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Special Session 140: Symmetry and Overdetermined problems

A rigidity result for the overdetermined problems with the mean curvature of the graph of solutions operator in the plane

Yuanyuan Lian

Department of Mathematical Analysis, University of Granada
Spain

Co-Author(s): Yuanyuan Lian; Pieralberto Sicbaldi

Abstract:

Let

$\Omega \subset \mathbb{R}^2$ be a

$C^{1,\alpha}$ domain whose boundary is unbounded and connected. Suppose that

$f:[0,+\infty) \to \mathbb{R}$ is

$C^1$ and there exists a nonpositive prime

$F$ of

$f$ such that

$F(0)=\sqrt{2}/2-1$ . If there exists a positive bounded solution

$u\in C^3$ with bounded

$\nabla u$ to the overdetermined problem

$\begin{equation*} \left\{\begin{array} {ll} \mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) + f(u) = 0 & \mbox{in }\; \Omega,\ u= 0 & \mbox{on }\; \partial \Omega, \ \frac{\partial u}{\partial \vec{\nu}}=1 &\mbox{on }\; \partial \Omega, \end{array}\right. \end{equation*}$ we prove that

$\Omega$ is a half-plane. It means that a positive capillary graph whose mean curvature depends only on the height of the graph is a half-plane.

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