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
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 , 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 .