Regularity Results for Eikonal-Type Equations with Nonsmooth Coefficients

TL;DR

This paper establishes local semiconcavity of solutions to certain Eikonal-type Hamilton-Jacobi equations with nonsmooth coefficients, advancing understanding of their regularity properties.

Contribution

It proves local semiconcavity with a power-like modulus for solutions where the Hamiltonian is Hölder continuous and convex, and demonstrates ${ m C}^{1,eta}$ regularity of extremal trajectories.

Findings

Solutions are locally semiconcave with a power-like modulus.

Extremal trajectories are ${ m C}^{1,eta}$ regular.

Regularity results hold under Hölder continuity and convexity assumptions.

Abstract

Solutions of the Hamilton-Jacobi equation $H(x,-Du(x))=1$, with $H(\cdot,p)$ H\"older continuous and $H(x,\cdot)$ convex and positively homogeneous of degree 1, are shown to be locally semiconcave with a power-like modulus. An essential step of the proof is the ${\mathcal C}^{1,\alpha}$-regularity of the extremal trajectories associated with the multifunction generated by $D_pH$.