Overdetermined problems involving the fractional Laplacian

Abstract: Overdetermined problems are a class of partial differential boundary value problem where "too many" boundary conditions are imposed on the solution. Such problems appear in physics, biology, and other applied sciences due to their strong connection with shape optimisation. The mathematical objective is to understand and classify the regions that admit solutions that satisfy all the required properties.

A classical result in this area is Serrin's problem, which says that a bounded domain in $\mathbb R^n$ that admits a function with constant Laplacian, zero Dirichlet data, and constant Neumann data must be a ball. We will consider two generalisations of this problem, both driven by the fractional Laplacian, and give an overview of the current literature and open problems.

Mathematics

Audience: advanced learners

Barcelona Mathematics Informal Seminar (SIMBa)

Series comments: SIMBa is a youth mathematics seminar organized by graduate students of the Barcelona area. It is aimed towards graduate and last course undergraduate students. Our goals are divulging the knowledge from different branches of mathematics for those interested and promote networking between the attendants.

This seminar is backed by the Faculty of Mathematics and Computer Science at Universitat de Barcelona, Faculty of Mathematics and Statistics at Universitat Politècnica de Catalunya, the Department of Mathematics from Universitat Autònoma de Barcelona, CRM, IMUB and BGSMath.