Eigenvalue inequalities for the Laplacian with mixed boundary conditions. (English) Zbl 1366.35106

On a bounded connected Lipschitz domain \(\Omega \subset {\mathbb R}^d\) denote by \( 0 < \lambda_1 < \lambda_2 \leq \lambda_3 \leq \cdots\) the eigenvalues of the (negative) Laplacian subject to a Dirichlet boundary condition on the boundary \(\partial \Omega\) and by \(0 = \mu_1 < \mu_2 \leq \mu_3 \leq \cdots\) the eigenvalues corresponding to a Neumann condition. The present paper focuses on Laplacian eigenvalues \(\lambda_k^\Gamma\) for the mixed case of a Dirichlet boundary condition on a nonempty part \(\Gamma_{\mathrm D}\) of \(\partial \Omega\) and a Neumann condition on the complement \(\Gamma_{\mathrm N}\) of \(\Gamma_{\mathrm D}\) in \(\partial \Omega\). It is assumed that \(\Gamma_{\mathrm D}\) and \(\Gamma_{\mathrm N}\) are two relatively open, non-empty subsets of \(\partial \Omega\) such that \(\Gamma_{\mathrm D} \cap \Gamma_{\mathrm N} = \emptyset\) and \(\partial \Omega \setminus (\Gamma_{\mathrm D} \cup \Gamma_{\mathrm N})\) has measure zero.
The tangential hyperplane \(T_{x'}\) exists for almost all \(x' \in \partial \Omega\). Define \(\hat \Gamma_{\mathrm N}\) to be the set of all \(x' \in \Gamma_{\mathrm N}\) such that \(T_{x'}\) exists. Define the linear subspace \[ {\mathcal S}(\Gamma_{\mathrm N}) := \bigcap_{x' \in \hat \Gamma_{\mathrm N}} T_{x'} \] of \({\mathbb R}^d\) consisting of all vectors being tangential to some \(x' \in \Gamma_{\mathrm N}\) away from a set of measure zero.
The authors prove that if \(\dim {\mathcal S}(\Gamma_{\mathrm N})\geq 1\) then \(\mu_{k+1}\leq \lambda_k^\Gamma\).
Assume that \(\Omega \subset {\mathbb R}^d\), \(d \geq 2\), is a polyhedral, convex, bounded domain. For this case the authors prove that \( \lambda_{k + \dim {\mathcal S} (\Gamma_{\mathrm D})}^\Gamma \leq \lambda_k\) holds for all \(k \in {\mathbb N}\).
The proofs are based on variational principles and proper choices of test functions.


35P15 Estimates of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text: DOI arXiv


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