Gradientbounds fornonlinear

strictly ellipticequations

with coerciveHamiltonians

Olivier Ley

Feb 2016

Gradient bounds for nonlinear strictlyelliptic equations with coerciveHamiltonians

Olivier Ley

IRMAR, INSA de Rennes, Francehttp://ley.perso.math.cnrs.fr

Collaboration with Vinh Duc Nguyen (Cardiff)

Nonlinear PDEs : Optimal Control, Asymptotic Problemsand Mean Field GamesOn the occasion of Martino Bardi’s 60th birthdayPadova, February 25-26, 2016

1/13

Gradientbounds fornonlinear

strictly ellipticequations

with coerciveHamiltonians

Olivier Ley

Feb 2016

Statement of the problem

(HJ) εφε − Tr(A(x)D2φε) + H(x ,Dφε) = 0

• x ∈ TN periodic setting

• ε > 0

• A(x) ≥ νI , ν > 0 strict ellipticity

• A(x) = σ(x)σ(x)T , σ ∈W 1,∞(TN)

• Assume there exists a continuous viscosity solution φε

ê Goal : To obtain gradient bounds |Dφε|∞ ≤ K

with K independent of ε

2/13

Gradientbounds fornonlinear

strictly ellipticequations

with coerciveHamiltonians

Olivier Ley

Feb 2016

Motivation for ε-independent bounds

It allows to solve the associated ergodic problem :

εφε → −c (ergodic constant), φε − φε(0)→ v as ε→ 0

and (c, v) ∈ R×W 1,∞(TN) is solution to

(HJerg) − Tr(A(x)D2v) + H(x ,Dv) = c .

[Lions-Papanicolau-Varadhan 86, Evans 89, Arisawa-Lions 98,Alvarez-Bardi 10, etc.]

Remark : |φε|∞ ∼ 1ε in general

3/13

Gradientbounds fornonlinear

strictly ellipticequations

with coerciveHamiltonians

Olivier Ley

Feb 2016

Motivation for Lipschitz bounds

ε-independent bounds for (HJ) leads in general totime-independent gradient bounds |Du(·, t)|∞ for the solutionof (HJevol)

∂tu − Tr(A(x)D2u) + H(x ,Du) = 0 (x , t) ∈ TN×(0,+∞)

u(x , 0) = u0(x) x ∈ TN

allowing a linearization procedure which permits to use theStrong Maximum Principle for viscosity solutions [Bardi-DaLio 99] to prove the large time behavior [Barles-Souganidis 01]

u(x , t) + ct → v(x) uniformly as t → +∞,

where (c , v) are solution to (HJerg).

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Gradientbounds fornonlinear

strictly ellipticequations

with coerciveHamiltonians

Olivier Ley

Feb 2016

Existing results for gradient bounds for (HJ)

• Elliptic regularity for classical solutions[Gilbarg-Trudinger] : |H(x , p)| ≤ C (1 + |p|2) (subquadratic)

• Ishii-Lions’ method :

[Ishii-Lions 90]

|H(x , p)− H(y , p)| ≤ C + ω(|x − y |)|x − y |τ |p|τ+2 τ ∈ [0, 1]

C , τ = 0, ω(|x − y |) = |x − y | ⇒ |DxH| ≤ |p|2

typical case : a(x)|p|2 (subquadratic)

[Barles 91]

|H(x , p)− H(y , p)| ≤ C + C |x − y ||p|3 + C |p|2little more than quadratic but restriction on the growth of H in p

These bounds depend on |φε|∞

5/13

Gradientbounds fornonlinear

strictly ellipticequations

with coerciveHamiltonians

Olivier Ley

Feb 2016

Gradient bounds for (HJ) with arbitrary growth

Bernstein method [Bernstein 1910]

• classical solutions [Gilbarg-Trudinger]

• viscosity solutions, weak Bernstein method : [Barles 91]Need of structural assumptions of “convexity type”[Barles-Souganidis 01] (A(x) ≡ I )

∃L > 0 : ∀|p| ≥ L, x ∈ TN , L(Hp · p − H − |H(·, 0)|∞) ≥ |Hx |

typical case : H(x , p) = a(x)|p|1+β + f (x),

β>0, a,f ∈W 1,∞(TN), a(x)>0

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Gradientbounds fornonlinear

strictly ellipticequations

with coerciveHamiltonians

Olivier Ley

Feb 2016

Remarks

• 1st order equations : εφε + H(x ,Dφε) = 0 with coerciveHamiltonian (of arbitrary growth) : immediate gradients boundsindependent of ε for subsolutions

• Holder bounds for possibly degenerate (HJ)[Capuzzo Dolcetta-Leoni-Porretta 10]Very general result for subsolutions :

if H(x , p) ≥ 1C |p|

k − C , k > 2 then φε ∈ C 0, k−2k−1 (TN)

with a Holder bound independent of ε

7/13

Gradientbounds fornonlinear

strictly ellipticequations

with coerciveHamiltonians

Olivier Ley

Feb 2016

Our slight extensions : “subquadratic case”

Theorem 1. |Dφε|∞ ≤ K for (HJ) when

(H1) ∃L > 1 : ∀x , y ∈ TN , if |p| = L then

H(x , p) ≥ |p|(H(y , p

|p|) + |H(·, 0)|∞ + N|x − y ||σx |2∞)

(H2) ∃α,C > 0 : ∀x , y ∈ TN ,|H(x , p)− H(y , p)| ≤ C |x − y |α|p|α+2 + C (1 + |p|2)

Comments :(H1) holds when H(x ,p)

|p| → +∞ as |p| → +∞ (superlinearity)

(H2) reduces to [Barles 91] when α = 1

Example : εφε−Tr(A(x)D2φε) + |Σ(x)Dφε|m + K (x ,Dφε) = 0

Σ∈C 0,γ(TN), m≤2+γ,K (x , p)

|p|→

|p|→+∞+∞, |K (x , p)|≤C (1+|p|2)

8/13

Gradientbounds fornonlinear

strictly ellipticequations

with coerciveHamiltonians

Olivier Ley

Feb 2016

Our slight extensions : “arbitrary growth”

Theorem 2. |Dφε|∞ ≤ K for (HJ) when

(H3) H(x , p) ≥ 1C |p|

k − C , k > 2

(H4) ∃α,β>0,B(p) such that β<(k−1)α+k andB(p)

|p|k→

|p|→+∞0

|H(x , p)− H(y , p)| ≤ C |x − y |α|p|β + B(p)

Comments :By (H3), φε ∈ C 0, k−2

k−1 (TN) [Capuzzo Dolcetta-Leoni-Porretta 10]

(H4) : no convexity-type assumptions, k , α can be big

Example : εφε − Tr(A(x)D2φε) + |Σ(x)Dφε|m + K (x ,Dφε) = 0

0 < Σ∈C 0,γ(TN), k ≤m≤ (k − 1)α + k ,K(x , p)

|p|k→

|p|→+∞0

(nonconvex) εφε − Tr(A(x)D2φε) + a(x)G (Dφε) + K (x ,Dφε) = 0

0 < a∈C 0,γ(TN), |p|k

C ≤ G (p) ≤ C |p|β , K(x , p)

|p|k→

|p|→+∞0

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Gradientbounds fornonlinear

strictly ellipticequations

with coerciveHamiltonians

Olivier Ley

Feb 2016

Ideas of proof (1) : Oscillation bound

Lemma 1. [L-Nguyen 15] Under (H1) (“superlinearity”),osc(φε) := maxTNφε −minTNφε ≤ O independent of ε.

Proof very simple, OK for degenerate and nonlocal equations.Will allow some “localization arguments” in the proof of regularity

Proof. Choose L >> 1 so that (H1) holds for |p| = L. Consider

M = maxx,y∈TN

φε(x)− Lφε(y) + (L− 1)minφε − L|x − y |

If M ≤ 0 we are done ; otherwise M > 0 and at maximum, x 6= y .Writing the viscosity inequalities with p = L x−y

|x−y | leads to

ε(φε(x)− Lφε(y))︸ ︷︷ ︸|εφε| ≤ |H(·, 0)|∞maximum principle

−Tr(A(x)X−A(y)Y )︸ ︷︷ ︸≤N|σx |2∞L|x−y |

+ H(x , p)− LH(y ,p

L)︸ ︷︷ ︸

|p| = L >> 1 sofrom (H1) bigger than

L(|H(·, 0)|∞ + N|σx |2∞|x − y |)

≤ 0

Contradiction 210/13

Gradientbounds fornonlinear

strictly ellipticequations

with coerciveHamiltonians

Olivier Ley

Feb 2016

Ideas of proof (2) : Ishii-Lions’ method

Lemma 2. Let ψ a concave smooth function with ψ(0) = 0.If maxx ,y∈TN{φε(x)− φε(y)− ψ(|x − y |)} > 0

is achieved at x , y with x 6= y and q = x−y|x−y | , then

−4νψ′′(|x − y |)︸ ︷︷ ︸>0 if ψ strict.concave

−C |x − y |ψ′(|x − y |)+H(x , ψ′(|x − y |)q)− H(y , ψ′(|x − y |)q) < 0

Idea : use ψ strictly concave s.t. −ψ′′(r)− Crψ′(r) >> 1

1) Holder bounds : ψ(r) = Krγ , γ ∈ (0, 1)2) Lipschitz bounds : ψ(r) = r − Kr1+γ , γ ∈ (0, 1)

Need to have r = |x − y | ≤ r0 small.For instance, in 2), r0 ≤ ((1 + γ)K )−γ , K very big

ê crucial use of Lemma 1 (oscillation bound)

11/13

Gradientbounds fornonlinear

strictly ellipticequations

with coerciveHamiltonians

Olivier Ley

Feb 2016

Ideas of proof (3) : Theorem 1

Step 1. (Holder estimate) |φε|C 0,γ ≤ K for some γ ∈ (0, 1).M = max{φε(x)− φε(y)− ψ(|x − y |}, ψ(r) = Lrγ .(H1)+Lemma 1 ê osc(φε) ≤ O independent of ε.Choosing K , r0 s.t. Krγ0 = O + 1 we have r = |x − y | < r0.If M > 0, then |x − y | 6= 0 and (H2)+Lemma 2 imply

−4νψ′′(r)− Crψ′(r)− Crαψ′(r)α+2 − Cψ′(r)2 − C < 0

...Contradiction for γ small enough.

Step 2. (Lipschitz estimate)M =max{φε(x)−φε(y)−ψ(|x − y |)}, ψ(r)=A1(A2r−(A2r)1+γ).We earn something, “Revenge of the ellipticity” : nonlinearity can bestronger than 2nd order terms but Holder estimate weaken thenonlinear terms :

rψ′(r) ≤︸︷︷︸concavity

ψ(r) <︸︷︷︸max.point

φε(x)− φε(y) ≤︸︷︷︸Step 1

Krγ

ê Better estimate of ψ′ : ψ′(r) ≤ Krγ−1

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Gradientbounds fornonlinear

strictly ellipticequations

with coerciveHamiltonians

Olivier Ley

Feb 2016

Ideas of proof (4) : Theorem 2

Step 1. |φε|C0,γ ≤ K for γ = k−2k−1

[Capuzzo Dolcetta-Leoni-Porretta 10]

Give a first improvement of the first derivative.

Step 2. |φε|C0,γ ≤ K for every γ ∈ (0, 1).Use ψ(r) = Krγ with the above improvement.

Step 3. Improvement to Lipschitz continuity as in Step 2 ofTheorem 1.

13/13