Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains

Abstract

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.

41 pages, 5 figures, LaTeX; some revision of exposition, misprints corrected, the version to appear in Commun. Math. Phys

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