Hopf-Lax formula and generalized characteristics

TL;DR

This paper investigates the differential properties of viscosity solutions to Hamilton-Jacobi equations using the Hopf-Lax formula, introducing a generalized characteristic concept and analyzing the solution's differentiability in specific regions.

Contribution

It introduces a generalized notion of characteristics for Hamilton-Jacobi equations and studies the differentiability strip of viscosity solutions derived from the Hopf-Lax formula.

Findings

Characterization of the differentiability strip of viscosity solutions.

Introduction of a generalized characteristic framework.

Insights into the structure of solutions for Hamilton-Jacobi equations.

Abstract

We study some differential properties of viscosity solution for Hamilton - Jacobi equations defined by Hopf-Lax formula $u(t,x)=\min_{y\in \R^n} \big\{\sigma (y)+tH^*\big (\frac {x-y}{t}\big)\big \}.$ A generalized form of characteristics of the Cauchy problems is taken into account the context. Then we examine the strip of differentiability of the viscosity solution given by the function $u(t,x).$