## Differential and Integral Equations

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

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

#### 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

**Permanent link to this document**

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