Limit solutions of the Chern-Simons equation

TL;DR

This paper studies the limits of solutions to the scalar Chern-Simons equation when solutions do not exist for certain measures, providing explicit formulas for the maximal solutions and analyzing the behavior for systems with multiple measures.

Contribution

It introduces a method to approximate measures and characterizes the largest solution in the absence of solutions, with explicit formulas and analysis for systems with multiple measures.

Findings

Solutions converge to a maximal measure solution $$

Explicit formula for the maximal measure $$ in terms of $$

Behavior of solutions depends on how measures charge singletons

Abstract

We investigate the scalar Chern-Simons equation $-\Delta u + e^u(e^u-1) = \mu$ in cases where there is no solution for a given nonnegative finite measure $\mu$. Approximating $\mu$ by a sequence of nonnegative $L^1$ functions or finite measures for which this equation has a solution, we show that the sequence of solutions of the Dirichlet problem converges to the solution with largest possible datum $\mu^# \le \mu$ and we derive an explicit formula of $\mu^#$ in terms of $\mu$. The counterpart for the Chern-Simons system with datum $(\mu, \nu)$ behaves differently and the conclusion depends on how much the measures $\mu$ and $\nu$ charge singletons.