

Preprint 81/2001
Concentration phenomena for the volume functional in unbounded domains: Identification of concentration points
Adriana Garroni and Stefan Müller
Contact the author: Please use for correspondence this email.
Submission date: 09. Nov. 2001
Pages: 19
published in: Journal of functional analysis, 199 (2003) 2, p. 386-410
DOI number (of the published article): 10.1016/S0022-1236(02)00062-9
Bibtex
MSC-Numbers: 35J20, 35B40
Keywords and phrases: variational problem, concentration, robin function, unbounded domains
Download full preprint: PDF (413 kB), PS ziped (182 kB)
Abstract:
We study the variational problem
in possibly unbounded domains , where
,
and F satisfies
and is upper semicontinuous. Extending earlier results for bounded domains we show that (almost) maximizers of
concentrate at a harmonic center, i.e. a minimum point of the Robin function
(the regular part of the Green function restricted to the diagonal). Moreover we obtain the asymptotic expansion
where and
depend only on F but not on
and can be computed from radial maximizers of the corresponding problem in
. The crucial point is to find a suitable definition of
. Interestingly the correct definition may be different from the lower semicontinuous extension of
to
, at least for
.