Regularity of solutions to Hamilton-Jacobi equations for...
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Regularity of solutions to Hamilton-Jacobi
equations for Tonelli Hamiltonians
Ludovic Rifford
Universite Nice Sophia Antipolis&
Institut Universitaire de France
Nonlinear Analysis and OptimizationRoyal Society, London, 2015
Ludovic Rifford Solutions of Hamilton-Jacobi equations
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In honour of Francis and Richard
Ludovic Rifford Solutions of Hamilton-Jacobi equations
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Other students and friends
Thierry Champion, Cyril Imbert, Olivier Ley and Victor Filipe Martins-Da Rocha
Sebastien Cambier and Nicolas Deniau
Ludovic Rifford Solutions of Hamilton-Jacobi equations
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Other students and friends
Thierry Champion, Cyril Imbert, Olivier Ley and Victor Filipe Martins-Da Rocha
Sebastien Cambier and Nicolas Deniau
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 5: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/5.jpg)
Other students and friends
Thierry Champion, Cyril Imbert, Olivier Ley and Victor Filipe Martins-Da Rocha
Sebastien Cambier and Nicolas Deniau
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 6: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/6.jpg)
Setting
Let M be a smooth compact manifold of dimension n ≥ 2 befixed. Let H : T ∗M → R be a Hamiltonian of class C 2
satisfying the following properties:
(H1) Superlinear growth:For every K ≥ 0, there is C ∗(K ) ∈ R such that
H(x , p) ≥ K |p|+ C ∗(K ) ∀(x , p) ∈ T ∗M .
(H2) Uniform convexity:For every (x , p) ∈ T ∗M , ∂2H
∂p2 (x , p) is positive definite.
For sake of simplicity, we may assume that M = Tn, that isthat H : Rn ×Rn → R satisfies (H1)-(H2) and is periodic withrespect to the x variable.
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 7: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/7.jpg)
Setting
Let M be a smooth compact manifold of dimension n ≥ 2 befixed. Let H : T ∗M → R be a Hamiltonian of class C 2
satisfying the following properties:
(H1) Superlinear growth:For every K ≥ 0, there is C ∗(K ) ∈ R such that
H(x , p) ≥ K |p|+ C ∗(K ) ∀(x , p) ∈ T ∗M .
(H2) Uniform convexity:For every (x , p) ∈ T ∗M , ∂2H
∂p2 (x , p) is positive definite.
For sake of simplicity, we may assume that M = Tn, that isthat H : Rn ×Rn → R satisfies (H1)-(H2) and is periodic withrespect to the x variable.
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 8: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/8.jpg)
A first result of regularity
We are concerned with the regularity properties of viscositysolutions of the Hamilton-Jacobi equation
H(x , dxu) = 0 ∀x ∈ M (HJ).
Theorem (Rifford 2007)
Let H : T ∗M → R be a Hamiltonian of class C 2 satisfying(H1)-(H2) and u : M → R be a viscosity solution of (HJ).Then the function u is semiconcave on M. Moreover, thesingular set of u is nowhere dense in M and u is C 1,1
loc on the
open dense set M \ Σ(u).
Reminder:
Σ(u) ={
x ∈ M | u not diff. at x}
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 9: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/9.jpg)
A first result of regularity
We are concerned with the regularity properties of viscositysolutions of the Hamilton-Jacobi equation
H(x , dxu) = 0 ∀x ∈ M (HJ).
Theorem (Rifford 2007)
Let H : T ∗M → R be a Hamiltonian of class C 2 satisfying(H1)-(H2) and u : M → R be a viscosity solution of (HJ).Then the function u is semiconcave on M. Moreover, thesingular set of u is nowhere dense in M and u is C 1,1
loc on the
open dense set M \ Σ(u).
Reminder:
Σ(u) ={
x ∈ M | u not diff. at x}
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 10: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/10.jpg)
A first result of regularity
We are concerned with the regularity properties of viscositysolutions of the Hamilton-Jacobi equation
H(x , dxu) = 0 ∀x ∈ M (HJ).
Theorem (Rifford 2007)
Let H : T ∗M → R be a Hamiltonian of class C 2 satisfying(H1)-(H2) and u : M → R be a viscosity solution of (HJ).Then the function u is semiconcave on M. Moreover, thesingular set of u is nowhere dense in M and u is C 1,1
loc on the
open dense set M \ Σ(u).
Reminder:
Σ(u) ={
x ∈ M | u not diff. at x}
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 11: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/11.jpg)
Semiconcave functions
A function u : M → R is called semiconcave if it can bewritten locally (in charts) as
u = g + h,
the sum of a smooth function g and a concave function h witha universal upper bound on the C 2-norm of g .
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 12: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/12.jpg)
Graph of a semiconcave function
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 13: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/13.jpg)
Characterization of viscosity solutions
Let L : TM → R be the Tonelli Lagrangian associated with Hby Legendre-Fenchel duality, that is
L(x , v) := maxp∈T∗
x M
{p · v − H(x , p)
}∀(x , v) ∈ TM .
Proposition
The function u : M → R is a viscosity solution of (HJ) iff:
(i) For every Lipschitz curve γ : [a, b]→ M, we have
u(γ(b)
)− u(γ(a)
)≤∫ b
a
L(γ(t), γ(t)
)ds.
(ii) ∀x ∈ M, there is a curve γx : (−∞, 0]→ M such that
u(γ(b)
)− u(γ(a)
)=
∫ b
a
L(γ(t), γ(t)
)ds ∀a < b < 0.
Ludovic Rifford Solutions of Hamilton-Jacobi equations
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Semiconcavity of critical solutions
b
b
b
γx x
x′
z = γx(−1)
γ′
u(x) = u(z) +
∫ 0
−1
L(γx(t), γx(t)
)dt
u(x ′) ≤ u(z) +
∫ 0
−1
L(γ′(t), γ′(t)
)dt
Thus
u(x ′) ≤ u(x) +
∫ 0
−1
L(γ′(t), γ′(t)
)− L(γx(t), γx(t)
)dt
Ludovic Rifford Solutions of Hamilton-Jacobi equations
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Semiconcavity of critical solutions
b
b
b
γx x
x′
z = γx(−1)
γ′
u(x) = u(z) +
∫ 0
−1
L(γx(t), γx(t)
)dt
u(x ′) ≤ u(z) +
∫ 0
−1
L(γ′(t), γ′(t)
)dt
Thus
u(x ′) ≤ u(x) +
∫ 0
−1
L(γ′(t), γ′(t)
)− L(γx(t), γx(t)
)dt
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 16: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/16.jpg)
Semiconcavity of critical solutions
b
b
b
γx x
x′
z = γx(−1)
γ′
u(x) = u(z) +
∫ 0
−1
L(γx(t), γx(t)
)dt
u(x ′) ≤ u(z) +
∫ 0
−1
L(γ′(t), γ′(t)
)dt
Thus
u(x ′) ≤ u(x) +
∫ 0
−1
L(γ′(t), γ′(t)
)− L(γx(t), γx(t)
)dt
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 17: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/17.jpg)
Semiconcavity of critical solutions
b
b
b
γx x
x′
z = γx(−1)
γ′
u(x) = u(z) +
∫ 0
−1
L(γx(t), γx(t)
)dt
u(x ′) ≤ u(z) +
∫ 0
−1
L(γ′(t), γ′(t)
)dt
Thus
u(x ′) ≤ u(x) +
∫ 0
−1
L(γ′(t), γ′(t)
)− L(γx(t), γx(t)
)dt
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 18: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/18.jpg)
Rest of the proof
We can repeat the previous argument to show that for everyx ∈ M , every semi-calibrated curve γx : (−∞, 0]→ M andevery t > 0, the graph of u at γx(−t) admits a supportfunction of class C 2 from below.
Moreover, we can show that
for every x ∈ M , there is a one-to-one correspondencebetween the limiting differential of u at x ,
d∗u(x) := {lim du(xk) | xk → x , u diff at xk} ,and the set of semi-calibrated curves.
b
‘
x
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 19: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/19.jpg)
Rest of the proof
We can repeat the previous argument to show that for everyx ∈ M , every semi-calibrated curve γx : (−∞, 0]→ M andevery t > 0, the graph of u at γx(−t) admits a supportfunction of class C 2 from below. Moreover, we can show that
for every x ∈ M , there is a one-to-one correspondencebetween the limiting differential of u at x ,
d∗u(x) := {lim du(xk) | xk → x , u diff at xk} ,and the set of semi-calibrated curves.
b
‘
x
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 20: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/20.jpg)
Rest of the proof
We can repeat the previous argument to show that for everyx ∈ M , every semi-calibrated curve γx : (−∞, 0]→ M andevery t > 0, the graph of u at γx(−t) admits a supportfunction of class C 2 from below. Moreover, we can show that
for every x ∈ M , there is a one-to-one correspondencebetween the limiting differential of u at x ,
d∗u(x) := {lim du(xk) | xk → x , u diff at xk} ,and the set of semi-calibrated curves.
b
‘
x
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 21: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/21.jpg)
A beautiful result by Arnaud
Theorem (Arnaud ’08)
Let M = T2 and H : T ∗M → R be an Hamiltonian of class C 2
satisfying (H1)-(H2). Let u : M → R be a solution of (HJ) ofclass C 1 without singularities. Then u is C 1,1
and C 2 almosteverywhere.
We call singularity any equilibrium of the characteristic flowof u, that is any x ∈ M such that
∂H
∂p(x , dxu) = 0.
Remark
Note that u C 1 ⇒ u C 1,1 and the graph of x 7→ dxu is aLipschitz Lagrangian submanifold of T ∗M which is invariantby the Hamiltonian flow φH
t .
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 22: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/22.jpg)
A beautiful result by Arnaud
Theorem (Arnaud ’08)
Let M = T2 and H : T ∗M → R be an Hamiltonian of class C 2
satisfying (H1)-(H2). Let u : M → R be a solution of (HJ) ofclass C 1 without singularities. Then u is C 1,1 and C 2 almosteverywhere.
We call singularity any equilibrium of the characteristic flowof u, that is any x ∈ M such that
∂H
∂p(x , dxu) = 0.
Remark
Note that u C 1 ⇒ u C 1,1 and the graph of x 7→ dxu is aLipschitz Lagrangian submanifold of T ∗M which is invariantby the Hamiltonian flow φH
t .
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 23: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/23.jpg)
A beautiful result by Arnaud
Theorem (Arnaud ’08)
Let M = T2 and H : T ∗M → R be an Hamiltonian of class C 2
satisfying (H1)-(H2). Let u : M → R be a solution of (HJ) ofclass C 1 without singularities. Then u is C 1,1 and C 2 almosteverywhere.
We call singularity any equilibrium of the characteristic flowof u, that is any x ∈ M such that
∂H
∂p(x , dxu) = 0.
Remark
Note that u C 1 ⇒ u C 1,1 and the graph of x 7→ dxu is aLipschitz Lagrangian submanifold of T ∗M which is invariantby the Hamiltonian flow φH
t .
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 24: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/24.jpg)
A beautiful result by Arnaud
Theorem (Arnaud ’08)
Let M = T2 and H : T ∗M → R be an Hamiltonian of class C 2
satisfying (H1)-(H2). Let u : M → R be a solution of (HJ) ofclass C 1 without singularities. Then u is C 1,1 and C 2 almosteverywhere.
We call singularity any equilibrium of the characteristic flowof u, that is any x ∈ M such that
∂H
∂p(x , dxu) = 0.
Remark
Note that u C 1 ⇒ u C 1,1 and the graph of x 7→ dxu is aLipschitz Lagrangian submanifold of T ∗M which is invariantby the Hamiltonian flow φH
t .
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 25: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/25.jpg)
Green bundles I
b
θ
Vθ
b
VφH−t
(θ)
DφH−t
(θ)φHt
For every θ = (x , dxu) ∈ T ∗M and every t ∈ R, we define theLagrangian subspace G t
θ ⊂ TθT∗M by (Vθ ' {0} × R2)
G tθ :=
(φHt
)∗
(VφH−t(θ)
).
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 26: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/26.jpg)
Green bundles II
Definition
For every θ = (x , dxu), we define the positive and negativeGreen bundles at θ as
G +θ := lim
t→+∞G tθ and G−θ := lim
t→−∞G tθ
b
θ
Vθ
G−θ
G+θ
Ludovic Rifford Solutions of Hamilton-Jacobi equations
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Green bundles III
The following properties hold:
For every θ = (x , dxu), G−θ � G +θ .
The function x ∈ M 7→ G +(x ,dxu) is upper-semicontinuous.
The function x ∈ M 7→ G−(x ,dxu) is lower-semicontinuous.
So, if G +(x ,dxu) = G−(x ,dxu) for some x then both functions
are continuous at x .
For every x ∈ M , we have
G−(x ,dxu) � HessCu(x) � G +(x ,dxu),
where HessCu(x) denotes the Clarke generalized Hessianof u at x .
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 28: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/28.jpg)
Green bundles III
The following properties hold:
For every θ = (x , dxu), G−θ � G +θ .
The function x ∈ M 7→ G +(x ,dxu) is upper-semicontinuous.
The function x ∈ M 7→ G−(x ,dxu) is lower-semicontinuous.
So, if G +(x ,dxu) = G−(x ,dxu) for some x then both functions
are continuous at x .
For every x ∈ M , we have
G−(x ,dxu) � HessCu(x) � G +(x ,dxu),
where HessCu(x) denotes the Clarke generalized Hessianof u at x .
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 29: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/29.jpg)
Green bundles III
The following properties hold:
For every θ = (x , dxu), G−θ � G +θ .
The function x ∈ M 7→ G +(x ,dxu) is upper-semicontinuous.
The function x ∈ M 7→ G−(x ,dxu) is lower-semicontinuous.
So, if G +(x ,dxu) = G−(x ,dxu) for some x then both functions
are continuous at x .
For every x ∈ M , we have
G−(x ,dxu) � HessCu(x) � G +(x ,dxu),
where HessCu(x) denotes the Clarke generalized Hessianof u at x .
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 30: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/30.jpg)
Green bundles III
The following properties hold:
For every θ = (x , dxu), G−θ � G +θ .
The function x ∈ M 7→ G +(x ,dxu) is upper-semicontinuous.
The function x ∈ M 7→ G−(x ,dxu) is lower-semicontinuous.
So, if G +(x ,dxu) = G−(x ,dxu) for some x then both functions
are continuous at x .
For every x ∈ M , we have
G−(x ,dxu) � HessCu(x) � G +(x ,dxu),
where HessCu(x) denotes the Clarke generalized Hessianof u at x .
Ludovic Rifford Solutions of Hamilton-Jacobi equations
![Page 31: Regularity of solutions to Hamilton-Jacobi equations for ...rifford/Papiers_en_ligne/Royal_Society_RIFFORD_2015.pdf · Regularity of solutions to Hamilton-Jacobi equations for Tonelli](https://reader034.fdocuments.us/reader034/viewer/2022042215/5ebb6cc1a541b34d0f713014/html5/thumbnails/31.jpg)
Green bundles III
The following properties hold:
For every θ = (x , dxu), G−θ � G +θ .
The function x ∈ M 7→ G +(x ,dxu) is upper-semicontinuous.
The function x ∈ M 7→ G−(x ,dxu) is lower-semicontinuous.
So, if G +(x ,dxu) = G−(x ,dxu) for some x then both functions
are continuous at x .
For every x ∈ M , we have
G−(x ,dxu) � HessCu(x) � G +(x ,dxu),
where HessCu(x) denotes the Clarke generalized Hessianof u at x .
Ludovic Rifford Solutions of Hamilton-Jacobi equations
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Ludovic Rifford Solutions of Hamilton-Jacobi equations