Liouville theorems and optimal regularity in elliptic equations

TL;DR

This paper links optimal regularity of solutions to elliptic PDEs with measurable coefficients to Liouville properties at infinity, using monotonicity formulas and blow-up techniques across dimensions.

Contribution

It introduces new monotonicity formulas and techniques to establish optimal regularity and Liouville properties for elliptic equations in various dimensions.

Findings

Proved an Alt-Caffarelli-Friedman type monotonicity formula in 2D.

Established H"older regularity of solutions via an almost-monotonicity formula.

Connected growth at infinity with regularity exponents through blow-up and G-convergence.

Abstract

The objective of this paper is to establish a connection between the problem of optimal regularity among solutions to elliptic PDEs with measurable coefficients and the Liouville property at infinity. Initially, we address the two-dimensional case by proving an Alt-Caffarelli-Friedman type monotonicity formula, enabling the proof of optimal regularity and the Liouville property for multiphase problems. In higher dimensions, we delve into the role of monotonicity formulas in characterizing optimal regularity. By employing a hole-filling technique, we present a distinct "almost-monotonicity" formula that implies H$\"{o}$lder regularity of solutions. Finally, we explore the interplay between the least growth at infinity and the exponent of regularity by combining blow-up and $G$-convergence arguments.