Symmetry for solutions of two-phase semilinear elliptic equations on hyperbolic space

TL;DR

This paper proves that bounded solutions to a semilinear elliptic equation on hyperbolic space, with symmetric boundary conditions, are necessarily one-dimensional, and establishes the existence of such solutions.

Contribution

It demonstrates symmetry reduction of solutions under certain conditions and constructs explicit one-dimensional solutions on hyperbolic space.

Findings

Solutions are symmetric and depend on one variable under specified conditions.

Existence of one-dimensional solutions with prescribed boundary behavior.

Conditions on the Lipschitz constant influence solution symmetry.

Abstract

Assume that $f(s) = F'(s)$ where $F$ is a double-well potential. Under certain conditions on the Lipschitz constant of $f$ on $[-1,1]$, we prove that arbitrary bounded global solutions of the semilinear equation $\Delta u = f(u)$ on hyperbolic space $\HH^n$ must reduce to functions of one variable provided they admit asymptotic boundary values on the infinite boundary of $\HH^n$ which are invariant under a cohomogeneity one subgroup of the group of isometries of $\HH^n$. We also prove existence of these one-dimensional solutions.