An Overdetermined Neumann boundary value problem with a general driving force

Abstract

In this paper, we prove the existence of a family of non trivial compact subdomains $\tilde{A}˜$ in the manifold $\mathcal{M}={\mathbb{R}}^N\times \mathbb{R}/2\ddot{I}€\mathbb{Z},N\ge 2$ for which the overdetermined Neumann boundary value problem \begin{align}\label{Neumann1} \left \{ \begin{aligned} $-\D w&=μg(w) && \qquad \text{in $ Ω$,}$ \frac{\partial w}{\partialη} &=0 &&\qquad \text{on $\partial \hat{I}\text{c◯},$} w&=c\ne 0 &&\qquad \text{on $\partial \hat{I}\text{c◯}$,} \end{aligned} \right. \end{align} admits solutions for some $\hat{I}¼>0$ and a $C^{1,\hat{I}\pm }$ function $g:\mathbb{R}\to \mathbb{R}.$ 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.