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Existence and regularity for an energy maximization problem in two dimensions. (English) Zbl 1110.81083
Summary: 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 an S-curve (in the sense of Gonchar-Rakhmanov). The above problem appears in the theory of the semiclassical limit of the integrable focusing nonlinear Schrödinger equation. In particular, its solution provides a justification of a crucial step in the asymptotic theory of nonlinear steepest descent for the inverse scattering problem of the associated linear non-self-adjoint Zakharov-Shabat operator and the equivalent Riemann-Hilbert factorization problem.

MSC:
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35Q40 PDEs in connection with quantum mechanics
49J20 Existence theories for optimal control problems involving partial differential equations
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