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.

MSC:

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

References:

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