Multidimensional entire solutions for an elliptic system modelling phase separation

TL;DR

This paper introduces a novel method for constructing entire solutions to a semilinear elliptic system modeling phase separation, extending previous 2D results to higher dimensions with explicit symmetry and growth characterizations.

Contribution

The paper develops a new approach to construct solutions in higher dimensions, linking symmetry groups, sphere partitions, and solution growth, with improved asymptotic estimates and monotonicity formulas.

Findings

Extended existence results to dimensions N ≥ 3

Established explicit relations between symmetries and solutions

Provided new asymptotic estimates and sharp monotonicity formulas

Abstract

For the system of semilinear elliptic equations \[ \Delta V_i = V_i \sum_{j \neq i} V_j^2, \qquad V_i > 0 \qquad \text{in $\mathbb{R}^N$} \] we devise a new method to construct entire solutions. The method extends the existence results already available in the literature, which are concerned with the 2-dimensional case, also in higher dimensions $N \ge 3$. In particular, we provide an explicit relation between orthogonal symmetry subgroups, optimal partition problems of the sphere, the existence of solutions and their asymptotic growth. This is achieved by means of new asymptotic estimates for competing system and new sharp versions for monotonicity formulae of Alt-Caffarelli-Friedman type.