H\"older continuity for non-coercive Hamilton-Jacobi equations associated to linear control systems

TL;DR

This paper proves H"older continuity for solutions to certain degenerate Hamilton-Jacobi equations linked to linear control systems, using a geometric approach tailored to the system's structure.

Contribution

It introduces a novel geometric method to establish regularity for non-coercive, non-convex Hamilton-Jacobi equations associated with linear control systems.

Findings

H"older estimates are anisotropic and depend on the control system's geometry.

Results apply to equations with unbounded source terms.

The approach extends regularity theory to degenerate, hypoelliptic settings.

Abstract

In this paper we establish H\"older continuity estimates for viscosity solutions to first order Hamilton-Jacobi equations linked to linear control systems satisfying the Kalman rank condition. Our model Hamiltonians are non-convex in the generalised momentum variable and - more importantly - they lack coercivity in certain directions. Therefore, all previously available results from the literature cannot be applied to these degenerate settings. In order to overcome these obstructions, we design a geometric argument, dictated by the linear control system. As a result of this, the obtained H\"older estimates are quantified in an anisotropic way within this geometric framework. The estimates hold true for unbounded source terms, for which one part of our analysis is inspired by a recent result on De Giorgi type methods for hypoelliptic operators.