Preprint 7/2001

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
Pages: 28
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 tex2html_wrap_inline585, tex2html_wrap_inline587, is a bounded domain, tex2html_wrap_inline589 and F satisfies tex2html_wrap_inline593 and is upper semicontinuous. We show that to second order in tex2html_wrap_inline595 the value tex2html_wrap_inline597 only depends on two ingredients. The geometry of tex2html_wrap_inline599 enters through the Robin function tex2html_wrap_inline601 (the regular part of the Green's function) and F enters through a quantity tex2html_wrap_inline605 which is computed from (radial) maximizers of the problem in tex2html_wrap_inline607. The asymptotic expansion becomes
Using this we deduce that a subsequence of (almost) maximizers of tex2html_wrap_inline611 must concentrate at a harmonic center of tex2html_wrap_inline599, i.e., tex2html_wrap_inline615, where tex2html_wrap_inline617 is a minimum point of tex2html_wrap_inline601.

24.11.2021, 02:11