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).
4/13
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
6/13
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
9/13
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
12/13
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
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