Uniqueness of semi-concave weak solutions for Hamilton-Jacobi equations

TL;DR

This paper demonstrates that for Hamilton-Jacobi equations with strictly convex nonlinearity, any semi-concave weak solution coincides with the viscosity solution, extending the understanding of solution uniqueness beyond semi-convex cases.

Contribution

It establishes the uniqueness of semi-concave weak solutions as viscosity solutions for a class of Hamilton-Jacobi equations with convex nonlinearity.

Findings

Semi-concave weak solutions are viscosity solutions under certain conditions.

The result extends the class of solutions known to be unique for Hamilton-Jacobi equations.

Motivated by spin glass problems, the paper links solution concepts in PDE theory.

Abstract

It is well known that when the nonlinearity is convex, the Hamilton-Jacobi PDE admits a unique semi-convex weak solution, which is the viscosity solution. In this paper, motivated by problems arising from spin glasses, we show that if the Hamilton-Jacobi PDE with strictly convex nonlinearity and regular enough initial condition admits a semi-concave weak solution, then this solution is the viscosity solution.