**
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

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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