Regularity Of Minimal Surfaces

Author : Ulrich Dierkes,Stefan Hildebrandt,Anthony Tromba
Publisher : Springer Science & Business Media
Page : 634 pages
File Size : 50,7 Mb
Release : 2010-08-16
Category : Mathematics
ISBN : 9783642117008

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Regularity of Minimal Surfaces by Ulrich Dierkes,Stefan Hildebrandt,Anthony Tromba Pdf

Regularity of Minimal Surfaces begins with a survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and H-surfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general Gauss-Bonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to derive a priori estimates. Therefore regularity proofs for non-minimizers have to be based on indirect reasoning using monotonicity formulas. This is followed by a long chapter discussing geometric properties of minimal and H-surfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateau ́s problem for H-surfaces in a Riemannian manifold. A natural generalization of the isoperimetric problem is the so-called thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed. The final chapter on branch points presents a new approach to the theorem that area minimizing solutions of Plateau ́s problem have no interior branch points.

Author : Giusti
Publisher : Springer Science & Business Media
Page : 250 pages
File Size : 43,9 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9781468494860

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Minimal Surfaces and Functions of Bounded Variation by Giusti Pdf

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].

Author : Jon T. Pitts
Publisher : Princeton University Press
Page : 337 pages
File Size : 46,7 Mb
Release : 2014-07-14
Category : Mathematics
ISBN : 9781400856459

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Existence and Regularity of Minimal Surfaces on Riemannian Manifolds. (MN-27) by Jon T. Pitts Pdf

Mathematical No/ex, 27 Originally published in 1981. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Author : Ulrich Dierkes,Stefan Hildebrandt,Albrecht Küster,Ortwin Wohlrab
Publisher : Springer Science & Business Media
Page : 435 pages
File Size : 44,9 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9783662087763

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Minimal Surfaces II by Ulrich Dierkes,Stefan Hildebrandt,Albrecht Küster,Ortwin Wohlrab Pdf

Minimal Surfaces I is an introduction to the field of minimal surfaces and a presentation of the classical theory as well as of parts of the modern development centered around boundary value problems. Part II deals with the boundary behaviour of minimal surfaces. Part I is particularly apt for students who want to enter this interesting area of analysis and differential geometry which during the last 25 years of mathematical research has been very active and productive. Surveys of various subareas will lead the student to the current frontiers of knowledge and can also be useful to the researcher. The lecturer can easily base courses of one or two semesters on differential geometry on Vol. 1, as many topics are worked out in great detail. Numerous computer-generated illustrations of old and new minimal surfaces are included to support intuition and imagination. Part 2 leads the reader up to the regularity theory for nonlinear elliptic boundary value problems illustrated by a particular and fascinating topic. There is no comparably comprehensive treatment of the problem of boundary regularity of minimal surfaces available in book form. This long-awaited book is a timely and welcome addition to the mathematical literature.

Author : Ulrich Dierkes,Stefan Hildebrandt,Albrecht Küster,Ortwin Wohlrab
Publisher : Springer Science & Business Media
Page : 528 pages
File Size : 50,8 Mb
Release : 2013-11-27
Category : Mathematics
ISBN : 9783662027912

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Minimal Surfaces I by Ulrich Dierkes,Stefan Hildebrandt,Albrecht Küster,Ortwin Wohlrab Pdf

Minimal surfaces I is an introduction to the field of minimal surfaces and apresentation of the classical theory as well as of parts of the modern development centered around boundary value problems. Part II deals with the boundary behaviour of minimal surfaces. Part I is particularly apt for students who want to enter this interesting area of analysis and differential geometry which during the last 25 years of mathematical research has been very active and productive. Surveys of various subareas will lead the student to the current frontiers of knowledge and can alsobe useful to the researcher. The lecturer can easily base courses of one or two semesters on differential geometry on Vol. 1, as many topics are worked out in great detail. Numerous computer-generated illustrations of old and new minimal surfaces are included to support intuition and imagination. Part 2 leads the reader up to the regularity theory fornonlinear elliptic boundary value problems illustrated by a particular and fascinating topic. There is no comparably comprehensive treatment of the problem of boundary regularity of minimal surfaces available in book form. This long-awaited book is a timely and welcome addition to the mathematical literature.

Author : Tobias Holck Colding,William P. Minicozzi II
Publisher : American Mathematical Society
Page : 330 pages
File Size : 50,6 Mb
Release : 2024-01-18
Category : Mathematics
ISBN : 9781470476403

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A Course in Minimal Surfaces by Tobias Holck Colding,William P. Minicozzi II Pdf

Minimal surfaces date back to Euler and Lagrange and the beginning of the calculus of variations. Many of the techniques developed have played key roles in geometry and partial differential equations. Examples include monotonicity and tangent cone analysis originating in the regularity theory for minimal surfaces, estimates for nonlinear equations based on the maximum principle arising in Bernstein's classical work, and even Lebesgue's definition of the integral that he developed in his thesis on the Plateau problem for minimal surfaces. This book starts with the classical theory of minimal surfaces and ends up with current research topics. Of the various ways of approaching minimal surfaces (from complex analysis, PDE, or geometric measure theory), the authors have chosen to focus on the PDE aspects of the theory. The book also contains some of the applications of minimal surfaces to other fields including low dimensional topology, general relativity, and materials science. The only prerequisites needed for this book are a basic knowledge of Riemannian geometry and some familiarity with the maximum principle.

Author : Jon T. Pitts
Publisher : Unknown
Page : 338 pages
File Size : 51,6 Mb
Release : 1981
Category : Electronic
ISBN : 0598051546

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Existence and Regularity of Minimal Surfaces on Riemannian Manifolds by Jon T. Pitts Pdf

Author : Ulrich Dierkes,Stefan Hildebrandt,Anthony Tromba
Publisher : Springer
Page : 623 pages
File Size : 43,5 Mb
Release : 2010-09-30
Category : Mathematics
ISBN : 364211699X

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Regularity of Minimal Surfaces by Ulrich Dierkes,Stefan Hildebrandt,Anthony Tromba Pdf

Regularity of Minimal Surfaces begins with a survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and H-surfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general Gauss-Bonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to derive a priori estimates. Therefore regularity proofs for non-minimizers have to be based on indirect reasoning using monotonicity formulas. This is followed by a long chapter discussing geometric properties of minimal and H-surfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateau ́s problem for H-surfaces in a Riemannian manifold. A natural generalization of the isoperimetric problem is the so-called thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed. The final chapter on branch points presents a new approach to the theorem that area minimizing solutions of Plateau ́s problem have no interior branch points.

Author : Ulrich Dierkes,Stefan Hildebrandt,Anthony Tromba
Publisher : Springer Science & Business Media
Page : 547 pages
File Size : 44,7 Mb
Release : 2010-08-16
Category : Mathematics
ISBN : 9783642117060

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Global Analysis of Minimal Surfaces by Ulrich Dierkes,Stefan Hildebrandt,Anthony Tromba Pdf

Many properties of minimal surfaces are of a global nature, and this is already true for the results treated in the first two volumes of the treatise. Part I of the present book can be viewed as an extension of these results. For instance, the first two chapters deal with existence, regularity and uniqueness theorems for minimal surfaces with partially free boundaries. Here one of the main features is the possibility of "edge-crawling" along free parts of the boundary. The third chapter deals with a priori estimates for minimal surfaces in higher dimensions and for minimizers of singular integrals related to the area functional. In particular, far reaching Bernstein theorems are derived. The second part of the book contains what one might justly call a "global theory of minimal surfaces" as envisioned by Smale. First, the Douglas problem is treated anew by using Teichmüller theory. Secondly, various index theorems for minimal theorems are derived, and their consequences for the space of solutions to Plateau ́s problem are discussed. Finally, a topological approach to minimal surfaces via Fredholm vector fields in the spirit of Smale is presented.

Author : Robert Osserman
Publisher : Courier Corporation
Page : 224 pages
File Size : 48,5 Mb
Release : 2013-12-10
Category : Mathematics
ISBN : 9780486167695

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A Survey of Minimal Surfaces by Robert Osserman Pdf

Newly updated accessible study covers parametric and non-parametric surfaces, isothermal parameters, Bernstein’s theorem, much more, including such recent developments as new work on Plateau’s problem and on isoperimetric inequalities. Clear, comprehensive examination provides profound insights into crucial area of pure mathematics. 1986 edition. Index.

Author : Luis A. Caffarelli,Xavier Cabré
Publisher : American Mathematical Soc.
Page : 114 pages
File Size : 42,6 Mb
Release : 1995
Category : Mathematics
ISBN : 9780821804377

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Fully Nonlinear Elliptic Equations by Luis A. Caffarelli,Xavier Cabré Pdf

The goal of the book is to extend classical regularity theorems for solutions of linear elliptic partial differential equations to the context of fully nonlinear elliptic equations. This class of equations often arises in control theory, optimization, and other applications. The authors give a detailed presentation of all the necessary techniques. Instead of treating these techniques in their greatest generality, they outline the key ideas and prove the results needed for developing the subsequent theory. Topics discussed in the book include the theory of viscosity solutions for nonlinear equations, the Alexandroff estimate and Krylov-Safonov Harnack-type inequality for viscosity solutions, uniqueness theory for viscosity solutions, Evans and Krylov regularity theory for convex fully nonlinear equations, and regularity theory for fully nonlinear equations with variable coefficients.

Author : Ulrich Dierkes
Publisher : Springer
Page : 454 pages
File Size : 47,8 Mb
Release : 1992
Category : Mathematics
ISBN : UOM:39015029207563

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Minimal Surfaces: Boundary regularity by Ulrich Dierkes Pdf

Author : Tobias H. Colding,William P. Minicozzi
Publisher : American Mathematical Soc.
Page : 330 pages
File Size : 48,9 Mb
Release : 2011
Category : Mathematics
ISBN : 9780821853238

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A Course in Minimal Surfaces by Tobias H. Colding,William P. Minicozzi Pdf

"Minimal surfaces date back to Euler and Lagrange and the beginning of the calculus of variations. Many of the techniques developed have played key roles in geometry and partial differential equations. Examples include monotonicity and tangent cone analysis originating in the regularity theory for minimal surfaces, estimates for nonlinear equations based on the maximum principle arising in Bernstein's classical work, and even Lebesgue's definition of the integral that he developed in his thesis on the Plateau problem for minimal surfaces. This book starts with the classical theory of minimal surfaces and ends up with current research topics. Of the various ways of approaching minimal surfaces (from complex analysis, PDE, or geometric measure theory), the authors have chosen to focus on the PDE aspects of the theory. The book also contains some of the applications of minimal surfaces to other fields including low dimensional topology, general relativity, and materials science."--Publisher's description.

Author : Ulrich Dierkes,Stefan Hildebrandt,Anthony Tromba
Publisher : Springer
Page : 623 pages
File Size : 46,8 Mb
Release : 2010-11-05
Category : Mathematics
ISBN : 3642117538

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Regularity of Minimal Surfaces by Ulrich Dierkes,Stefan Hildebrandt,Anthony Tromba Pdf

Regularity of Minimal Surfaces begins with a survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and H-surfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general Gauss-Bonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to derive a priori estimates. Therefore regularity proofs for non-minimizers have to be based on indirect reasoning using monotonicity formulas. This is followed by a long chapter discussing geometric properties of minimal and H-surfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateau ́s problem for H-surfaces in a Riemannian manifold. A natural generalization of the isoperimetric problem is the so-called thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed. The final chapter on branch points presents a new approach to the theorem that area minimizing solutions of Plateau ́s problem have no interior branch points.

Author : Ulrich Dierkes,Stefan Hildebrandt,Friedrich Sauvigny
Publisher : Springer
Page : 692 pages
File Size : 52,7 Mb
Release : 2010-10-01
Category : Mathematics
ISBN : 3642116973

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Minimal Surfaces by Ulrich Dierkes,Stefan Hildebrandt,Friedrich Sauvigny Pdf

Minimal Surfaces is the first volume of a three volume treatise on minimal surfaces (Grundlehren Nr. 339-341). Each volume can be read and studied independently of the others. The central theme is boundary value problems for minimal surfaces. The treatise is a substantially revised and extended version of the monograph Minimal Surfaces I, II (Grundlehren Nr. 295 & 296). The first volume begins with an exposition of basic ideas of the theory of surfaces in three-dimensional Euclidean space, followed by an introduction of minimal surfaces as stationary points of area, or equivalently, as surfaces of zero mean curvature. The final definition of a minimal surface is that of a nonconstant harmonic mapping X: \Omega\to\R^3 which is conformally parametrized on \Omega\subset\R^2 and may have branch points. Thereafter the classical theory of minimal surfaces is surveyed, comprising many examples, a treatment of Björling ́s initial value problem, reflection principles, a formula of the second variation of area, the theorems of Bernstein, Heinz, Osserman, and Fujimoto. The second part of this volume begins with a survey of Plateau ́s problem and of some of its modifications. One of the main features is a new, completely elementary proof of the fact that area A and Dirichlet integral D have the same infimum in the class C(G) of admissible surfaces spanning a prescribed contour G. This leads to a new, simplified solution of the simultaneous problem of minimizing A and D in C(G), as well as to new proofs of the mapping theorems of Riemann and Korn-Lichtenstein, and to a new solution of the simultaneous Douglas problem for A and D where G consists of several closed components. Then basic facts of stable minimal surfaces are derived; this is done in the context of stable H-surfaces (i.e. of stable surfaces of prescribed mean curvature H), especially of cmc-surfaces (H = const), and leads to curvature estimates for stable, immersed cmc-surfaces and to Nitsche ́s uniqueness theorem and Tomi ́s finiteness result. In addition, a theory of unstable solutions of Plateau ́s problems is developed which is based on Courant ́s mountain pass lemma. Furthermore, Dirichlet ́s problem for nonparametric H-surfaces is solved, using the solution of Plateau ́s problem for H-surfaces and the pertinent estimates.