Concentration of low energy extremals: Identification of concentration points
Martin Flucher, Adriana Garroni,and Stefan Müller
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Submission date: 11. Mar. 2001
published in: Calculus of variations and partial differential equations, 14 (2002) 4, p. 483-516
DOI number (of the published article): 10.1007/s005260100112
MSC-Numbers: 35J20, 35B40
Keywords and phrases: variational problem, concentration, robin function
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We study the variational problem
where , , is a bounded domain, and F satisfies and is upper semicontinuous. We show that to second order in the value only depends on two ingredients. The geometry of enters through the Robin function (the regular part of the Green's function) and F enters through a quantity which is computed from (radial) maximizers of the problem in . The asymptotic expansion becomes
Using this we deduce that a subsequence of (almost) maximizers of must concentrate at a harmonic center of , i.e., , where is a minimum point of .