The exceptional nonlocal geometry of overdetermined boundary conditions

Abstract:

Overdetermined boundary conditions arise in the search of

optimal shapes for a broad range problems e.g. in fluid mechanics, the

theory of elasticity, electrostatics and integral geometry. Due to their

relevance, the resulting overdetermined boundary value problems are

addressed in prominent conjectures. The Berestycki-Caffarelli-Nirenberg

conjecture from 1997, disproved by Sicbaldi in 2010, has lead to various

recent results on the existence and classification of extremal unbounded

domains. These unbounded optimal shapes can be regarded as analogues of

constant mean curvature surfaces governed by nonlocal effects.

Schiffer’s conjecture, and the related Pompeiu problem in integral

geometry from 1929, is still open. In my talk, I will discuss a choice

of classical and recent results on overdetermined boundary value problems.