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Extremal Viscosity Solutions of Almost Periodic Hamilton-Jacobi Equations

Commun. Math. Anal.
Volume 18, Number 2 (2015), 86 - 105

Extremal Viscosity Solutions of Almost Periodic Hamilton-Jacobi Equations

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Abstract

This paper deals with viscosity solutions of Hamilton-Jacobi equations in which the Hamiltonian $H$ is weakly monotone with respect to the zero order term: this leads to non-uniqueness of solutions, even in the class of \emph{periodic} or \emph{almost periodic} (briefly a.p.) functions. The lack of uniqueness of a.p. solutions leads to introduce the notion of minimal (maximal) a.p. solution and to study its properties. The classes of \emph{asymptotically almost periodic} (briefly a.a.p.) and \emph{pseudo almost periodic} (briefly p.a.p.) functions are also considered.