Global semiconcavity of solutions to first-order Hamilton-Jacobi equations with state constraints

TL;DR

This paper proves that solutions to certain first-order Hamilton-Jacobi equations with state constraints are globally semiconcave, using techniques from weak KAM theory and Euler-Lagrange equations, with conditions on the Hamiltonian and its derivatives.

Contribution

It establishes conditions under which solutions are globally semiconcave, extending local results and providing optimal conditions on the derivatives of the Hamiltonian.

Findings

Solutions are locally semiconcave with boundary-dependent constants.

Under certain conditions, solutions are globally semiconcave.

The conditions on Df are shown to be essentially optimal in 1D.

Abstract

We focus on the global semiconcavity of solutions to first-order Hamilton--Jacobi equations with state constraints, especially for the Hamiltonian $H(x, \beta):=|\beta|^p-f(x)$ with $p \in (1, 2]$. We first show that the solution is locally semiconcave, and the semiconcavity constant at each point depends on the first time a corresponding minimizing curve emanating from this point hits the boundary. Then, with appropriate conditions on $Df$, we prove that for any such minimizing curve, the time it takes to hit the boundary of the domain is $+\infty$, and as a consequence, the solution is globally semiconcave. Moreover, the condition on $Df$ is essentially optimal with examples in one-dimensional space. The proofs employ the Euler-Lagrange equations and techniques in weak KAM theory.