Differential and Integral Equations

Bounded-from-below viscosity solutions of Hamilton-Jacobi equations

Olivier Alvarez

Abstract

We establish a uniqueness and existence theorem for bounded-from-below viscosity solutions of Hamilton-Jacobi equations of the form $u+H(Du)=f$ in $\mathbb{R}^N$. More precisely, we show that there is a unique solution $u$ such that $u^-$ grows at most linearly, when $f^-$ behaves analogously and when $H$ is convex and nonlinear. We sharpen this result when the behavior of the Hamiltonian is known at infinity. We also discuss some extensions to more general Hamiltonians and to the Dirichlet problem in an unbounded open set.

Article information

Source
Differential Integral Equations, Volume 10, Number 3 (1997), 419-436.

Dates
First available in Project Euclid: 2 May 2013

https://projecteuclid.org/euclid.die/1367525660

Mathematical Reviews number (MathSciNet)
MR1744854

Zentralblatt MATH identifier
0890.35026

Subjects
Primary: 49L25: Viscosity solutions
Secondary: 35F20: Nonlinear first-order equations

Citation

Alvarez, Olivier. Bounded-from-below viscosity solutions of Hamilton-Jacobi equations. Differential Integral Equations 10 (1997), no. 3, 419--436. https://projecteuclid.org/euclid.die/1367525660