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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.