Extremal Steklov-Neumann Eigenvalues

Abstract

Let $Ω$ be a bounded open planar domain with smooth connected boundary, $Γ$, that has been partitioned into two disjoint components, $Γ= Γ_S \sqcup Γ_N$. We consider the Steklov-Neumann eigenproblem on $Ω$, where a harmonic function is sought that satisfies the Steklov boundary condition on $Γ_S$ and the Neumann boundary condition on $Γ_N$. We pose the extremal eigenvalue problems (EEPs) of minimizing/maximizing the $k$-th non-trivial Steklov-Neumann eigenvalue among boundary partitions of prescribed measure. We formulate a relaxation of these EEPs in terms of weighted Steklov eigenvalues where an $L^\infty(Γ)$ density replaces the boundary partition. For these relaxed EEPs, we establish existence and prove optimality conditions. We also prove a homogenization result that allows us to use solutions to the relaxed EEPs to infer properties of solutions to the original EEPs. For a disk, we provide numerical and asymptotic evidence that the minimizing arrangement of $Γ_S\sqcup Γ_N$ for the $k$-th eigenvalue consists of $k+1$ connected components that are symmetrically arranged on the boundary. For a disk, for $k = 1$, the constant density is a maximizer for the relaxed problem; we also provide numerical and asymptotic evidence that for $k\ge 2$, the maximizing density for the relaxed problem is a non-trivial function; a sequence of rapidly oscillating Steklov/Neumann boundary conditions approach the supremum value.

Figures (6)

• Figure 1: Minimizing bang-bang densities, $\rho_k^\bigtriangledown$, and associated eigenfunctions for \ref{['e:RelaxMin']} with $\Omega = \mathbb{D}$, $\alpha = 0.5$, and odd $k = 1, 3, 5, 7, 9, 11$ (values indicated). The red part of the boundary is where $\rho=1$, i.e., where the Steklov boundary condition is imposed. See Section \ref{['s:DiskMin']}.
• Figure 2: Minimizing bang-bang densities, $\rho_k^\bigtriangledown$, and associated eigenfunctions for \ref{['e:RelaxMin']} with $\Omega = \mathbb{D}$, $\alpha = 0.5$, and even $k=2,4,12$ (values indicated). The red part of the boundary is where $\rho=1$, i.e., where the Steklov boundary condition is imposed. Each eigenvalue has multiplicity 2 and a basis for the eigenspace is plotted. See Section \ref{['s:DiskMin']}.
• Figure 3: Solutions for \ref{['e:RelaxMin']} with $\Omega = \mathbb{D}$, $\alpha = 0.5$, and $k = 1,2,3,4$ (values indicated). In the left hand panels, the minimizing bang-bang density $\rho_k^\bigtriangledown$ is plotted in blue and the corresponding eigenfunction(s) are plotted in red. For $k$ odd, the eigenvalue is simple and the right hand panel shows that the optimality condition \ref{['e:minRhoCondMult']} is satisfied. For $k$ even, the eigenvalue has multiplicity $2$ and the right hand panel shows that the optimality condition \ref{['e:minRhoCondMult']} is satisfied for the two eigenfunctions plotted in the left hand panel. See Section \ref{['s:DiskMin']}.
• Figure 4: (Left) Maximizing densities, $\rho_k^\triangle$, and associated eigenfunctions for \ref{['e:RelaxMax']} with $\Omega = \mathbb{D}$, $\alpha = 0.5$, and even $k=2,4,6,8$ (values indicated). Interestingly, the eigenfunctions are constant on intervals where $\rho \neq 1$. (Right) The eigenfunctions are plotted on $\Omega =\mathbb{D}$. The red part of the boundary is where $\rho=1$, i.e., where the Steklov boundary condition is imposed. See Section \ref{['s:DiskMax']}.
• Figure 5: Solutions for \ref{['e:RelaxMax']} with $\Omega = \mathbb{D}$, $\alpha = 0.5$, and $k = 1,2,3,4$ (values indicated). In the left hand panels, the maximizing density $\rho_k^\triangle$ is plotted in blue and the corresponding eigenfunction(s) are plotted in red. For $k$ even, the eigenvalue is simple and the right hand panel shows that the optimality condition \ref{['e:maxRhoCondMult']} is satisfied. For $k$ odd, the eigenvalue has multiplicity $2$ and the right hand panel shows that the optimality condition \ref{['e:maxRhoCondMult']} is satisfied for the two eigenfunctions plotted in the left hand panel. See Section \ref{['s:DiskMax']}.

Theorems & Definitions (18)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 1.5
• Conjecture 1.6
• Theorem 1.7
• Lemma 2.1: Friedland_1977
• Lemma 2.2
• proof