Flipping regularity via the Harnack approach and applications to nonlinear elliptic problems

TL;DR

This paper introduces an abstract framework linking one-sided geometric control to two-sided estimates for functions, with applications to nonlinear elliptic equations, regularity, and viscosity solutions, offering new insights into the De Giorgi-Nash-Moser theory.

Contribution

It develops a novel abstract approach connecting geometric control to estimates and regularity in nonlinear elliptic problems, extending classical theories.

Findings

Establishes two-sided estimates from one-sided geometric control.

Provides regularity criteria for viscosity solutions.

Derives $L^pL^ Infty$-estimates as a converse to classical results.

Abstract

We prove an abstract result ensuring that one-sided geometric control yields two-sided estimates for functions satisfying general conditions. Our findings resonate in the context of nonlinear elliptic problems, including supersolutions to fully nonlinear elliptic equations and functions in the De Giorgi class. Among the consequences of our abstract results are regularity estimates, and conditions for a continuous function to be in the class of viscosity solutions. We also prove that one-sided geometric control yields $L^pL^\infty$-estimates. It provides a converse to the implication in the De Giorgi-Nash-Moser theory.