
## 1. Singular Steklov-type problems

1. ### Convergence of the polygonal Steklov spectrum to the spectrum of a smooth domain

#### Problem 1.1.

[Nilima Nigam] Let $\mathbb{D}$ be a disk with $P_n$ a regular $n$-gon inscribed in $\mathbb{D}$ or circumscribing $\mathbb{D}$. Estimate the difference between their $k$th Steklov eigenvalues $$\left|\sigma_k(\mathbb{D}) - \sigma_k(P_n)\right|$$ in terms of $k$ and $n$.
• ### Oscillating behaviour of eigenfunctions

Furthermore, can we isolate in higher dimension the oscillating behaviour of the eigenfunctions from a slowly oscillating part that depends on the geometry of the boundary?

#### Problem 1.2.

[Oscar Bruno] Capture geometrically the oscillatory behaviour of eigenfunctions in higher dimension. Can we factor out the oscillatory behaviour from the eigenfunctions like it is done in dimension $2$?
• ### Steklov asymptotics for generalised cylinders and cones

In any dimension, in a recent paper of A. Girouard, J. Lagacé, I. Polterovich and A. Savo find two-terms asymptotics for the Steklov spectrum on cubes. Can one do the same for other similarly singular domains like cylinders or cones?

#### Problem 1.3.

[Leonid Parnovski] Calculate Steklov asymptotics for domains with corners, or specifically for cylinders and cones in term of the Laplace spectrum of the base manifold?
• ### The Steklov spectrum of a Helmholtz problem

The Steklov problem can be defined for other equations than the Poisson equation. Can we understand the spectrum in such cases?

#### Problem 1.4.

[Fioralba Cakoni] For a bounded function $\rho$, and fixed $\lambda$, consider the problem $$\begin{cases} \Delta u = \lambda \rho u \\ \frac{\partial u }{\partial n} = \sigma u \end{cases}$$

a) What is the relation between $\rho$ and the number of negative eigenvalues?

b) What are the asymptotics of the solutions $u_j$ corresponding to negative $\sigma_j$? as $\lambda$ converges to a fixed Dirichlet eigenvalue. Can we get further asymptotic terms?

c) (Leonid Parnovski) Can one describe asymptotics for the related sloping beach problem?

d) What happens when one allows for $\lambda$ or $\rho$ to be complex-valued?
• ### Heat trace asymptotics for the Steklov problem on orbifolds, polygons and polyhedra

One can find complete asymptotics for the Steklov problem on manifolds with smooth boundary. Can one do the same in those singular settings. This could allow one to get geometric invariants out of them.

#### Problem 1.5.

[Emily Dryden] a) Prove existence of heat trace asymptotics for orbifolds, polygons or polyhedrons.

b) Compute the heat trace coefficient in those settings.
• ### Weyl law for Lipschitz domain

#### Problem 1.6.

[M. Agranovich] Prove Weyl’s law for the Steklov problem on domains with Lipschitz boundary.
• ### Properties of the single and double layer operators

The Dirichlet-to-Neumann map is a combination of single-layer and double-layer operators. As such, understanding them means understanding the Steklov problem.

#### Problem 1.7.

[G. Rozenblyum] Explore the relation between the spectral properties of the Dirichlet-to-Neumann map, the single layer and the double layer operators in the presence of corners.

Does inverse simple layer have similar contributions from the corners as those exhibited in the Steklov problem
• #### Problem 1.8.

[Nilima Nigam] Can one find the same relations for the Helmholtz case? For the Steklov-Maxwell case?

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