An Overdetermined Neumann boundary value problem with a general driving force

TL;DR

This paper proves the existence of specific nontrivial domains in a manifold where an overdetermined Neumann boundary value problem admits solutions, extending previous results to nonlinear functions and more general domains.

Contribution

It introduces a method to construct domains with nonconstant curvature for overdetermined Neumann problems, generalizing prior linear cases to nonlinear functions g.

Findings

Existence of solutions in constructed domains for some ta > 0

Domains have nonconstant principal curvature, not isoparametric or homogeneous

Method applies to both linear and nonlinear functions g

Abstract

In this paper, we prove the existence of a family of non trivial compact subdomains $\O$ in the manifold $\mathcal{M}=\R^N\times \R/2\pi\Z, N\geq 2$ for which the overdetermined Neumann boundary value problem \begin{align}\label{Neumann1} \left \{ \begin{aligned} $-\D w&=\mu g(w) && \qquad \text{in $ \Omega$,}$ \frac{\partial w}{\partial\eta} &=0 &&\qquad \text{on $\partial \Omega,$} w&=c\ne 0 &&\qquad \text{on $\partial \Omega$,} \end{aligned} \right. \end{align} admits solutions for some $\mu > 0$ and a $C^{1, \alpha}$ function $g:\R \rightarrow \R.$ The domains we construct have nonconstant principal curvature, and therefore are not isoparametric nor homogeneous. The argument we develop applies for both linear and non-linear functions $g$. By this, we generalise a recent result obtained by Fall, Weth and the first named author in \cite{Fall-MinlendI-Weth4}, where the overdetermined Neumann eigenvalue problem for the Laplacian was considered.