Calculus of Variations and Geometric Measure Theory

G. Tortone

Liouville theorems and optimal regularity in elliptic equations

created by tortone on 01 Nov 2023



Inserted: 1 nov 2023

Year: 2022

ArXiv: 2204.10772 PDF


The objective of this paper is the connection between the problem of optimal regularity among solutions to elliptic divergence equations with measurable coefficients with the Liouville property at infinity. Initially, we address the two-dimensional case by proving an Alt-Caffarelli-Friedman type monotonicity formula for non-negative subsolutions with disjoint supports, which allows to prove optimal regularity and the Liouville property for multiphase problems. In the higher-dimensional case we discuss the role of the monotonicity formula in the characterization of the least growth at infinity and the exponent of regularity as well. Finally, we establish this connection by combining blow-up and $G$-convergence argument.