Abstract
For all , we find smooth entire epigraphs in , namely, smooth domains of the form , which are not half-spaces and in which a problem of the form in has a positive, bounded solution with Dirichlet boundary data and constant Neumann boundary data on . 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
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
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