# Remarks on the thin obstacle problem and constrained Ginibre ensembles

## TL;DR

This paper links the constrained Ginibre eigenvalue problem to a thin obstacle problem, proving well-posedness in a Sobolev space and advancing understanding of eigenvalue distributions with constraints.

## Contribution

It establishes the well-posedness of the obstacle problem in H^1(R^2), improving prior results limited to local Sobolev spaces, and relates eigenvalue constraints to potential theory.

## Key findings

- The obstacle problem is well posed in H^1(R^2).
- The coincidence set has two well-separated components.
- The results improve previous H^1_{loc} results.

## Abstract

We consider the problem of constrained Ginibre ensemble with prescribed portion of eigenvalues on a given curve $\Gamma\subset \mathbb R^2$ and relate it to a thin obstacle problem. The key step in the proof is the $H^1$ estimate for the logarithmic potential of the equilibrium measure. The coincidence set has two components: one in $\Gamma$ and another one in $\mathbb R^2\setminus \Gamma$ which are well separated. Our main result here asserts that this obstacle problem is well posed in $H^1(\mathbb R^2)$ which improves previous results in $H^1_{loc}(\mathbb R^2)$.