SIMBa: Overdetermined problems involving the fractional Laplacian

Dimecres 18 d’octubre se celebrarcelebrarà una xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Conferenciant: Jack Thompson.
Universitat: The University of Western Australia.

Data: Dimecres, 18 d’octubre de 2023.
Hora: 13:00, pausa pel cafè; 13:20, xerrada.
Lloc: UB i Zoom.
Idioma: Anglès.

Títol: Overdetermined problems involving the fractional Laplacian.

Resum: 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.