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## 4. Shape optimisation for Steklov

1. ### Homogeneisation of the Steklov-Neumann problem

#### Problem 4.1.

[Nilima Nigam] Let $\Omega$ be a domain in $\mathbb R^2$ and fix a fraction of the perimeter. Consider partitions of its boundary into $\Gamma = \Gamma_N \cup \Gamma_S$. Put Neumann boundary conditions on $\Gamma_N$ and Steklov on $\Gamma_S$.

a) How should one separate the boundary to maximise the first non-zero eigenvalue?

b) What happens under homogeneisation?

c) What happens if one instead puts a weight on the boundary, i.e. amongst all functions $\rho$ on the boundary with fixed integal and value between $0$ and $1$, find the $\rho$ that optimises $\sigma_1$ in the eigenvalue problem $$\begin{cases} \Delta u = 0 \\ \frac{\partial u }{\partial n} = \sigma \rho u \end{cases} .$$
• ### Symmetry of Steklov optimisers

Numerical work from E. Akhmetgaliev, C. Y. Kao and B. Osting has indicated that the maximiser in planar domains for the $k$th Steklov eigenvalue under area constraint exhibits $2\pi/k$ rotation symmetries.

#### Problem 4.2.

[Iosif Polterovich] Show analytically that the actual maximisers to the Steklov problem do exhibit those symmetries.
• ### Eigenvalue degeneration for Riemannian metrics

#### Problem 4.3.

[Ailana Fraser] In dimension $d = 3$, can one find a sequence of Riemannian metrics such that both the boundary and interior volume stay bounded but such that $\sigma_1 \to \infty$.
• ### Eigenvalue optimisation in dimension $2$

Weinstock’s theorem shows that $\sigma_1 |\partial \Omega| \le 2 \pi$ and a paper of Fraser and Schoen shows that for any Planar domain $\sigma_1 |\partial \Omega| \le 4 \pi$. Are these bounds sharp?

#### Problem 4.4.

[Jean Lagacé and Iosif Polterovich] a) Is the $4 \pi$ bound sharp? If not, find a sharp bound.

b) Find the maximising domain or show that such a domain does not exist.
• ### Shape optimisation in the Riemannian setting under volume constraint.

#### Problem 4.5.

[Misha Karpukhin] Let $\mathbb B^d$ be a ball of unit volume with metric $g$ in the conformal class of the Euclidean metric.

Is it true that the Euclidean ball maximises $\sigma_1$?

Is $\sigma_1$ bounded under such constraint?
• ### Pólya-Szegö conjecture for the Steklov problem

#### Problem 4.6.

[Édouard Oudet] Do regular $n$-gons maximise $\sigma_1$ among all $n$-gons for given area or perimeter?
There is numerical evidence for triangles (Dominguez, Nigam and Shahriari).

Amongst rectangles, the square optimises (Girouard, Lagacé, Polterovich, Savo)?

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