Vol. LXXVI, 1 (2007)
p. 77 - 83

Overdetermined Problems and the p-Laplacian
B. Kawohl

Abstract.  In this lecture I report on essentially two results for overdetermined boundary value problems and the p-Laplace operator. The first one is joint work with H. Shahgholian on Bernoulli type free boundary problems that model for instance galvanization processes. For this family of problems the limits p ® ¥ and p ® 1 lead to interesting analytical and surprising geometric questions. In particular for the case p ® 1 I add more recent results, that are not contained in [12]. The second one is joint work with F. Gazzola and I. Fragala [6]. It provides an alternative and more geometric proof of Serrin's seminal symmetry result for positive solutions to overdetermined boundary value problems. As a byproduct I give an analytical proof for the geometric statement that a closed plane curve of curvature not exceeding K must enclose a disk of radius 1/K.

Keywords: Overdetermined boundary problem, free boundary, Bernoulli problem, symmetry of solutions, degenerate elliptic operators.

AMS Subject classification:  35J60, 35J70, 49J45, 35R35, 35B50.

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Acta Mathematica Universitatis Comenianae
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Comenius University
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