Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons

We obtain asymptotic formulae for the Steklov eigenvalues and eigenfunctions curvilinear polygons in terms of their side lengths and angles. These are quite precise: the errors tend to zero as the spectral parameter to infinity. The Steklov problem on planar domains with corners is linked to the classical sloshing and sloping beach problems in ; as we show it is also related to quantum graphs. Somewhat , the arithmetic properties of the angles of a curvilinear polygon a significant effect on the boundary behaviour of the Steklov . Our proofs are based on an explicit construction of quasimodes. use a variety of methods, including ideas from spectral geometry, layer analysis, and some new techniques tailored to our problem.