Helena Nussenzveig Lopes (Rio de Janeiro)
“Vanishing viscosity, inviscid dissipation and anomalous dissipation”

Clément Mouhout (Cambridge)

Giuseppe Savaré (Bocconi)
“Variational principles for evolution problems”

Tobias Weth (Frankfurt)
“The rigidity and nonrigidity of overdetermined boundary value problems”

Overdetermined boundary value problems for elliptic PDE arise in the search of optimal shapes in a broad range of problems, e.g., in fluid mechanics, electrostatics, and the theory of elasticity. Due to their relevance, these problems are addressed in prominent conjectures. The Berestycki-Caffarelli-Nirenberg conjecture from 1997 has lead to numerous results on the existence and classification of extremal unbounded domains where associated overdetermined Dirichlet problems admit positive solutions. These unbounded optimal shapes can be regarded as analogues of constant mean curvature surfaces, but they are influenced by nonlocal effects. Schiffer’s conjecture addresses an overdetermined Neumann problem and is closely related to the Pompeiu problem in integral geometry. It is still open, despite recent progress in the functional analytic theory of overdetermined Neumann problems. In my lectures, I will present both classical and recent results on overdetermined boundary value problems and discuss underlying methods of independent interest.