Calculus of Variations and Geometric Measure Theory

G. Tortone

Liouville theorems and optimal regularity in elliptic equations

created by tortone on 01 Nov 2023
modified on 25 Jan 2025

[BibTeX]

Published Paper

Inserted: 1 nov 2023
Last Updated: 25 jan 2025

Journal: Proceedings of the London Mathematical Society
Year: 2024
Doi: 10.1112/plms.12587

ArXiv: 2204.10772 PDF

Abstract:

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.