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We prove the global convergence of an analytical trial freeboundary algorithm, called the operator method, in the context of a very large class of multiple-free-boundary problems in R , N ≥ 2. We study the general case of a finite number of annular flow-layers, having a nested family of closed, (N − 1)-dimensional hypersurfaces as interfaces. Each interface is characterized by a general non-linear joining condition relating the normal derivatives of the stream functions in the two adjoining layers.

On the successive approximation of classical solutions for multi-surface free-boundary problems with general non-linear joining conditions