Extremal Steklov-Neumann Eigenvalues
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908882587090944 |
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| author | Kao, Chiu-Yen Osting, Braxton Tan, Chee Han Viator, Robert |
| author_facet | Kao, Chiu-Yen Osting, Braxton Tan, Chee Han Viator, Robert |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15975 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extremal Steklov-Neumann Eigenvalues Kao, Chiu-Yen Osting, Braxton Tan, Chee Han Viator, Robert Optimization and Control Spectral Theory 31A25, 35P15, 49M41, 65K10, 65N25, 49R05 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. |
| title | Extremal Steklov-Neumann Eigenvalues |
| topic | Optimization and Control Spectral Theory 31A25, 35P15, 49M41, 65K10, 65N25, 49R05 |
| url | https://arxiv.org/abs/2509.15975 |