Limit solutions of the Chern-Simons equation

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 $μ^# \le μ$ and we derive an explicit formula of $μ^#$ 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.