Uniqueness Results for Mean Field Equations with Singular Data

Abstract

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].

Keywords:

• Mean field equation
• Singular data
• Uniqueness results

Mathematics Subject Classification:

• 35J60
• 35J99
• 35P15
• 35R05