On the eigenvalue problem for a bulk/surface elliptic system

2024/03/28 by Vitillaro, Enzo · 1 citation
#35J05 #35J20 #35J25 #35J57 #35L05 #35L10 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2403.19759

Abstract

The paper addresses the doubly elliptic eigenvalue problem \begincases -Δu=λu amp;in Ω,
u=0 amp;on Γ₀,
-Δ_Γu +∂_νu =λu amp;on Γ₁, \endcases where Ω is a bounded open subset of ℝ^N (N≥ 2) with C¹ boundary Γ=Γ₀∪Γ₁, Γ₀∩Γ₁=∅, Γ₁ being nonempty and relatively open on Γ. Moreover H^N-1(Γ₀∩Γ₁)=0 and H^N-1(Γ₀)>0. We recognize that L²(Ω)× L²(Γ₁) admits a Hilbert basis of eigenfunctions of the problem and we describe the eigenvalues. Moreover, when Γ is at least C² and Γ₀∩Γ₁=∅, we give several qualitative properties of the eigenfunctions.

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