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Currents in metric spaces. (English) Zbl 0984.49025
In this remarkable paper the authors present an extension of the concept of currents introduced by Federer and Fleming to complete metric spaces without any differentiable structure. As the authors say the starting point for their investigations has been the short note of E. De Giorgi [Atti Sem. Mat. Fis. Univ. Modena 43, No. 2, 285-292 (1995; Zbl 0862.49028)] in which he raised some natural questions about the existence of solutions of the generalized Plateau problem in Banach and Hilbert spaces. Now, in the first sections of their paper the authors introduce the basic tools which are necessary for putting De Giorgi’s problem in a suitable analytical framework, i.e., the definitions of currents, rectifiable currents and normal currents for complete metric spaces are given. Among the main results of the paper is the closure theorem and the boundary-rectifiability theorem for integer-rectifiable currents in general spaces which is quite surprising since the classical proofs use the structure of \(\mathbb{R}^n\) as underlying space in one or the other way. In final sections the reader will find an approach to the generalized Plateau problem in the case that the underlying space is a Banach space. Moreover, the Euclidean isoperimetric inequality is extended to a more general setting.

MSC:
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
49J45 Methods involving semicontinuity and convergence; relaxation
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