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Does the viscosity solution of the Hamilton Jacobi equation satisfy the equation pointwise a.e.?

Rosy
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1 Answers1

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1) A viscosity solution satisfies pointwise the equation at every point of differentiability.

Namely, let $u\in C(\Omega)$ be a viscosity solution of $$ F(x, u(x), Du(x)) = 0\qquad \text{in}\ \Omega, $$ i.e., for every $x\in\Omega$, $$ (1)\quad F(x, u(x), p) \leq 0 \quad \forall p\in D^+ u(x), \qquad F(x, u(x), p) \geq 0 \quad \forall p\in D^- u(x). $$ Assume that $u$ is differentiable at some point $x_0\in\Omega$. This means that $$ D^+ u(x_0) = D^- u(x_0) = \{\nabla u (x_0)\}, $$ hence, using (1), $$ 0\leq F(x_0, u(x_0), \nabla u(x_0)) \leq 0. $$

EDIT: proof with test functions. If $u$ is differentiable at $x_0$, then $\nabla\varphi(x_0) = \nabla u(x_0)$ for every test function $\varphi$ touching $u$ at $x_0$ from above or from below. Hence $$ 0\leq F(x_0, u(x_0), \nabla u(x_0)) \leq 0. $$

2) For many HJ equations, solutions are differentiable a.e.

For example, if you consider an HJ equation of the form $$ u + H(x, Du(x)) = 0 $$ with $H(x, p) \to +\infty$ as $|p|\to +\infty$, then any bounded and continuous solution $u$ is also locally Lipschitz continuous (so that, by Rademacher's theorem, it is differentiable a.e.). See e.g. Bardi-Capuzzo Dolcetta, "Optimal Control...", Section 4.1.

Rigel
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  • What are these sets $D^+(x)$ and D^-(x)? are they set of all points where u-v has local maximum and local minimum respectively? i am using Definition of Viscosity solution given in L.C.Evans, PDE, Chapter 10. – Rosy Dec 11 '17 at 05:31
  • Sorry!! i am not able to connect your proof and the definition given in Evans PDE book... Could u please explain? – Rosy Dec 11 '17 at 05:40
  • I have added the same proof with test functions. – Rigel Dec 11 '17 at 06:29
  • I have one doubt in the edited proof: how to show that given a lipschitz continuous function which is differentiable at $x_0$ there exists a smooth function touching u at $x_0$ – Rosy Dec 12 '17 at 11:44
  • This fact can be proved with some work. It is not immediate. (You can find it in the cited book by Bardi and Capuzzo Dolcetta.) – Rigel Dec 12 '17 at 12:50
  • i got it!! Thanks a lot – Rosy Dec 13 '17 at 13:40