Regularity for Doubly Nonlinear Equations in the Mixed Regime

TL;DR

This paper establishes local H"older continuity for nonnegative solutions to doubly nonlinear equations, using a novel integral Harnack estimate to handle both singular and degenerate cases within specific parameter ranges.

Contribution

Introduces a new technique and integral Harnack estimate to prove regularity of solutions in challenging singular and degenerate regimes.

Findings

Proves local H"older continuity for solutions in mixed regimes.

Develops a new integral $L^1$-$L^1$ Harnack estimate.

Handles both singular and degenerate cases up to Barenblatt numbers.

Abstract

We study the local H\"older continuity of nonnegative solutions to doubly nonlinear equations by introducing a new technique that allows us to treat the cases where the equation is both singular and degenerate, up to specific Barenblatt numbers. Our argument relies on a new integral $L^1$-$L^1$ Harnack estimate, of independent interest.