Minimal Surfaces and Functions of Bounded VariationSpringer Science & Business Media, 14 mar 2013 - 240 pagine 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]. |
Sommario
3 | |
Traces of BV Functions | 30 |
The Reduced Boundary | 42 |
Regularity of the Reduced Boundary | 52 |
Some Inequalities | 63 |
Approximation of Minimal Sets I | 74 |
Approximation of Minimal Sets II | 85 |
Regularity of Minimal Surfaces | 97 |
Classical Solutions of the Minimal Surface Equation | 137 |
The a priori Estimate of the Gradient | 151 |
Direct Methods | 160 |
Boundary Regularity | 172 |
A Further Extension of the Notion of NonParametric Minimal Surface | 182 |
The Bernstein Problem | 201 |
Appendix A | 218 |
Appendix B | 224 |
Minimal Cones | 104 |
The First and Second Variation of the Area | 115 |
The Dimension of the Singular Set | 128 |
NonParametric Minimal Surfaces | 135 |
Appendix C | 226 |
235 | |
239 | |
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approximate B₁ ball Bernstein's theorem boundary datum bounded open set bounded variation BR(XO BV(N BV(Q Caccioppoli set chapter classical solution Co(R coinciding compact set conclude converges define definition denote diam Dirichlet problem DnQE(Y DOE(Y Du² E₁ estimate follows function f Giorgi gradient graph Hausdorff measure hence holds hyperplane hypersurface inequality integral least perimeter Lemma Let Q lim sup Lipschitz Lipschitz-continuous maximum principle mean curvature minimal set minimal surface equation minimizes the area minimum Moreover non-negative mean curvature non-parametric minimal surfaces obtain open set particular Proof Proposition prove quasi-solution Radon measure reduced boundary regularity Remark result satisfies singular set smooth subgraph subsequence supersolution Suppose T(Dv Theorem 1.9 vector wdHn ΘΩ ΤΩ