Semi-concave singularities and the Hamilton-Jacobi equation

TL;DR

This paper investigates the Hamilton-Jacobi equation with semiconcave initial data, establishing inequalities between weak solution types and providing conditions for explicit semi-concave functions to be viscosity solutions.

Contribution

It generalizes the entropy inequality for scalar conservation laws to the Hamilton-Jacobi context, linking variational and viscosity solutions for semiconcave initial conditions.

Findings

Proves an inequality between variational and viscosity solutions.

Provides conditions for semi-concave functions to be viscosity solutions.

Extends entropy inequality concepts to Hamilton-Jacobi equations.

Abstract

We study the Cauchy problem for the Hamilton-Jacobi equation with a semiconcave initial condition. We prove an inequality between two types of weak solutions emanating from such an initial condition (the variational and the viscosity solution).We also give conditions for an explicit semi-concave function to be a viscosity solution. These conditions generalize the entropy inequality characterizing piecewise smooth solutions of scalar conservation laws in dimension one.