Symmetry results for stable and monotone solutions to fibered systems of PDEs

TL;DR

This paper investigates symmetry properties of solutions to certain elliptic PDE systems with fibered structures, establishing new symmetry results for stable and monotone solutions using a Poincaré-type formula.

Contribution

The paper introduces a novel Poincaré-type formula and applies it to prove symmetry results for stable and monotone solutions of fibered elliptic systems.

Findings

Established symmetry results for stable solutions.

Proved symmetry for monotone solutions.

Developed a new Poincaré-type formula for systems.

Abstract

We study the symmetry properties for solutions of elliptic systems of the type {ll}-\dive(a_1(x,|\nabla u^1|(X))\nabla u^1(X))=F_{1}(x, u^1(X),..., u^n(X)), ... -\dive(a_n(x,|\nabla u^n|(X))\nabla u^n(X))=F_{n}(x, u^1(X),..., u^n(X)), where $x\in \R^m$ with $1\leq m< N$, $X=(x,y)\in \R^m\times \R^{N-m}$, and $F_{1},..., F_{n}$ are the derivatives with respect to $\xi^1,..., \xi^n$ of some $F=F(x,\xi^1,..., \xi^n)$ such that for any $i=1,..., n$ and any fixed $(x,\xi^1,..., \xi^{i-1},\xi^{i+1},..., \xi^n)\in \R^m\times \R^{n-1}$ the map $\xi^i\to F(x,\xi^1,...,\xi^i,..., \xi^n)$ belongs to $C^2(\R)$. We obtain a Poincar\'e-type formula for the solutions of the system and we use it to prove a symmetry result both for stable and for monotone solutions.