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Minimal Surfaces and Functions of Bounded Variation (Paperback, 1984 ed.)
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Minimal Surfaces and Functions of Bounded Variation (Paperback, 1984 ed.)
Series: Monographs in Mathematics, 80
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The problem of finding minimal surfaces, i. e. of finding the
surface of least area among those bounded by a given curve, was one
of the first considered after the foundation of the calculus of
variations, and is one which received a satis factory solution only
in recent years. Called the problem of Plateau, after the blind
physicist who did beautiful experiments with soap films and
bubbles, it has resisted the efforts of many mathematicians for
more than a century. It was only in the thirties that a solution
was given to the problem of Plateau in 3-dimensional Euclidean
space, with the papers of Douglas [DJ] and Rado [R T1, 2]. The
methods of Douglas and Rado were developed and extended in
3-dimensions by several authors, but none of the results was shown
to hold even for minimal hypersurfaces in higher dimension, let
alone surfaces of higher dimension and codimension. It was not
until thirty years later that the problem of Plateau was
successfully attacked in its full generality, by several authors
using measure-theoretic methods; in particular see De Giorgi [DG1,
2, 4, 5], Reifenberg [RE], Federer and Fleming [FF] and Almgren
[AF1, 2]. Federer and Fleming defined a k-dimensional surface in
IR" as a k-current, i. e. a continuous linear functional on
k-forms. Their method is treated in full detail in the splendid
book of Federer [FH 1].
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