# Viscosity solution of Hamilton Jacobi equation

In the book Partial Differential Equations by L.C. Evans there are two solution concepts for Hamilton Jacobi equations

A. VISCOSITY SOLUTION(which is defined in chapter 10)

B. WEAK SOLUTION(which is defined in chapter 3)

I have the following doubts....

1)Is the visocotiy solution lipschitz continuous? if so does it satatisfy the equation pointwise a.e.

2)Are these two solutions same?

if so what is the justification?

Definition of Viscosity solution(L.C.Evans, PDE, Chapter 10): Assume that $u$ is bounded and uniformly continuous on $R^n \times [0,T]$,for each $T\geq 0$. We say that u is viscosity solution of the initial value problem

$u_t+H(Du,x)=0$ in $R^n \times (0,\infty)$,

u=g on $R^n \times \{t=0\}$ provided

A)$u=g$ on $R^n \times (0,\infty)$, and

B)for each v $\in C^ \infty (R^n \times (0,\infty))$,

if $u-v$ has a local maximum at a point $(x_0,t_0) \in R^n \times (0,\infty)$ then $v_t(x_0,t_0)+H(Dv(x_0,t_0),x_0)\leq 0$

and if

if $u-v$ has a local minimum at a point $(x_0,t_0) \in R^n \times (0,\infty)$ then $v_t(x_0,t_0)+H(Dv(x_0,t_0),x_0)\geq 0$

• Could you explain what is meant with the viscosity solution? – md2perpe Dec 8 '17 at 8:56
• i have edited the definition of viscosity solution in the above question – Rosy Dec 8 '17 at 10:25

## 1 Answer

Yes, wherever a viscosity solution is differentiable, it satisfies the PDE. In many cases the viscosity solution is Lipschitz (e.g., it is Lipschitz in the setting of Evans Chapter 10), but there are circumstances where the viscosity solution is less regular (continuous, or even discontinuous).

Viscosity solution is more general than weak solution. The weak solution from Chapter 3 is a viscosity solution, but the weak solution is required to be semi-concave, which requires $H$ to be convex. In more general settings (say, $H$ not convex), the notion of weak solution is not applicable, but we still have viscosity solution.

• Jeff, I only see in Evan's text that viscosity solutions are bounded and uniformly continuous by definition. Where is the result in Ch 10 that indicates they are also Lipschitz? Is it given implicitly by some other property? – Chester Jan 22 '18 at 12:11
• I don't have a copy of Evans on me, but there is a lemma or proposition somewhere in Evans Ch10 showing that the value function is Lipschitz (given the assumptions on the cost, dynamics, etc.). This is not at all necessary, and is just a convenience for making the uniqueness proof a bit easier. – Jeff Jan 22 '18 at 23:44
• I see. I glossed over that part of Ch 10. Thanks! – Chester Jan 22 '18 at 23:59