  In the present paper we develop an approach to obtain sharp spectral
asymptotics for Steklov type problems on planar domains with corners. Our main
focus is on the two-dimensional sloshing problem, which is a mixed
Steklov-Neumann boundary value problem describing small vertical oscillations
of an ideal fluid in a container or in a canal with a uniform cross-section. We
prove a two-term asymptotic formula for sloshing eigenvalues. In particular,
this confirms a conjecture posed by Fox and Kuttler in 1983. We also obtain
similar eigenvalue asymptotics for other related mixed Steklov type problems,
and discuss applications to the study of Steklov spectral asymptotics on
polygons.
