Seminario di Geometria
Aula 1E (pal. RM004)
Mikhail Karpukhin (Mc Gill University, Montreal)
Upper bounds for Steklov eigenvalues via conjugate harmonic forms
In 1975 Hersch, Payne and Schiffer used the concept of conjugate harmonic functions on the complex plane to prove a sharp upper bound for Steklov eigenvalues on simply connected domains. In this talk we will discuss a higher dimensional version of this concept defined for an arbitrary Riemannian manifold with boundary-conjugate harmonic forms. As a result, an inequality relating Steklov eigenvalues of the manifold with the Laplace eigenvalues of the boundary is obtained. This inequality is reduced to Hersch-Payne-Schiffer inequality in the case of simply connected domains and yields improved upper bounds even in a two-dimensional case.