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2021•arXiv (Cornell University)Open access

Lower bounds for Steklov eigenfunctions

Jeffrey Galkowski, John A. Toth

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Abstract

Let $(Ω,g)$ be a compact, analytic Riemannian manifold with analytic boundary $\partial Ω= M.$ We give $L^2$-lower bounds for Steklov eigenfunctions and their restrictions to interior hypersurfaces $H \subset Ω^{\circ}$ in a geometrically defined neighborhood of $M$. Our results are optimal in the entire geometric neighborhood and complement the results on eigenfunction upper bounds in the author's previous work.

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Let $(Ω,g)$ be a compact, analytic Riemannian manifold with analytic boundary $\partial Ω= M.$ We give $L^2$-lower bounds for Steklov eigenfunctions and their restrictions to interior hypersurfaces $H \subset Ω^{\circ}$ in a geometrically defined neighborhood of $M$. Our results are optimal in the entire geometric neighborhood and complement the results on eigenfunction upper bounds in the author's previous work.

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Available abstract

Let $(Ω,g)$ be a compact, analytic Riemannian manifold with analytic boundary $\partial Ω= M.$ We give $L^2$-lower bounds for Steklov eigenfunctions and their restrictions to interior hypersurfaces $H \subset Ω^{\circ}$ in a geometrically defined neighborhood of $M$. Our results are optimal in the entire geometric neighborhood and complement the results on eigenfunction upper bounds in the author's previous work.

Key concepts: Eigenfunction, Riemannian manifold, Mathematics, Omega, Boundary (topology), Complement (music), Manifold (fluid mechanics), Upper and lower bounds

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