ) (see [9], Sec. 17.8).We first prove that ω(x, ρ) in (7.4) satisfies
(7.6) d2+α(x)ρ−αω(x, ρ) ≤ εV2,α +N(ε)V2
for all x ∈ B+r ∪ Γ, ρ ∈ (0, d(x)], and ε > 0, where in accordance with (2.9),
d(x) =1
2dist(x, ∂Ω \ Γ), V2,α = [v]
(0)
2,α;B+r ∪Γ, V2 = [v]
(0)
2,0;B+r ∪Γ.
We will divide the proof of (7.6) into several cases.(a) x ∈ Γ, 0 < ρ ≤ d(x). In this case (7.6) follows immediately from
Lemma 7.1 with x0 = x, r = d(x).(b) d(x)/4 ≤ ρ ≤ d(x). Since dα(x)ρ−α < 4α < 4, the left hand side of
(7.6) does not exceed 4d2(x)ω(x, d(x)) ≤ 8V2, so this estimate is true evenwith ε = 0.
(c) xn ≤ ρ ≤ d(x)/4, where x = (x′, xn), x′ ∈ Γ ⊂ Rn
0 . In this casewe have B+
ρ (x) ⊂ B+2ρ(x
′). Moreover, (7.6) is valid for x = x′, hence 2ρ <d(x)/2 < d(x′), and in view of (a) we get
(d) 0 < ρ < ρ0 = min(d(x)/4, xn). First we apply the estimate (4.15); thisgives us ραω(x, ρ) ≤ Nρ−α
0 ω(x, ρ0), and then (7.6) follows from (b) or (c),depending whether ρ0 = d(x)/4 or ρ0 = xn < d(x)/4.
We have proved (7.6). By virtue of (2.19), we get
N−11 V2,α ≤ M
(0)2,α[v;B
+r ∪ Γ] ≤ εV2,α +N(ε)V2.
Setting ε = (2N1)−1, we get V2,α ≤ NV2. Finally, using the interpolation
inequalities (2.12), and then Corollary 3.3, we obtain
∥v∥(0)2,α;B+
r ∪Γ ≤ N · supB+
r
|v| = N · sup∂B+
r
|φ|,
so the desired inequality (7.2) is true with α = α = α(n, ν) ∈ (0, 1].
Theorem 7.2. Let F (x, u, ui, uij) satisfy Assumptions 3.1 with Ω = B+r0(x0)
and some constants K,K1 ≥ 0, ν ∈ (0, 1], α ∈ (0, α), where x0 ∈ Rn0 ,
r0 ∈ (0, 1], and α = α(n, ν) is the constant in Theorem 7.1. Let u0 ∈ C2,α(Γ),where Γ = Rn
0 ∩ Br0(x0) is identified with a ball in Rn0 = Rn−1. Then for any
function u ∈ C2,α;0(B+ ∪ Γ), satisfying the equalities
(7.7) F [u] = 0 in B+, u = u0 on Γ,
we have
(7.8) ∥u∥(0)2,α;B+∪Γ ≤ N(n, ν,K, α) ·[supB+
|u|+ r2+α0 (K1 + |u0|2,α;Γ)
].
Proof. Setting
u = u− u0, F (x, u, ui, uij) = F (x, u+ u0(x), ui +Diu0(x), uij +Diju0(x)),
one can see that the equalities (7.7) are equivalent to
F [u] = 0 in B+, u = 0 on Γ.
Moreover, F satisfies Assumptions 3.1 with a new constant K1 = N · (K1 +|u0|2,α;Γ) in place of K1. Thus the proof of (7.8) is reduced to the case u0 = 0.
As in the proof of Theorem 5.1, we introduce the notations
U2,α = [u](0)
2,α;B+∪Γ, Uk = [u](0)
k,0;B+∪Γ,
and then we fix
y = (y′, yn) ∈ B+ ∪ Γ, d = d(y) =1
2dist(y, ∂B+ \ Γ), ρ ∈ (0, d], ε ∈ (0, 1/2].
Then in the cases (a) ρ/ε ≤ minyn, d/8 and (b) ρ/ε > d/8, quite analo-gously to (5.5), we obtain the estimate(7.9)
In the remained case (c) yn < ρ/ε ≤ d/8, we take d′ = 12disty′, ∂B+ \Γ,
r = 4ρ/ε, φ = u − Ty′,2u, u = u(y′), ui = Diu(y′), uij = Diju(y
′), andF0(uij) = F (y′, u, ui, uij + uij) ∈ F(ν). It is easy to see that
B+ρ (y) ⊂ B+
r/2(y′), r ≤ d/2 ≤ d′.
Moreover, since u = 0 on Γ and y ∈ Γ, also φ = 0 on Γ. Define v as thesolution of the problem (7.1) in B+
r = B+r (y
′). By analogy to (5.2), relying onTheorem 7.1 in place of Theorem 4.1, we have
d2+αρ−2−αE2[v;B+ρ (y)] ≤ Nεα−αU2,α.
The other points of the proof of Theorem 5.1, yielding the estimate (7.9)in the case (c) and the desired estimate (7.8), are valid with minimal modifi-cations.
Remark 7.1. Theorem 7.2 together with Theorem 3.1 yield Theorem 3.2.Indeed, upon dividing ∂Ω into a finite number of small subsets and “flatteningof the boundary”, Theorem 7.2 gives C2,α- estimates near ∂Ω for solutions ofthe problem (3.4). These estimates and interior C2,α-estimate (3.6) constitutethe estimate (3.7) in Theorem 3.2.
8. Boundary C2,α− estimates: the oblique derivative problem
The formulation of the following Theorems 8.1 and 8.2 are similar to ones ofTheorems 7.1 and 7.2, only instead of the Dirichlet condition on Γ we nowhave Dnu = 0 on Γ.
Theorem 8.1. Let ν ∈ (0, 1], x0 ∈ Rn0 , r > 0, B+
r = B+r (x0), φ ∈ C(B+
r ) begiven, and the function F0(uij) ∈ F(ν). Then the equation
(8.1) F0[v] = F0(Dijv) = 0 in B+r
with the boundary conditions
(8.2) Dnv = 0 on Γ = Rn0 ∩Br(x0), v = φ on ∂B+
r \ Γ,
has a unique solution v ∈ C2,α;0(B+r ∪ Γ) ∩ C(B+
r ) , and
(8.3) ∥v∥(0)2,α;B+
r ∪Γ ≤ N · sup∂B+
r
|φ|,
where the constants α ∈ (0, 1], N > 0 depend only on n, ν.
Proof. We choose α ∈ (0, α) (for example, α = α/2), where α = α(n, ν) ∈(0, 1] be the smaller of the constants α in Theorem 4.1 and Lemma 6.1. Undersuch choice of α, we will prove (8.3).
Step 1. All the reasonings concerning the existence of the solution of theproblem (8.1), (8.2) are quite similar to ones in the previous section relatedto the problem (7.1). Therefore, we will assume the existence of solution v,
25
and moreover, we will consider smooth F0(uij), so that v ∈ C3(B+r ∪ Γ). By
Corollary 3.1, using the equalities (8.2), we get
(8.4) V0 = supB+
r
|v| = sup∂B+
r \Γ|v| ≤ sup
∂B+r
|φ|.
Following the lines of the proof of Theorem 7.1, we notice that it sufficesto prove the estimate
(8.5) d2+α(y)ρ−αω(y, ρ) ≤ N · (V2 + V0)
for all y ∈ B+r ∪ Γ, 0 < ρ ≤ d(y) = 1
2dist(y, ∂Ω \ Γ). This estimate is similar
to (7.6), and the cases (b)–(d)) of its proof remain valid, so we will consideronly the case (a) y ∈ Γ.
Step 2. Let us fix y ∈ Γ and d = d(y). Differentiating the equality (8.1)with respect to xn, we obtain aijDijDnv = 0 in B+
r , where aij = ∂F0/∂uij.Since Dnv = 0 on Γ, we can apply Corollary 6.1 to the function V = Dnv inB+
d (y) ⊂ B+r , that gives us the estimate
(8.6) [Dnnv]α;B0d/2
(y) ≤ Nd−α oscB+
d (y)Dnnv ≤ Nd−2−αV2.
Besides this, we have Dinv = 0 on Γ for i = 1, · · · , n − 1. Therefore,setting Ω = B0
d/2(y) ⊂ Rn0 = Rn−1, we see that the function v0(x
0 ∩Br0(x0), and let F (x, u, ui, uij) satisfy Assumptions 3.1 with Ω = B+ and some
26
constants K,K1 ≥ 0, α ∈ (0, 1). Then for any function u ∈ C2,α;0(B+ ∪ Γ)satisfying the equalities
(8.8) F [u] = 0 in B+, Dnu = 0 on Γ,
we have
(8.9) ∥u∥(0)2,α;B+∪Γ ≤ N ·(supB+
|u|+ r2+α0 K1
),
where N = N(n, ν,K, α), provided 0 < α < α for some constant α =α(n, ν) ∈ (0, 1).
Proof. We take as α = α(n, ν) ∈ (0, 1) the constant α in Theorem 8.1.Then we can reproduce almost literary the proof of Theorem 7.2, only in thecase (c) we define v as the solution of the problem (8.1), (8.2), and accordingly,we rely on Theorem 8.1 instead of Theorem 7.1
Repeating the reasonings in Step 2 of the proof of Theorem 5.1, we nowhave
with aij satisfying (3.2), and the function v(x) together with
φ(x) = u(x)− Ty′,2u(x), w(x) =Arα
2nν(r2 − |x− y′|2)
satisfy the relations
|L0(φ− v)| ≤ rαA = rα supB+
r (y′)
|F0[φ]| ≤ −L0w in B+r (y
′),
Dn(φ− v)| = Dnw = 0 on Γ ∩Br(y′), φ− v = w = 0 on ∂B+
r (y′) \ Γ.
Therefore, we can apply the comparison principle yielding
supB+
ρ (y′)
|φ− v| ≤ supB+
r (y′)
|φ− v| ≤ supB+
r (y′)
|w| = A
2nνr2+α.
The remained part of the proof is almost the same as in the proof ofTheorem 5.1 , so we obtain the estimate (8.9).
We will use the estimate (8.9) in the proof of the C2,α-estimate (3.10) inthe formulation of the Theorem 3.3. However, “flattening of the boundary”∂Ω ∈ C1,α in the general case would deteriorate C2,α-smoothness of solutions.Therefore, we first consider a special case of the boundary conditions which isreduced to (8.8). For this purpose, we need some auxiliary results concerningthe extension of functions from ∂Ω to Ω. The following lemma is contained in[8], Lemma 2.3.
27
Lemma 8.1. Let r > 0, B+r = Rn
+ ∩ Br(0), B0r = Rn
0 ∩ Br(0), and Φ0 ∈C1,α(B0
r ), 0 < α < 1. Then there exists a function Φ ∈ C∞(B+r ) ∩ C1,α(B+
r )such that Φ = Φ0 on B0
r ,
(8.10) |Φ|1,α;B+r≤ N(n, α) · |Φ0|1,α;B0
r,
and for any k = 2, 3, · · · , we have
(8.11) max|l|=k
supB+
r
yk−1−αn |DlΦ(y)| ≤ N(k, n, α) · [Φ0]1,α;B0
r.
Let us fix an arbitrary point x0 ∈ ∂Ω ∈ C1,α. Using Definition 3.1 and theimplicit theorem, we cam choose an orthonormal coordinate system centeredat x0 and r0 ∈ (0, 1] such that Ωr0(x0) = Ω ∩ Br0(x0) is represented in theform (2.23) with ψ0 ∈ C1,α(B0
r0), where B0
r0is the projection of Br0(x0) onto
Rn0 .Without loss of generality we now take x0 = 0. Applying Lemma 8.1
with Φ0 = ψ0, 0 < r ≤ r, we obtain the existence of a smooth functionψ ∈ C∞(B+
r ) ∩ C1,α(B+r ) such that ψ = ψ0 on B0
r , and (8.10), (8.11) are truefor Φ = ψ,Φ0 = ψ0. Moreover, replacing ψ(y) by ψ(y) + Nyn if necessary,we may assume that Dnψ ≥ 1 on B+
r . Now we introduce the new coordinatesx = x(y) ∈ C1,α(B+
r ) by the mapping
x′ = y′, xn = ψ(y), y ∈ B+r .
We have det ∂x/∂y = Dnψ ≥ 1, therefore, the inverse mapping y = y(x),where
y′ = x′, yn = η(x), x ∈ Ωr = x(B+r ),
has the same smoothness as x = x(y). It is easy to see that
N−1yn ≤ d(x) =1
2dist(x, ∂Ω) ≤ Nyn = Nη(x), x ∈ Ωr,
with N = N(Ω). Therefore, for yn = η(x) we have
(8.12) max|l|=k|Dlη(x)| ≤ N(k, α,Ω) · y1+α−k
n , x ∈ Ωr, k = 2, 3, · · · .
Using all these properties, one can obtain the following lemma as a conse-quence of Lemma 8.1.
Lemma 8.2. Let Ω be a bounded domain in Rn with ∂Ω ∈ C1,α, 0 < α < 1.Under the previous assumptions, let r ∈ (0, r] be chosen small enough, so thatΩr = x(B+
r ) ⊂ Ω, and let a function ϕ0 ∈ C1,α(Ωr) be given. Then thereexists a function ϕ ∈ C∞(Ωr)∩C1,α(Ωr) such that ϕ = ϕ0 on γr = ∂Ω∩ ∂Ωr,
(8.13) |ϕ|1,α;Ωr ≤ N(α,Ω) · |ϕ0|1,α;Ωr ,
and for any k = 2, 3, · · · , we have
(8.14) max|l|=k
supΩr
dk−1−α(x)|Dlϕ(x)| ≤ N(k, α,Ω) · [ϕ0]1,α;Ωr .
28
Notice that in this construction, we can take r > 0 independent on x0 ∈ ∂Ω.Using then the standard partition of unity (see [9], Sec. 6.9), we arrive at thefollowing statement.
Corollary 8.1. If ϕ0 ∈ C1,α(Ω), then there exists a function ϕ ∈ C∞(Ω) ∩C1,α(Ω) such that ϕ = ϕ0 on ∂Ω,
(8.15) |ϕ|1,α;Ω ≤ N(α,Ω)|ϕ0|1,α;Ω,
and for any k = 2, 3, · · · , we have
(8.16) [ϕ](−1−α)k,o;Ω ≤ N(k, α,Ω)[ϕ0]1,α;Ω.
The following two lemmas serve as the intermediate steps in the proof of theestimate (3.10). As before, we fix x0 ∈ ∂Ω ∈ C1,α, 0 < α < 1, and a suitableC1,α-mapping x = x(y), so that some portion of Ω near x0 is represented inthe form Ωr = x(B+
r ), 0 < r ≤ r0, and γr = ∂Ω ∩ ∂Ωr = x(Γ), Γ = B0r .
Lemma 8.3. Let F (x, u, ui, uij) satisfy Assumptions 3.1 with some constantsK,K1 ≥ 0, 0 < α < 1. Then for any function u ∈ C2,α;0(Ωr ∪ γr) satisfyingthe equalities
(8.17) F0[u] = 0 in Ωr, Dnu = 0 on γr = ∂Ω ∩ ∂Ωr,
we have
(8.18) ∥u∥(0)2,α;Ωr∪γr ≤ N(n, ν,K, α,Ω) ·(supΩr
|u|+ r2+αK∗1
),
where K∗1 = K1 + |u|2,0;Ωr , provided 0 < α < α for some constant α =
α(n, ν,Ω) ∈ (0, 1).
Proof. Under the C1,α-diffeomorphism
(8.19) x ∈ Ωr ←→ y ∈ B+r , where x′ = y′, xn = ψ(y), yn = η(x),
(8.25) |Dlu0(x)| ≤ Ndα−1(x)G0 for all |l| = 3, x ∈ Qr.
If |l| = 3, ln > 0, then Dlu0(x) = DijDnu0(x) = Dijg1(x) for some i, j, and(8.25) gives us (8.22). If |l| = 3, ln = 0, then (8.24), (8.22) yield
|Dlu0(x)| =∣∣∣∣∫ h
xn
Dlg1(x′, t) dt
∣∣∣∣ ≤ NG0
∫ h
xn
dα−2(x′, t) dt,
and (8.25) follows from (8.23). Finally, applying Lemma 2.4 and using (8.25),we obtain:
[u0]2,α = maxi,j
[Diju0]α ≤ N maxi,j
[Diju0]−α1,0 ≤ NG0.
Moreover, Dnu0 = g1 = g0 on γr. By setting u = u − u0, as in the proofof Theorem 7.2, this lemma is reduced to Lemma 8.3.
Theorem 8.3. Under the assumptions of Theorem 3.3, let u ∈ C2,α(Ω) be asolution of the problem (3.9). Then the estimate (3.10) holds.
Proof. Let us fix x0 ∈ ∂Ω. In the previous construction, we can choosean orthonormal coordinate system with b(x0) = (b1(x0), · · · , bn(x0)) directedalong the positive xn- axis. By virtue of (3.8), we can impose the restrictionN−1 ≤ det ∂x/∂y ≤ N , where N = N(n, ν0), on the C1,α-diffeomorphism(8.19). Therefore, the constant α in Lemma 8.3 depends only on n, ν, ν0.
Dividing both sides of the condition biDiu + b0u = g by bn(x0) > 0, wecan reduce it to the case bi(x0) = δin. Next, we rewrite it in the form
Dnu = g0 = g − (bi − δin)Diu− b0u on γr = ∂Ω ∩ ∂Ωr.
Since |bi(x)− δin| = |bi(x)− bi(x0)| ≤ Nrα in Ωr, by virtue of (2.6) we get
By definition of [u]2,α, we can choose x, y ∈ Ω, and i, j, such that
(8.29) U2,α ≤ 2 |Diju(x)−Diju(y)|/|x− y|α.
We consider separately three cases.(a) |x − y| < δr/3, d(x) = dist(x, ∂Ω) < δr/3. In this case for some
x0 ∈ ∂Ω we have |x − x0| = 2d(x) < 2δr/3, hence x, y ∈ Ω ∩ Bδr(x0) = ωr.Now let us fix r > 0 such that N0r
α < 1/4 in (8.27). Since the right handside in (8.29) does not exceed [u]2,α;ωr , from (8.27) we obtain (8.28).
(b) |x − y| < δr/3, d(x) ≥ δr/3. We have y ∈ B(x) = Bd(x)(x), hence(8.28) follows from the interior estimate (3.6).
(c) |x− y| ≥ δr/3. Directly from (8.29) it follows U2,α ≤ N |u|2,0.We have proved (8.28). Finally, using the interpolation inequalities which
are true even for Lipschitz domains (see [20], Ch.5, Sec.33), we can replace|u|2,0 by U0 = supΩ |u| in (8.28), so that (8.28) turns into (3.10).
Remark 8.1. Under the assumptions of Theorem 3.3, U0 = supΩ |u| is easyestimated by the comparison principle (see [9] the proof of Theorem 6.31).Hence we have a priori estimates of solutions to the oblique derivative problem(3.9) in C2,α(Ω), depending only on the prescribed constants. On the groundsof these C2,α-estimates, the solvability of the problem (3.9) can be stated bymeans of the satndard continuity method (see [9], Sec. 17.9).
References
1. Anulova, S.V., Safonov, M.V.: Control of diffusion processes with the reflec-tion on the boundary. In: Statistics and Controlled Stochastic Processes,Steklov Seminars 1985-86, v.2, pp. 1–15. New York, Optim. Software Inc.,1989
2. Brezis, H., Evans, L.C.: A variational inequality approach to the Bellman-Dirichlet equation for two elliptic operators. Arch. Rational Mech. Anal.71, 1–13 (1979)
3. Caffarelli, L.A.: Interior a priori estimates for solutions of fully nonlinearequations. Ann. of Math. 130, 189–213 (1989)
4. Caffarelli, L.A., Nirenberg, L., Spruck.: The Dirichlet problem for nonlinearsecond order elliptic equations. 1. Monge-Ampere equaiton. Comm. PureAppl. Math. 38, 209–252 (1985)
32
5. Caffarelli, L.A., Kohn, J.J., Nirenberg, L., Spruck.: The Dirichlet problemfor nonlinear second order elliptic equations. 2. Complex Monge-Ampere,and uniformly elliptic, equaitons. Comm. Pure Appl. Math. 37, 369–402(1984)
6. Campanato, S.: Proprieta di una famiglia di spazi funzionali. Ann. ScuolaNorm. Sup. Pisa (3) 18, 137–160 (1964)
7. Evans, L.C.: Classical solutions of fuly nonlinear, convex, second-orderelliptic equations. Comm. Pure Appl. Math. 35, 333–363 (1982)
9. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of sec-ond Order. Berlin-Heidelberg-New York-Tokyo: Springer 1983 (second ed.)
10. Krylov, N.V.: Controlled Diffusion Processes. Moscow: Nauka 1977 inRussian; English transl.: Berlin-Heidelberg-New York: Springer 1980
11. Krylov, N.V.: Boundedly nonhomogeneous elliptic and parabolic equa-tions. Izv. Akad. Nauk SSSR, Ser. Mat. 46, 487– 523 (1982) in Russian;English transl. in: Math. USSR Izv. 20, 459–492 (1983)
12. Krylov, N.V.: Boundedly nonhomogeneous elliptic and parabolic equationsin a domain. Izv. Akad. Nauk SSSR, Ser. Mat. 47, 75–108 (1983) in Russian;English transl. in: Math. USSR Izv. 22, 67–97 (1984)
13. Krylov, N.V.: Nonlinear Elliptic and Parabolic Equations of Second Order.Moscow: Nauka 1985 in Russian; English transl.: Dordrecht: Reidel 1987
14. Krylov, N.V., Safonov, M.V.: A certain property of solutions of parabolicequations with measurable coefficients. Izv. Akad. Nauk SSSR, Ser. Mat.44, 161–175 (1980) in Russian; English transl. in: Math. USSR Izv. 16,151–164 (1981)
16. Lieberman, G.M., Trudinger, N.S.: Nonlinear oblique boundary value prob-lem for nonlinear elliptic equations. Trans. Amer. Math. Soc. 295, 509–546(1986)
17. Lions, P.L., Trudinger, N.S.: Linear oblique derivative problem for theuniformly elliptic Hamiltom-Jacobi-Bellman equation. Math. Zeit. 191, 1–15 (1986)
18. Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear andQuasilinear Equations of Parabolic Type. Moscow: Nauka 1967 in Rus-sian; English transl.: Amer. Math. Soc., Providence, R.I. 1968
33
19. Ladyzhenskaya, Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equa-tions. Moscow: Nauka 1973 (second ed.) in Russian; English transl. of firsted.: Academic Press 1968
21. Nirenberg, L.: On nonlinear elliptic partial difefrential equations andHolder continuity. Comm. Pure Appl. Math. 6 103–156, 395 (1953)
22. Safonov, M.V.: Harnack inequality for elliptic equations and the Holderproperty of their solutions. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst.Steklov (LOMI) 96, 272–287 (1980) in Russian; English translation in: J.Soviet Math. 21, 851–863 (1983).
23. Safonov, M.V.: Boundary C2+α-estimates for solutions of nonlinear equa-tions. Uspechi Mat. Nauk 38, 146–147 (1983) in Russian
24. Safonov, M.V.: On the classical solution of Bellman’s elliptic equations.278, 810–813 (1984) in Russian; English translation in: Soviet Math. Dokl.30, 482–485 (1984)
25. Safonov, M.V.: On smoothness near the boundary of solutions of ellipticBellman equations. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov(LOMI) 147, 151–154 (1985) in Russian; English translation in: J. SovietMath. 37, 885–888 (1987).
26. Safonov, M.V.: On the classical solution of nonlinear elliptic equation ofsecond order. Izv. Akad. Nauk SSSR, Ser. Mat. 52, 1272–1287 (1988) inRussian; English transl. in: Math. USSR Izv. 33, 597–612 (1989)
27. Safonov, M.V.: Unimprovability of estimates of Holder constants for so-lutions of linear elliptic equations with measurable coefficients. Mat. Sb.132, 272–288 (1987) in Russian; English transl. in: Math. USSR Sb. 60,269–281 (1988)
28. Trudinger, N.S.: Lectures on nonlinear second order elliptic equations.Nankai Institute of Mathematics. Tianjin, China 1985
29. Wang, L.: On the regularity theory of fully nonlinear parabolic equations.1–3. Comm. Pure Appl. Math. 45, 27–76, 141–178, 255–262 (1992)
