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.

They conjectured that "exceptional domains" in the plane are one of three types, the exterior of the disk, a half-plane, or a certain nontrivial example. Dmitry Khavinson, Razvan Teodorescu, and I recently confirmed this conjecture (preprint) under some additional assumptions.

Open Problem: Our additional assumptions were that the domain is simply connected (after adding the point at infinity if necessary). We also assumed the boundary passes through infinity finitely many times and is smooth at infinity.

Thus, it is an open problem to remove these assumptions. Perhaps new ideas will come from quadrature domain theory, as these exceptional domains are nothing other than arc-length null quadrature domains, and this question is then about characterizing arc-length null quadrature domains. M. Sakai characterized area null quadrature domains without additional assumptions on the topology using a generalized Cauchy transform.

Remark about ideal fluid dynamics: These exceptional domains can be interpreted in terms of classical ideal fluid dynamics. Indeed, 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 exterior of the domain is an air bubble 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. The first two examples of exceptional domains correspond to trivial fluid flows, but the nontrivial example seems interesting:

(i) Exterior of a disk = rotational flow with single hollow vortex. (ii) Halfplane = uniform flow along side an infinite, flat bubble. (iii) Nontrivial example = two infinitely elongated corotating hollow vortices with one stagnation point and no singularities.

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