
## 3. Geometry of Steklov eigenfunctions

1. ### Eigenfunctions on the ellipse

In a recent paper, it was seen that numerically, the nodal set of Steklov eigenfunctions doesn’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.