Serrin’s overdetermined problem and constant mean curvature surfaces

By Manuel del Pino, Frank Pacard, Jun-cheng Wei. 2013

64 citations recorded by the source corpus.

For all $N \geq 9$, we find smooth entire epigraphs in $\R^N$, namely smooth domains of the form $\Omega : = \{x\in \R^N\ / \ x_N > F (x_1,\ldots, x_{N-1})\}$, which are not half-spaces and in which a problem of the form $\Delta u + f(u) = 0 $ in $\Omega$ has a positive, bounded solution with 0 Dirichlet boundary data and constant Neumann boundary data on $\partial \Omega$. This answers negatively for large dimensions a question by Berestycki, Caffarelli and Nirenberg \cite{bcn2}. In 1971, Serrin \cite{serrin} proved 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 a sphere. In lower dimensions we succeed in providing examples for domains whose boundary is close to large dilations of a given CMC surface where Serrin's overdetermined problem is solvable.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/5defedc08507f2a705159930d3d9b6e773245a24

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In Applied Mathematics

1. D. Ruiz
2. A. Ros
3. P. Sicbaldi
4. W. H. Meeks
5. Xianhua Tang
6. Sitong Chen
7. Joaquín Pérez
8. Gongbao Li

Citations in the corpus

• A Rigidity Result for Overdetermined Elliptic Problems in the Plane (2015, background): “The domain in [13] has boundary close to a dilated Bombieri–De Giorgi– Giusti entire minimal graph [9].”
• Solutions to overdetermined elliptic problems in nontrivial exterior domains (2016, background): “Finally, in [8] Del Pino, Pacard and Wei consider problem (1.”