An anisotropic monotoncity formula, with applications to some segregation problems

TL;DR

This paper establishes a new anisotropic monotonicity formula for elliptic equations, enabling analysis of segregation phenomena and Liouville-type theorems for systems with different diffusion operators.

Contribution

It introduces an anisotropic monotonicity formula applicable to elliptic systems with varying ellipticity matrices, advancing understanding of segregation and Liouville theorems.

Findings

Proved an Alt-Caffarelli-Friedman monotonicity formula for anisotropic elliptic equations.

Derived Liouville-type theorems for subsolutions of elliptic systems.

Analyzed segregation phenomena in systems with different diffusion operators.

Abstract

We prove an Alt-Caffarelli-Friedman montonicity formula for pairs of functions solving elliptic equations driven by different ellipticity matrices in their positivity sets. As application, we derive Liouville-type theorems for subsolutions of some elliptic systems, and we analyze segregation phenomena for systems of equations where the diffusion of each density is described by a different operator.