Uniqueness Results for Mean Field Equations with Singular Data

Abstract

We prove uniqueness of solutions for mean field equations [10 Caglioti , E. , Lions , P.-L. , Marchioro , C. , Pulvirenti , M. ( 1995 ). A special class of stationary flows for two dimensional Euler equations: a statistical mechanics description. II . Comm. Math. Phys. 174 : 229 – 260 .[Crossref], [Web of Science ®] , [Google Scholar]] with singular data [5 Bartolucci , D. , Tarantello , G. ( 2002 ). Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory . Comm. Math. Phys. 229 : 3 – 47 .[Crossref], [Web of Science ®] , [Google Scholar]], 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 Chang , S. Y. A. , Chen , C. C. , Lin , C. S. ( 2003 ). Extremal functions for a mean field equation in two dimension . New Stud. Adv. Math. 2 : 61 – 93 . [Google Scholar]]. 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 Chang , S. Y. A. , Chen , C. C. , Lin , C. S. ( 2003 ). Extremal functions for a mean field equation in two dimension . New Stud. Adv. Math. 2 : 61 – 93 . [Google Scholar]].