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The purpose of the present paper is the solution of the boundary value problems for minimal surfaces when the boundaries are not, or not entirely-fixed Jordan curves but are free to move on prescribed manifolds. At the same time I shall present modifications and simplifications of my previous solution of the Plateau' and Douglas' problem for fixed boundary curves and prescribed topological structure and incidentally discuss certain features of the problem in order to clarify its relation to the theory of conformal mapping. Though based on previous publications, the paper may, except for some references, be read independently.

We study different classes of Jensen measures for plurisubharmonic functions, in particular the relation between Jensen measures for continuous functions and Jensen measures for upper bounded functions. We prove an approximation theorem for plurisubharmonic functions in$B$-regular domain. This theorem implies that the two classes of Jensen measures coincide in$B$-regular domains. Conversely we show that if Jensen measures for continuous functions are the same as Jensen measures for upper bounded functions and the domain is hyperconvex, the domain satisfies the same approximation theorem as above.