The first Steklov eigenvalue of a Riemannian manifold with boundary is, up to constant factors, equal to the isocapacitary constant Γ∂, and the same holds for the bottom of the Dirichlet-to-Neumann spectrum in the non-compact case.
Cheeger type inequalities associated with isocapacitary constants on graphs
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In this paper, we introduce Cheeger type constants via isocapacitary constants introduced by Maz'ya to estimate first Dirichlet, Neumann and Steklov eigenvalues on a finite subgraph of a graph. Moreover, we estimate the bottom of the spectrum of the Laplace operator and the Dirichlet-to-Neumann operator for an infinite subgraph. Estimates for higher-order Steklov eigenvalues on a finite or infinite subgraph are also proved.
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Cheeger type inequalities associated with isocapacitary constants on Riemannian manifolds with boundary
The first Steklov eigenvalue of a Riemannian manifold with boundary is, up to constant factors, equal to the isocapacitary constant Γ∂, and the same holds for the bottom of the Dirichlet-to-Neumann spectrum in the non-compact case.