Solutions to the overdetermined boundary problem for semilinear equations with position-dependent nonlinearities

TL;DR

This paper demonstrates the existence of nontrivial solutions for overdetermined boundary problems involving semilinear equations with position-dependent nonlinearities on Euclidean spaces and Riemannian manifolds, also deriving symmetry results.

Contribution

It extends the understanding of overdetermined boundary problems to include position-dependent nonlinearities and nonconstant curvature manifolds, providing new existence and symmetry results.

Findings

Existence of solutions on Euclidean space and Riemannian manifolds.

Rigidity and partial symmetry results for solutions.

Applicability to nonconstant curvature manifolds.

Abstract

We show that a wide range of overdetermined boundary problems for semilinear equations with position-dependent nonlinearities admits nontrivial solutions. The result holds true both on the Euclidean space and on compact Riemannian manifolds. As a byproduct of the proofs we also obtain some rigidity, or partial symmetry, results for solutions to overdetermined problems on Riemannian manifolds of nonconstant curvature.