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Abstract

We prove a lower bound for the $k$-th Steklov eigenvalues in terms of an
isoperimetric constant called the $k$-th Cheeger-Steklov constant in three
different situations: finite spaces, measurable spaces, and Riemannian
manifolds. These lower bounds can be considered as higher order Cheeger type
inequalities for the Steklov eigenvalues. In particular it extends the Cheeger
type inequality for the first nonzero Steklov eigenvalue previously studied by
Escobar in 1997 and by Jammes in 2015 to higher order Steklov eigenvalues. The
technique we develop to get this lower bound is based on considering a family
of accelerated Markov operators in the finite and mesurable situations and of
mass concentration deformations of the Laplace-Beltrami operator in the
manifold setting which converges uniformly to the Steklov operator. As an
intermediary step in the proof of the higher order Cheeger type inequality, we
define the Dirichlet-Steklov connectivity spectrum and show that the Dirichlet
connectivity spectra of this family of operators converges to (or bounded by)
the Dirichlet-Steklov spectrum uniformly. Moreover, we obtain bounds for the
Steklov eigenvalues in terms of its Dirichlet-Steklov connectivity spectrum
which is interesting in its own right and is more robust than the higher order
Cheeger type inequalities. The Dirichlet-Steklov spectrum is closely related to
the Cheeger-Steklov constants.