Nontrivial solutions to the relative overdetermined torsion problem in a cylinder

Abstract

Given a bounded regular domain $\omega \subset {\mathbb{R}}^{N-1}$ and the half-cylinder $\Sigma =\omega \times (0,+\infty )$, we consider the relative overdetermined torsion problem in $\Sigma$, i.e. \[\left\{ \begin{array}{ll} Δ{u}+1=0 &\mbox{in $\Omega$},\newline \partial_ηu = 0 &\mbox{on ${\tilde{\Gamma }}_{\Omega }$},\newline u=0 &\mbox{on ${\Gamma }_{\Omega }$},\newline \partial_νu =c &\mbox{on ${\Gamma }_{\Omega }$}. \end{array} \right. \] where $\Omega \subset \Sigma$, ${\Gamma }_{\Omega }=\partial \Omega \cap \Sigma$, ${\tilde{\Gamma }}_{\Omega }=\partial \Omega \setminus {\Gamma }_{\Omega }$, $\nu$ is the outer unit normal vector on ${\Gamma }_{\Omega }$ and $\eta$ is the outer unit normal vector on ${\tilde{\Gamma }}_{\Omega }$. We build nontrivial solutions to this problem in domains $\Omega$ that are the hypograph of certain nonconstant functions $v:\overline{\omega }\to (0,+\infty )$. Such solutions can be reflected with respect to $\omega$, giving nontrivial solutions to the relative overdetermined torsion problem in a cylinder. The proof uses a local bifurcation argument which, quite remarkably, works for any generic base $\omega$.

14 pages