Preprint 81/2001

Concentration phenomena for the volume functional in unbounded domains:
Identification of concentration points

Adriana Garroni and Stefan Müller

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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
MSC-Numbers: 35J20, 35B40
Keywords and phrases: variational problem, concentration, robin function, unbounded domains
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We study the variational problem
in possibly unbounded domains tex2html_wrap_inline606, where tex2html_wrap_inline608, tex2html_wrap_inline610 and F satisfies tex2html_wrap_inline614 and is upper semicontinuous. Extending earlier results for bounded domains we show that (almost) maximizers of tex2html_wrap_inline616 concentrate at a harmonic center, i.e. a minimum point of the Robin function tex2html_wrap_inline618 (the regular part of the Green function restricted to the diagonal). Moreover we obtain the asymptotic expansion
where tex2html_wrap_inline620 and tex2html_wrap_inline622 depend only on F but not on tex2html_wrap_inline626 and can be computed from radial maximizers of the corresponding problem in tex2html_wrap_inline628. The crucial point is to find a suitable definition of tex2html_wrap_inline630. Interestingly the correct definition may be different from the lower semicontinuous extension of tex2html_wrap_inline632 to tex2html_wrap_inline634, at least for tex2html_wrap_inline636.

03.07.2017, 01:40