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Molla Khalili, Ahmad | 2009

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 39793 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Ranjbar Motlagh, Alireza
  7. Abstract:
  8. The development of intrinsic theories for area-minimization problems was motivated in the 1950 by the diffichlty to prove, existence for the Plateau problem for surfaces in Euclidean spaces of dimension higher than two. After the pioneering work of R. Caccioppoli and E. De Giorgi on set with finite perimeter, W. H. Fleming and H. Federer developed the theory of currents, which leads to existence results for the Plateau problem for oriented surfaces of any dimension and codimension. The aim of this paper is to develop an existence of the Federer-Fleming theory to spaces without a differentiable structure, and virtually to any complete metric space. The starting point of our research has been a very short paper of De Giorgi amazingly, he was able to formulate a generalized Plateau problem in any metric space E using (necessarily) only the metric structure. The basic idea of De Giorgi has been to replace the duality of differential forms with the duality of (k+1)-tuples (f, ..., fk). Generalized Plateau problem is: min{M(T) : T ∈ Ik(E), ∂T = S} In the Euclidean theory an important class of currents, in connection with the Plateau problem, is the class of normal and rectifiable currents. We have also studied some situations in which certainly there are plenty of rectifiable currents; for instance if E is a Banach space
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  9. Keywords:
  10. Metric Space ; Current Theory ; Current Normal ; Current Rectifiable ; Plateau Problem ; Cone Construction

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