Year: 2025
Type: article
Abstract: The paper addresses the doubly elliptic eigenvalue problem\[ \begin{cases} -\Delta u=\lambda u & \text{in } \Omega,\\ \phantom{-}u=0 & \text{on } \Gamma_0,\\ -\Delta_\Gamma u +\partial_\nu u =\lambda u & \text{on } \Gamma_1, \end{cases} \]where $Ω$ is a bounded open subset of $\mathbb{R}^N$ $(N ≥ 2)$ with a $C^1$ boundary $Γ = Γ_0∪Γ_1,$ $Γ_0∩Γ_1 = ∅,$ $Γ_1$ being nonempty and relatively open on $Γ.$ Moreover, $\mathcal{H}^{N-1}(\overline{\Gamma}_0 \cap \overline{\Gamma}_1)=0$ and $\mathcal{H}^{N−1} (Γ_0) > 0.$ We prove that $L^2 (Ω)×L^2 (Γ_1)$ admits a Hilbert basis constituted by eigenfunctions and we describe the behavior of the eigenvalues. Moreover, when $Γ$ is at least $C^2$ and $\overline{Γ}_0∩\overline{Γ}_1=∅,$ we give several qualitative properties of the eigenfunctions.
Author: Enzo Vitillaro
Institution: University of Perugia
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
Funder: European Commission
Award: G972786147
OpenAlex ID: W4413229970