2. Eigenvalues
Questions about the optimization of eigenvalues of several operator, under geometric constraints.-
Curl-curl eigenvalue problem
Problem 2.1.
[Nilima Nigam] Let $\Omega$ be a smooth compact domain of $\mathbb{R}^3$, we define $(\vec{E_k},\lambda_k)_{k\geq 1}$ to be the eigenpair of the Maxwell equation in $\Omega$, as given by the equation
\begin{align*} \nabla \times \nabla \times \vec{E}_k & = \lambda_k \vec{E}_k \quad \text{in $\Omega$} \\ \nabla \cdot \vec{E}_k & = 0 \qquad \text{in $\Omega$} \\ \vec{E}_k \times \vec{\nu} & = 0 \qquad \text{on $\partial \Omega$} \end{align*} where $\vec{\nu}$ is the outward normal vector of $\Omega$. These eigenvalues and eigenvector may be defined variationally through the Rayleigh quotient \[\frac{\int_\Omega |\nabla \times \vec{E}|^2 \, dx}{\int_\Omega |\vec{E}|^2 \, dx}\] taken among vector fields $\vec{E}\in\mathcal{C}^1(\Omega,\mathbb{R}^3)$ verifying the constraints \[\nabla \cdot \vec{E}=0\text{ in }\Omega,\ \vec{E}\times \vec{\nu}=0\text{ in }\partial\Omega\] Few things seem to be known about the range of the first few eigenvalues of this operator depending on the geometric constraints on $\Omega$: Nilima Nigam raises the questions of the optimal shape for $\lambda_k(\Omega)$ for $k=1,2,3$, under either a simple volume constraint, or a volume constraint among convex sets, and whether the ball may be expected to be optimal or not. Note also that this problem is a simplified version of Maxwell’s equation, where permittivity (for instance) was not included. -
Curl eigenvalue
Problem 2.2.
[Iosif Polterovich] Let $\Omega \subset \mathbb{R}^3$ be a bounded Lipschitz set with volume $1$, consider the curl eigenvalue problem
\begin{align*} \nabla \times \vec{u} & = \mu \vec{u} \text{ on $\Omega$,} \\ \vec{\nu} \cdot \vec{u} & = 0 \text{ on $\partial \Omega$}.\\ \int_{\Omega}\vec u\cdot \vec w&=0 \text{ for any }\vec w\in L^2(\Omega)\text{ with }\nabla\times \vec w=0\\ \end{align*} More details on the definition may be found in [Enciso, Gerner, Peralta–Salas]. There exists a sequence of eigenpairs $(\mu_k,u_k)_{k\in\mathbb{Z}}$ such that \[...\leq \mu_{-2}(\Omega)\leq \mu_{-1}(\Omega)<0<\mu_1(\Omega)\leq \mu_2(\Omega)\leq ...\to\infty\]
We remind the variational definition \[\min(\mu_1(\Omega)^2,\mu_{-1}(\Omega)^2)=\inf\left\{\frac{\int_{\Omega}|\nabla \times u|^2}{\int_{\Omega}|u|^2}, \right\}\] where $u$ is taken in $\mathcal{C}^1(\Omega,\mathbb{R}^3)$ with the constraints $\vec{u}\cdot\vec{\nu}=0$ on the boundary, $\int_{\Omega}\vec{u}\cdot\vec{w}=0$ for any $\vec{w}$ with $\nabla\times\vec{w}=0$.
The goal is to minimize the first positive eigenvalue $\mu_1(\Omega)$ under volume constraint on $\Omega$. In the case where $\Omega$ is supposed to be convex, then it is known from [Enciso, Gerner, Peralta–Salas] that the ball is not optimal, and that there is an optimal domain that is not analytic. Without convexity constraint, the existence of optimal domain is only known (from [Enciso, Gerner, Peralta–Salas]) among uniformly Hölder sets, and remains to be investigated in general.
Moreover, it is known (from the works of the same authors) that if $\Omega$ is a $\mathcal{C}^{2,\alpha}$ optimal set, then $\Omega$ is not axisymmetric, and every connected component of $\partial\Omega$ is diffeomorphic to a torus. -
Exterior Robin problem
Problem 2.3.
[Lukas Bundrock] Let $n \geq 3$ and let $\Omega\subset\mathbb{R}^3$ be a smooth compact domain. For a given $\alpha \in \mathbb{R}$, we consider the Robin eigenvalue problem on the complement of $\Omega$, of unknown $(\lambda,u)$:
\begin{align*} - \Delta u & = \lambda u \qquad \text{in $\Omega^\text{ext} = \mathbb{R}^n \setminus \overline{\Omega}$,} \\ \frac{\partial u}{\partial \nu} & = \alpha u \qquad \text{on $\partial \Omega$.} \end{align*}
The problem has essential spectrum $[0,\infty)$, but does have some eigenvalues too, provided $\alpha<\alpha_*(n)<0$ where $\alpha_*(n)$ is some dimensionnal constant. The lowest eigenvalue is given by the Rayleigh quotient \[ \lambda_1^\alpha(\Omega^\text{ext}) = \min_{u \in W^{1,2}(\Omega^\text{ext})} \frac{\int_{\Omega^\text{ext}} |\nabla u|^2 \, dx + \alpha \int_{\partial \Omega} u^2 \, dS}{\int_{\Omega^\text{ext}} u^2 \, dx} . \] For $\alpha<0$, D. Krejcirik and V. Lotoreichik have shown that the ball maximizes $\lambda_1^\alpha(\Omega^\text{ext})$ in dimension $n=2$ among all smooth, bounded, simply connected open sets of given measure and among all smooth, bounded, simply connected open sets of given perimeter.
This is no longer true in higher dimensions for sufficiently negative $\alpha$: a counterexample is given by an ellipsoid, exploiting the asymptotic formula of $\lambda_1^\alpha(\Omega^{\text{ext}})$ that involves the maximum of the curvature of $\Omega$.
However the ball is still a local maximizer of $\lambda_1^\alpha(\Omega^\text{ext})$ among all nearly spherical domains of given measure. This situation raises two questions:- Does a global maximizer of $\lambda_1^\alpha(\Omega^\text{ext})$ (under volume constraint) exist in dimensions $n \geq 3$ ? If so, what shape does it have ?
- Since the ball is a local but not a global maximizer, does another critical domain exists as a consequence of Mountain Pass Theorem ? Must such a domain also be smooth ?
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Maximizing Steklov $\sigma_1$ on $N$-gons
Problem 2.4.
[Iosif Polterovich] One of the “well-known” conjecture of shape optimization is the optimization of $\lambda_1$ (first eigenvalue of the Laplacian with Dirichlet boundary condition) among polygonal sets of fixed measure. A question raised by Iosif Polterovich on which even less is known is for the Steklov eigenvalue: given some integer $n\geq 3$, what is the maximal value of the first Steklov eigenvalue $\sigma_1$ among $n$-sided polygonal set of either fixed area or fixed perimeter ?
Even the cases $n=3$ or $n=4$ are open, where one should be mindful of the behaviour of the corner in the shape derivative formula. -
Minimization of $p$-capacity
Problem 2.5.
[Dorin Bucur] Let $p\in (1,2)$, and $\Omega\subset \mathbb{R}^2$ a bounded convex set, we define its $p$-capacity
\[\text{cap}_p(\Omega)=\inf\left\{\int_{\R^2}|\nabla u|^p,\ u\in\mathcal{C}^1_c(\mathbb{R}^2),\text{ such that }u\geq 1\text{ in a neighbourhood of }\Omega\right\}\]
Note that this notion of capacity is not well-defined for $p=2$ (by the scale-invariance of $\int_{\R^2}|\nabla u|^2$ and the fact that points have zero $2$-capacity), and is usually replaced with the logarithmic capacity.
The question is the following: among convex sets $\Omega$ of fixed perimeter, is it true that $\text{cap}_p(\Omega)$ is minimal for segments ?
Note that the perimeter should be counted twice when $\Omega$ is a segment (or has unidimensional parts).
It is known from (Colesanti, Salani - 2003) that $\Omega\mapsto \text{cap}_p(\Omega)^{\frac{1}{2-p}}$ satisfy the Brunn-Minkowsky inequality, meaning that the search for optimal sets may be restricted to the extremal points for Brunn-Minkowsky sum, meaning triangles (includes segments).
It is however unclear how to conclude even in the case of triangles.
Cite this as: AimPL: Symmetry-breaking of optimal shapes, available at http://aimpl.org/symmetrybreaking.