3. Geometry of Steklov eigenfunctions

Eigenfunctions on the ellipse
In a recent paper, it was seen that numerically, the nodal set of Steklov eigenfunctions doesnâ€™t become dense on the interior of an ellipse. This would be a counterexample to an usual conjecture. Can one prove that this actually happens by calculating explicitly the eigenfunctions of the ellipse?Problem 3.1.
[Oscar Bruno] Calculate explicitly the eigenfunctions of the Steklov problem on the ellipse 
Obtaining the domain from polynomial eigenfunctions
Conjecture 3.2.
[Mihai Putinar] If all Steklov eigenfunctions are polynomials on a planar domain then that domain is a disk. 
Obtaining the domain from analytic continuation
Problem 3.3.
[Dima Khavinson] Suppose that all Steklov eigenfunctions of a smooth planar domain can be continued harmonically across the boundary to the whole plane. Does it follow that this domain is a disk?
Cite this as: AimPL: Steklov eigenproblems, available at http://aimpl.org/stekloveigen.