Geometry and topology of some overdetermined elliptic problems
Antonio Ros,Pieralberto Sicbaldi +1 more
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TLDR
In this article, necessary conditions on the geometry and the topology of domains in R2 that support a positive solution to a classical overdetermined elliptic problem are studied. But these conditions are not applicable to higher dimensions.About:
This article is published in Journal of Differential Equations.The article was published on 2013-09-01 and is currently open access. It has received 47 citations till now. The article focuses on the topics: Geometry and topology & Overdetermined system.read more
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Splitting Theorems, Symmetry Results and Overdetermined Problems for Riemannian Manifolds
TL;DR: In this article, a unified approach to splitting theorems, symmetry results and overdetermined elliptic problems is proposed, based on the existence of a stable solution to the semilinear equation on a Riemannian manifold with non-negative Ricci curvature.
Journal ArticleDOI
Classification of the Solutions to an Overdetermined Elliptic Problem in the Plane
TL;DR: In this paper, the authors classify the solutions to an overdetermined elliptic problem in the plane in the finite connectivity case by establishing a one-to-one correspondence between the solutions of this problem and a certain type of minimal surfaces.
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Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds
TL;DR: In this paper, a unified approach to splitting theorems, symmetry results and overdetermined elliptic problems is proposed, based on the existence of a stable solution to the semilinear equation $-\Delta u = f(u) on a Riemannian manifold with non-negative Ricci curvature.
Journal ArticleDOI
Serrin’s overdetermined problem and constant mean curvature surfaces
TL;DR: For all n ≥ 9, the authors showed that a bounded domain where such an overdetermined problem is solvable must be a ball, in analogy to a famous result by Alexandrov that states that an embedded compact surface with constant mean curvature (CMC) in Euclidean space must be either a sphere or a sphere.
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A rigidity result for overdetermined elliptic problems in the plane
TL;DR: In this paper, the authors obtained a partial answer to a question raised by H. Berestycki, L. Caffarelli and L. Nirenberg in 1997.
References
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Book ChapterDOI
Elliptic Partial Differential Equations of Second Order
Piero Bassanini,Alan R. Elcrat +1 more
TL;DR: In this paper, a class of partial differential equations that generalize and are represented by Laplace's equation was studied. And the authors used the notation D i u, D ij u for partial derivatives with respect to x i and x i, x j and the summation convention on repeated indices.
Journal ArticleDOI
Symmetry and related properties via the maximum principle
TL;DR: In this paper, the authors show that positive solutions of second order elliptic equations are symmetric about the limiting plane, and that the solution is symmetric in bounded domains and in the entire space.
Book
Eigenvalues in Riemannian geometry
TL;DR: The Dirichlet Heat Kernel for Regular Domains as mentioned in this paper is a heat kernel for non-compact manifolds that is based on the Laplacian on forms (LFP).
Journal ArticleDOI
Bifurcation from simple eigenvalues
TL;DR: In this article, a general version of the main problem of bifurcation theory, given p ϵ C, determine the structure of G−1{0} in some neighborhood of p.
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