Uniqueness Results for Mean Field Equations with Singular Data

We prove uniqueness of solutions for mean field equations [ 10 ] with singular data [ 5 ], arising in the analysis of two-dimensional turbulent Euler flows. In this way, we generalize to the singular case some uniqueness results obtained by Chang, Chen and the second author [ 11 ]. In particular, by using a sharp form of an improved Alexandrov-Bol's type isoperimetric inequality, we are able to exploit the role played by the singularities and then obtain uniqueness under weaker boundary regularity assumptions than those assumed in [ 11 ].