Geometry and Topology of some overdetermined elliptic problems

Details Geometry and Topology of some overdetermined elliptic problems Geometry and Topology of some overdetermined elliptic problems

2012-02-23

arxiv.org/abs/1202.5167v1

Mathematics

Analysis of PDEs

Scientific paper

We prove some geometric and topological properties for regular (unbounded) domains $\Omega \subset \mathbb{R}^{2}$ that support a positive solution of the equation $\Delta u + f(u) = 0$, with 0 Dirichlet and constant Neumann boundary conditions, where $f$ has Lipschitz regularity. In particular, we prove that if $\Omega$ has finite topology and there exists a positive constant $R$ such that $\Omega$ does not contain balls of radius $R$, then any end of $\Omega$ stays at bounded distance from a straight line. As a corollary of this result, we prove that under the hypothesis that $\mathbb{R}^{2}\backslash \bar{\Omega}$ is connected and there exists a positive constant $\lambda$ such that $f(t)\geq \lambda t$ for all $t\geq0$, then $\Omega$ is a ball. We study also some boundedness and symmetry properties of the function $u$ and the domain $\Omega$. In particular, if $u$ is bounded and the domain lies in a half-plane, then we obtain that $\Omega$ is either a ball or a half-plane. Some of these geometric properties that we prove are established also for $\Omega \subset \mathbb{R}^{n}$.

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