Remarks on the thin obstacle problem and constrained Ginibre ensembles

We consider the problem of constrained Ginibre ensemble with prescribed portion of eigenvalues on a given curve Γ⊂R2 and relate it to a thin obstacle problem. The key step in the proof is the H1 estimate for the logarithmic potential of the equilibrium measure. The coincidence set has two components: one in Γ and another one in R2∖Γ which are well separated. Our main result here asserts that this obstacle problem is well posed in H1(R2) which improves previous results in H1loc(R2).