Asymptotics of Steklov eigenvalues for curvilinear polygons
Abstract: I will discuss sharp asymptotics of large Steklov eigenvalues for planar curvilinear polygons. The asymptotic expressions for eigenvalues are given in terms of roots of some trigonometric polynomials which depend explicitly on the side lengths and angles of the polygon.
The results are somewhat surprising — both the eigenvalue asymptotics and the corresponding quasimodes depend non-trivially on the arithmetic properties of the angles of the polygon, and are also related to the eigenvalues of a particular quantum graph. The proofs involve some classical hydrodynamics results related to a sloping beach problem, and a sloshing problem. I’ll also state some open questions. The talk will be based on joint works with Leonid Parnovski, Iosif Polterovich, and David Sher, see arXiv:1908.06455 and arXiv:1709.01891.