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Minimal Surfaces and Functions of Bounded Variation

  • Authors
  • Enrico Giusti

Part of the Monographs in Mathematics book series (MMA, volume 80)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Parametric Minimal Surfaces

    1. Front Matter
      Pages 1-1
    2. Enrico Giusti
      Pages 30-41
    3. Enrico Giusti
      Pages 42-51
    4. Enrico Giusti
      Pages 52-62
    5. Enrico Giusti
      Pages 63-73
    6. Enrico Giusti
      Pages 74-84
    7. Enrico Giusti
      Pages 85-96
    8. Enrico Giusti
      Pages 97-103
    9. Enrico Giusti
      Pages 104-114
    10. Enrico Giusti
      Pages 115-127
    11. Enrico Giusti
      Pages 128-134
  3. Non-Parametric Minimal Surfaces

    1. Front Matter
      Pages 135-135
    2. Enrico Giusti
      Pages 151-159
    3. Enrico Giusti
      Pages 160-171
    4. Enrico Giusti
      Pages 172-181
    5. Enrico Giusti
      Pages 201-217
  4. Back Matter
    Pages 218-240

About this book

Introduction

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].

Keywords

Area Calc Factor Minimal surface Simple Object Access Protocol approximation calculus calculus of variations equation form function functional functions minimum

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