  We study the integrable structure of the Dirichlet boundary problem in two
dimensions and extend the approach to the case of planar multiply-connected
domains. The solution to the Dirichlet boundary problem in multiply-connected
case is given through a quasiclassical tau-function, which generalizes the
tau-function of the dispersionless Toda hierarchy. It is shown to obey an
infinite hierarchy of Hirota-like equations which directly follow from
properties of the Dirichlet Green function and from the Fay identities. The
relation to multi-support solutions of matrix models is briefly discussed.
