$\newcommand{\Cat}{{\rm Cat}}$ $\newcommand{\A}{\mathcal A}$ $\newcommand{\freestar}{ \framebox[7pt]{\star} }$

## 5. Misc.

1. ### Eigenvalue multiplicity in dimension two

#### Problem 5.1.

[Richard Laugesen and Leonid Parnovski] a) Prove that the spectrum in the sloshing problem is always simple (Conjecture of Kozlov-Kuznetsov-Motygin)

b) Does there exist a simply connected planar domain with a Steklov eigenvalue of multiplicity greater or equal to $3$ If you can, are the multiplicty uniformly bounded?
1. Remark. [Braxton Osting] Numerically, it seems like optimisers for the $k$th Steklov eigenvalue have multiplicity $3$ (paper of E. Akhmetgaliyev, C. Y. Kao and B. Osting.)
• ### Isospectrality

One can construct from laplace-isospectral manifolds without boundary pairs of Steklov isospectral domains. Similarly, in higher dimensions it is possible to find two different metrics on balls that are Steklov isospectral in higher dimension. Can this be extended to euclidean domains, or to manifolds in dimension $2$.

#### Problem 5.2.

[Leonid Parnovski] a) In any dimension, either find two Euclidean domains with the same Steklov spectrum or show that this is impossible.

b) In dimension 2, the same question for surfaces with boundary.
• ### Spectrum of infinite domains

Consider a ball with Steklov boundary condition with a thin infinite cylinder attached with Neumann boundary condition. What can one say about the spectrum in this case?

#### Problem 5.3.

[Stephen Shipman] a) Does the analytic continuation of the resolvent have poles? If so, characterize them.

b) What is the convergence of the spectrum for truncated domains converging to this infinite domain?
• ### Inverse problems for polygons

Spectral determination is a classic problem in spectral geometry. For planar domains with smooth boundary, it is hopeless to do so with only spectral asymptotics. Can one do it in the singular case.

#### Problem 5.4.

[Leonid Parnovski] Reconstruct a polygon from Steklov spectral asymptotics modulo $o(1)$.
• ### Can one hear the genus of a surface from its Steklov spectrum

Determining geometric invariants from the spectrum is a decades old problem in spectral geometry. What can one say knowing the Steklov spectrum

#### Problem 5.5.

[Iosif Polterovich and Leonid Parnovski] a) Given a Riemannian surface with boundary, can one obtain its genus from the Steklov spectrum?

b)? Given a closed surface embedded in $\mathbb{R}^3$, can one determine the genus of the boundary from the Steklov problem in its interior?

Cite this as: AimPL: Steklov eigenproblems, available at http://aimpl.org/stekloveigen.