Differential and Integral Equations

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

Olivier Alvarez

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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

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

First available in Project Euclid: 2 May 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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