Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.
Existence and Regularity for an Energy Maximization Problem in Two Dimensions
Spyridon Kamvissis and Evguenii Rakhmanov
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.