A Free Boundary Problem for the Laplace Equation: Motivated by some questions in differential geometry, L. Hauswirth, F. H\'elein, and F. Pacard posed a problem to find domains (called "exceptional domains") that admit a "roof function", a positive harmonic function with zero Dirichlet data and constant Neumann data.

Exceptional domains in the plane at first seemed to be classified as one of three types: the exterior of the disk, a half-plane, or a certain nontrivial example. It turns out that there is another (in my opinion more even more interesting) infinitely-connected example that appeared in fluid dynamics literature in 1976.

To see how exceptional domains can be interpreted in terms of classical ideal fluid dynamics, view the roof function as a stream function for an ideal fluid flow. Then the zero Dirichlet data implies that the boundary is a stream line. Let us imagine that the holes (exterior of the domain) are air bubbles at constant pressure. Then the constant Neumann data implies by Bernoulli's law that the pressure exerted by the fluid on the air bubble is constant, thus the bubble pressure is in equilibrium with the Bernoulli's law pressure.


An infinite-genus exceptional domain with a periodic array of holes that can be viewed as air bubbles in a hollow vortex equilibrium.

Dmitry Khavinson, Razvan Teodorescu, and I recently made progress on classifying exceptional domains (preprint) using H^p theory.

Open Problem: We made assumptions on the topology of the domain. It is an open problem to remove this assumption.

Breaking news: M. Traizet has mostly settled this problem. He classified finitely-connected exceptional domains using a beautiful nontrivial correspondence to complete minimal surfaces embedded in R^3.

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