preprint

Inserted: 1 nov 2023

Year: 2022

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.