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Mathematics Colloquium: Donato Cianci (University of Michigan)

Title: Large gaps for Steklov eigenvalues under boundary constraints

Abstract: This talk will focus on the spectral geometry of Steklov eigenvalues. Steklov eigenvalues are eigenvalues of the Dirichlet-to-Neumann operator which, for a bounded domain in Euclidean space, maps a function on the boundary of the domain to the outward normal derivative of the harmonic extension of that function. Such operators are important in the study of electric impedance tomography (the Dirichlet-to-Neumann operator is often called to voltage-to-current map) and fluid mechanics. Recently, geometers have started studying Steklov eigenvalues on Riemannian manifolds with boundary. The main question of interest is: To what extent does the geometry of the underlying manifold influence the Steklov eigenvalues? Since the Dirichlet-to-Neumann operator is defined on the boundary, one might expect that the Steklov eigenvalues are closely related to the geometry of the boundary. However, after introducing the Steklov eigenvalues on a Riemannian manifold and reviewing some of their geometric properties, I?ll discuss some recent work which demonstrates that this is not always the case.

Contact

Pierre Clare

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