Existence and Regularity for an Energy Maximization Problem in Two Dimensions

TL;DR

This paper proves the existence and regularity of solutions for an energy maximization problem involving charges in a half-plane, with implications for nonlinear Schrödinger equations and Riemann-Hilbert problems.

Contribution

It establishes the existence and regularity of solutions, showing that under certain conditions, solutions are S-curves, advancing the understanding of energy maximization in complex analysis.

Findings

Solution exists and is regular under specified conditions.

Solutions are characterized as S-curves in the sense of Gonchar-Rakhmanov.

Results support the asymptotic analysis of nonlinear Schrödinger equations.

Abstract

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 Schr\"odinger 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.