Seminari del Dipartimento

Analisi Matematica

This talk is concerned with the singular mean field problem of Liouville type on compact surfaces. This equation appears in the prescribed Gaussian curvature problem in Geometry and also arises in the Electroweak Theory and in the abelian Chern-Simons-Higgs model in Physics. We will focus on the existence and compactness of solutions for this type of problems, which have been extensively studied for positive potentials. However, the case of sign changing potentials has not been much considered in the literature. For the latter case, we establish some results on the solvability by means of variational methods, considering also the case of singularities with negative order. In addition, we will derive a suitable compactness criterion. Joint works with Francesca De Marchis (Roma I) and David Ruiz (Granada).