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OVERDETERMINED PROBLEMS AND THE p-LAPLACIAN

By B. Kawohl

Abstract

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. Fragalá [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. 1. Bernoulli problems It is well-known that minimizing the functional ∫ ( ) p 1 |∇v | p − 1 Ep(v) = p a p χ {v>0} dx R n on the set {v ∈ W 1,p (Rn); v ≡ 1onK} leads to the following Euler-Lagrange equation with overdetermined Bernoulli-type boundary condition ∆pup =div(|∇up | p−2 (1.1) ∇up) = 0 in {up> 0}\K

Topics: Key words and phrases. overdetermined boundary problem, free boundary, Bernoulli problem, symmetry of solutions, degenerate elliptic operators. 78 B. KAWOHL
Year: 2005
OAI identifier: oai:CiteSeerX.psu:10.1.1.212.5090
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