Characterization of $f$-extremal disks

Abstract

We show uniqueness for overdetermined elliptic problems defined on topological disks $\Omega$ with $C^2$ boundary, i.e., positive solutions $u$ to $\Delta u+f(u)=0$ in $\Omega \subset (M^2,g)$ so that $u=0$ and $\frac{\partial u}{\partial \vec{\eta }}=cte$ along $\partial \Omega$, $\vec{\eta }$ the unit outward normal along $\partial \Omega$ under the assumption of the existence of a candidate family. To do so, we adapt the Gálvez-Mira generalized Hopf-type Theorem to the realm of overdetermined elliptic problem. When $(M^2,g)$ is the standard sphere ${\mathbb{S}}^2$ and $f$ is a $C^1$ function so that $f(x)>0$ and $f(x)\ge x{f}^{\prime }(x)$ for any $x\in {\mathbb{R}}_{+}^{\ast }$, we construct such candidate family considering rotationally symmetric solutions. This proves the Berestycki-Caffarelli-Nirenberg conjecture in ${\mathbb{S}}^2$ for this choice of $f$. More precisely, this shows that if $u$ is a positive solution to $\Delta u+f(u)=0$ on a topological disk $\Omega \subset {\mathbb{S}}^2$ with $C^2$ boundary so that $u=0$ and $\frac{\partial u}{\partial \vec{\eta }}=cte$ along $\partial \Omega$, then $\Omega$ must be a geodesic disk and $u$ is rotationally symmetric. In particular, this gives a positive answer to the Schiffer conjecture D for the first Dirichlet eigenvalue and classifies simply-connected harmonic domains, also called {\it Serrin Problem}) in ${\mathbb{S}}^2$.

The main modification is on page 17 (Corollary 4.5 and the paragraph before) where we explain that our result applies to the Allen-Cahn non linearity