Recent progress on Overdetermined Elliptic Problems

In this talk we will survey recent progress on the Beresticky-Caffarelli-Nirenberg Conjecture in Space Forms; that is, let $\Omega$ be an open connected domain of a complete connected Riemannian manifold ($M,g$) and consider the OEP given by
\begin{cases} \Delta u + f(u)=0& \mbox{in}\ \Omega, \\ u>0& \mbox{in}\ \Omega,\\ u=0& \mbox{on}\ \partial\Omega,\\ \langle \triangledown u, \overrightarrow{\eta} \rangle g=\alpha& \mbox{on}\ \partial\Omega,\\ \label{eq:0.1}\tag{0.1} \end{cases} \begin{cases} \Delta u + f(u)=0& \mbox{in}\ \Omega, \\ u>0& \mbox{in}\ \Omega,\\ u=0& \mbox{on}\ \partial\Omega,\\ \langle \triangledown u, \overrightarrow{\eta} \rangle g=\alpha& \mbox{on}\ \partial\Omega,\\ \label{eq:0.1}\tag{0.1} \end{cases} @where $\overrightarrow{\eta}$ is the unit outward normal vector along $ \partial\Omega,\alpha$ a negative constant and $f : \mathbb{R}\rightarrow\mathbb{R}$ is a continuous function. A domain $\Omega \subset M$ that supports a solution to (0.1)\eqref{eq:0.1} is called an $f$--extremal domain. Berestycki, Caffarelli and Nirenberg considered the problem (0.1)\eqref{eq:0.1} when $\Omega \subset \mathbb{R}^n$ is an unbounded domain and also its complement, such problem appears naturally in the regularity of free boundary solutions at a boundary point. They formulated the following:

BCN conjecture:If $f$ is Lipschitz, $\Omega\subset\mathbb{R}^n$ is a smooth (in fact, Lipschitz) connected domain with $\mathbb{R}^n\backslash\Omega$ connected where the OEP (0.1) admits a bounded solution, then $\Omega$ is either a ball, a half-space, a cylinder $\mathbb{B}^k\times\mathbb{R}^{n-k} (\mathbb{B}^k$ is a ball of $\mathbb{R}^n)$ or the complement of one of them.

P. Sicbaldi [7] gave a counterexample of the BCN conjecture when $n\geq3$. Nevertheless, the BCN conjecture motivated interesting works. Recently, important contributions have been made in dimension $n = 2$. First, Ros-Sicbaldi [4] exploited the analogy between OEPs and constant mean curvature surfaces (in short, CMC surfaces) which allowed them to prove the BCN conjecture in dimension 2 under some extra hypothesis. Second, Ros-Ruiz-Sicbaldi [5] proved that the BCN conjecture is true in dimension 2 for unbounded domains whose complement is unbounded, such domain must be a half-space. Also, Ros-Ruiz-Sicbaldi [6] constructed exteriors domains different from the exterior of a geodesic ball in $\mathbb{R}^2$ for particular choices of the Lipschitz function f, this gives a counterexample to the BCN conjecture in $\mathbb{R}^2$ in all its generality.

In [1, 2], we extended the above results for extremal domains in the Hyperbolic Space; exploding the analogy with properly embedded CMC hypersurfaces in the Hyperbolic Space; mostly using Serrin’s Reflection (the analogous to Alexandrov’s Reflection for hypersurfaces) and certain type curvature estimates.

In [3], we showed uniqueness for overdetermined elliptic problems defined on topological disks on Riemannian surfaces. To do so, we adapt the G´alvez-Mira generalized Hopf-type 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 $f(x)\geq xf^{\prime}(x)$ for any $x\in\mathbb{R}^{*}_{+}$, we proved 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 $\frac{\partial u}{\partial u}=cte$ 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 in $\mathbb{S}^2$.

In the first hour; we will overview the above results and their connection to CMC hypersurfaces. In the second hour of the seminar, we will focus on the proof of the main results in [1, 2] based on Serrin’s reflection and curvature estimates; and the main technique in [3], which is a Hopf’s type Theorem for OEPs.

References

[1] J. M. Espinar, A. Farina and L. Mazet, $f$--extremal Domains in Hyperbolic Space. Preprint.
[2] J. M. Espinar and J. Mao, Extremal Domains on Hadamard Manifolds. To appear in J. Diff. Eq.
[3] J. M. Espinar and L. Mazet, Characterization of f-extremal disks. To appear in J. Diff. Eq.
[4] A. Ros and P. Sicbaldi, Geometry and topology of some overdetermined elliptic problems, J. Differential Equations, 255 (2013) 951–977.
[5] A. Ros, D. Ruiz and P. Sicbaldi, A rigidity result for overdetermined elliptic problems in the plane, Comm. Pure And Appl. Math., 70} (2017) no. 7}, 1223–1252.
[6] A. Ros, D. Ruiz and P. Sicbaldi, Solutions to overdetermined elliptic problems in nontrivial exterior domains. To appear in JEMS.
[7] P. Sicbaldi, New extremal domains for the first eigenvalue of the Laplacian in flat tori, Calc. Var. Partial Differential Equations, 37 (2010) 329–344.