Geometric inequalities and symmetry results for elliptic systems

TL;DR

This paper develops geometric inequalities and symmetry results for elliptic systems, using Poincaré formulas and level set analysis to identify one-dimensional symmetry in solutions, with applications to phase separation conjectures.

Contribution

It introduces general Poincaré-type formulas and applies level set analysis to establish symmetry of solutions in broad elliptic systems, including cases related to De Giorgi's conjecture.

Findings

Established Poincaré-type formulas for elliptic systems

Proved one-dimensional symmetry of monotone and stable solutions

Applied results to phase separation problems in 2

Abstract

We obtain some Poincar\'{e} type formulas, that we use, together with the level set analysis, to detect the one-dimensional symmetry of monotone and stable solutions of possibly degenerate elliptic systems of the form {eqnarray*} {{array}{ll} div(a(|\nabla u|) \nabla u) = F_1(u, v), div(b(|\nabla v|) \nabla v) = F_2(u, v), {array}. {eqnarray*} where $F\in C^{1,1}_{loc}(\R^2)$. Our setting is very general, and it comprises, as a particular case, a conjecture of De Giorgi for phase separations in $\R^2$.