ACTA MATHEMATICA UNIVERSITATIS COMENIANAE
Vol. LXXVI, 1 (2007)
p. 77 - 83
Overdetermined Problems and the p-Laplacian
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 .
The second one is joint work with F. Gazzola and I. Fragala .
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.
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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