Minimal Surfaces and Functions of Bounded Variation

1. Front Matter

   Pages i-xii

   PDF

2. Parametric Minimal Surfaces

   1. Front Matter

      Pages 1-1

      PDF

   2. Functions of Bounded Variation and Caccioppoli Sets

      • Enrico Giusti

      Pages 3-29

   3. Traces of BV Functions

      • Enrico Giusti

      Pages 30-41

   4. The Reduced Boundary

      • Enrico Giusti

      Pages 42-51

   5. Regularity of the Reduced Boundary

      • Enrico Giusti

      Pages 52-62

   6. Some Inequalities

      • Enrico Giusti

      Pages 63-73

   7. Approximation of Minimal Sets (I)

      • Enrico Giusti

      Pages 74-84

   8. Approximation of Minimal Sets(II)

      • Enrico Giusti

      Pages 85-96

   9. Regularity of Minimal Surfaces

      • Enrico Giusti

      Pages 97-103

   10. Minimal Cones

       • Enrico Giusti

       Pages 104-114

   11. The First and Second Variation of the Area

       • Enrico Giusti

       Pages 115-127

   12. The Dimension of the Singular Set

       • Enrico Giusti

       Pages 128-134

3. Non-Parametric Minimal Surfaces

   1. Front Matter

      Pages 135-135

      PDF

   2. Classical Solutions of the Minimal Surface Equation

      • Enrico Giusti

      Pages 137-150

   3. The a priori Estimate for the Gradient

      • Enrico Giusti

      Pages 151-159

   4. Direct Methods

      • Enrico Giusti

      Pages 160-171

   5. Boundary Regularity

      • Enrico Giusti

      Pages 172-181

   6. A Further Extension of the Notion of Non-Parametric Minimal Surface

      • Enrico Giusti

      Pages 182-200

   7. The Bernstein Problem

      • Enrico Giusti

      Pages 201-217

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].

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

Policies and ethics