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1 November 2015 Serrin’s overdetermined problem and constant mean curvature surfaces
Manuel Del Pino, Frank Pacard, Juncheng Wei
Duke Math. J. 164(14): 2643-2722 (1 November 2015). DOI: 10.1215/00127094-3146710

Abstract

For all N9, we find smooth entire epigraphs in RN, namely, smooth domains of the form Ω:={xRN|xN>F(x1,,xN1)}, which are not half-spaces and in which a problem of the form Δu+f(u)=0 in Ω has a positive, bounded solution with 0 Dirichlet boundary data and constant Neumann boundary data on Ω. This answers negatively for large dimensions a question by Berestycki, Caffarelli, and Nirenberg. In 1971, 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.

Citation

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Manuel Del Pino. Frank Pacard. Juncheng Wei. "Serrin’s overdetermined problem and constant mean curvature surfaces." Duke Math. J. 164 (14) 2643 - 2722, 1 November 2015. https://doi.org/10.1215/00127094-3146710

Information

Received: 15 October 2013; Revised: 9 November 2014; Published: 1 November 2015
First available in Project Euclid: 26 October 2015

zbMATH: 1342.35188
MathSciNet: MR3417183
Digital Object Identifier: 10.1215/00127094-3146710

Subjects:
Primary: 35J25
Secondary: 35J67

Rights: Copyright © 2015 Duke University Press

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Vol.164 • No. 14 • 1 November 2015
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