On the improvement of H\"older seminorms in superquadratic Hamilton-Jacobi equations

TL;DR

This paper establishes maximal L^q-regularity for viscous Hamilton-Jacobi equations with superquadratic growth, using new Hölder estimates derived from decay properties of nonlinear quotients, extending regularity results.

Contribution

It introduces novel Hölder estimates and oscillation techniques to prove maximal regularity for superquadratic Hamilton-Jacobi equations with unbounded data.

Findings

Maximal L^q-regularity holds for q > (N+2)(γ-1)/γ.

New Hölder estimates are derived from decay of nonlinear quotients.

Oscillation estimates lead to Liouville type results for entire solutions.

Abstract

We show in this paper that maximal $L^q$-regularity for time-dependent viscous Hamilton-Jacobi equations with unbounded right-hand side and superquadratic $\gamma$-growth in the gradient holds in the full range $ q > (N+2)\frac{\gamma-1}\gamma$. Our approach is based on new $\frac{\gamma-2}{\gamma-1}$-H\"older estimates, which are consequence of the decay at small scales of suitable nonlinear space and time H\"older quotients. This is obtained by proving suitable oscillation estimates, that also give in turn some Liouville type results for entire solutions.