{"ID":2868839,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.15975","arxiv_id":"2509.15975","title":"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.","short_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 bounda...","url_abs":"https://arxiv.org/abs/2509.15975","url_pdf":"https://arxiv.org/pdf/2509.15975v2","authors":"[\"Chiu-Yen Kao\",\"Braxton Osting\",\"Chee Han Tan\",\"Robert Viator\"]","published":"2025-09-19T13:35:51Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.SP\"]","methods":"[]","has_code":false}