## Advances in Differential Equations

- Adv. Differential Equations
- Volume 22, Number 3/4 (2017), 258-304.

### A well posedness result for generalized solutions of Hamilton-Jacobi equations

#### Abstract

We study the Dirichlet problem for stationary Hamilton-Jacobi equations $$ \begin{cases} H(x, u(x), \nabla u(x)) = 0 & \ \textrm{in} \ \Omega \\ u(x)=\varphi(x) & \ \textrm{on} \ \partial \Omega. \end{cases} $$ We consider a Caratheodory hamiltonian $H=H(x,u,p)$, with a Sobolev-type (but not continuous) regularity with respect to the space variable $x$, and prove existence and uniqueness of a Lipschitz continuous maximal generalized solution which, in the continuous case, turns out to be the classical viscosity solution. In addition, we prove a continuous dependence property of the solution with respect to the boundary datum $\varphi$, completing in such a way a well posedness theory.

#### Article information

**Source**

Adv. Differential Equations, Volume 22, Number 3/4 (2017), 258-304.

**Dates**

First available in Project Euclid: 18 February 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1487386869

**Mathematical Reviews number (MathSciNet)**

MR3611507

**Zentralblatt MATH identifier**

06723005

**Subjects**

Primary: 35F21: Hamilton-Jacobi equations 35F20: Nonlinear first-order equations 49L25: Viscosity solutions 46B50: Compactness in Banach (or normed) spaces 35F30: Boundary value problems for nonlinear first-order equations

#### Citation

Zagatti, Sandro. A well posedness result for generalized solutions of Hamilton-Jacobi equations. Adv. Differential Equations 22 (2017), no. 3/4, 258--304. https://projecteuclid.org/euclid.ade/1487386869