Existence and Regularity for an Energy Maximization Problem in Two Dimensions
Spyridon Kamvissis and Evguenii Rakhmanov
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Submission date: 30. Sep. 2004 (revised version: January 2005)
published in: Journal of mathematical physics, 46 (2005) 8, art-no. 083505
DOI number (of the published article): 10.1063/1.1985069
Keywords and phrases: semiclassical nls, maximal equilibrium energy problem, s-curve
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We consider the variational problem of maximizing the weighted equilibrium Green's energy of a distribution of charges free to move in a subset of the upper half-plane, under a particular external field. We show that this problem admits a solution and that, under some conditions, this solution is regular in some strictly defined sense: it is an S-curve. The above problem appears in the theory of weak dispersive limits of integrable equations. In particular, its solution provides a justification of a crucial step in the asymptotic theory of steepest descent for the associated Riemann-Hilbert problems.