Shape optimization with surface interactions

June 17 to June 21, 2019

at the

American Institute of Mathematics, San Jose, California

organized by

Evans Harrell, David Krejcirik, and Vladimir Lotoreichik

Original Announcement

This workshop will be devoted to identifying and attacking "hot" open problems in the spectral shape optimization characterized by an interplay between the geometry and singularly supported potentials. The models considered include but are not limited to Robin Laplacians, Schreodinger operators with Dirac-delta interactions on manifolds, and magnetic Hamiltonians modelling surface superconductivity.

The organization of the workshop is motivated by current open problems in spectral geometry. An example of such an open problem is a generalization of the well-known geometric fact that among all domains of fixed area the disk has the smallest perimeter. This geometric fact was anticipated in ancient times, but a rigorous proof appeared only in the 19th century. The more recent physical counterpart that among all planar membranes of a given area the circular membrane produces the lowest fundamental tone has had an interesting history, too. It took a half-century to establish the result for membranes with fixed edges and more than hundred years for more general repulsive boundary conditions. As the latest progress in this research field, there is an interesting observation that the disk is no more the optimizer for an analogous problem with attractive boundary conditions, and the optimal geometry still remains unknown in that situation.

In the problem above, the shape of the membrane plays the role of geometry and the type of boundary conditions realizes diverse curve-supported interactions. Related open problems involve higher dimensions, different constraints, interface conditions on submanifolds, optimization of other spectral quantities coming from various fields of modern physics, etc.

The main topics for the workshop are:

Geometric optimization of eigenvalues of elliptic operators (Bareket's conjecture).
Qualitative properties of eigenfunctions (nodal-line and hot-spots conjectures).
Singular interactions (Dirac-delta potential, interface conditions, strong magnetic field).

This workshop aims to bring together experts in spectral theory, harmonic analysis, partial differential equations, geometric analysis, and mathematical physics, whose areas of expertise complement and enrich each other in order to make progress in the topics listed above.