Seminar Analysis: Daniele Bartolucci (University of Rome Tor Vergata)

Critical Mean Field Equations on multiply connected domains

The uniqueness of solutions of the (Liouville) mean field-type equation on a simply connected domain and in the sub critical regime λ∈(0,8π) was first proved by T. Suzuki (1992).  This result has been later improved by S.Y.A. Chang, C.C. Chen and C.S. Lin (2003) [CCL] to cover the critical value λ∈(0,8π]. The case where the domain is not simply connected has been a long-standing open problem which we have finally solved in a recent paper in collaboration with C.S. Lin. Our proof is based on a new generalization of a P.D.E. version of the Alexandrov-Bol's isoperimetric inequality on multiply connected domains. Another delicate problem is to understand the existence/non-existence of solutions for this equation on multiply connected domains at the critical parameterλ=8π. Criticality here means that the variational functional whose critical points are solutions of the equation is not anymore coercive for λ=8π, which implies in particular in this situation that existence/non existence of solutions depend on the geometry of the domain. I will discuss our generalization of a result in [CCL] which yield necessary and sufficient conditions for the existence of solutions for the mean field equation at the critical parameter λ=8π.

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