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

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

2008-06-18

arxiv.org/abs/0806.2952v1

Mathematics

Analysis of PDEs

24 pages

Scientific paper

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.

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