Razvan Teodorescu: Exceptional domains for over-determined boundary-value problems

Is it possible to find an unbounded domain D and a harmonic function on it with null Dirichlet data and constant Neumann data almost everywhere on its boundary?

A superficial analysis would identify only (1) the exterior of a ball and (2) half-spaces as candidates for D (domains which are called "exceptional"), yet in two dimensions another non-trivial solution exists. A complete classification of exceptional domains is still an open problem, leading to questions relevant not only to analysis, but also to differential geometry and operator theory. Perhaps even more intriguing are the emerging connections with the Loewner equation and Dyson random-matrix ensemble.

Work done in collaboration with D. Khavinson and E. Lundberg.