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March/April 2017 A well posedness result for generalized solutions of Hamilton-Jacobi equations
Sandro Zagatti
Adv. Differential Equations 22(3/4): 258-304 (March/April 2017).

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.

Citation

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Sandro Zagatti. "A well posedness result for generalized solutions of Hamilton-Jacobi equations." Adv. Differential Equations 22 (3/4) 258 - 304, March/April 2017.

Information

Published: March/April 2017
First available in Project Euclid: 18 February 2017

zbMATH: 06723005
MathSciNet: MR3611507

Subjects:
Primary: 35F20, 35F21, 35F30, 46B50, 49L25

Rights: Copyright © 2017 Khayyam Publishing, Inc.

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Vol.22 • No. 3/4 • March/April 2017
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