Regularity Of Minimal Surfaces

Author : Ulrich Dierkes
Publisher : Springer Science & Business Media
ISBN 13 : 3642117007
Total Pages : 634 pages
Book Rating : 4.08/5 ( download)

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Download or read book Regularity of Minimal Surfaces written by Ulrich Dierkes and published by Springer Science & Business Media. This book was released on 2010-08-16 with total page 634 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : Jon T. Pitts
Publisher : Princeton University Press
ISBN 13 : 1400856450
Total Pages : 337 pages
Book Rating : 4.59/5 ( download)

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Download or read book Existence and Regularity of Minimal Surfaces on Riemannian Manifolds. (MN-27) written by Jon T. Pitts and published by Princeton University Press. This book was released on 2014-07-14 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : Giusti
Publisher : Springer Science & Business Media
ISBN 13 : 1468494864
Total Pages : 250 pages
Book Rating : 4.60/5 ( download)

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Download or read book Minimal Surfaces and Functions of Bounded Variation written by Giusti and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : Ulrich Dierkes
Publisher : Springer Science & Business Media
ISBN 13 : 3662087766
Total Pages : 435 pages
Book Rating : 4.63/5 ( download)

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Download or read book Minimal Surfaces II written by Ulrich Dierkes and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 435 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : Tobias Holck Colding
Publisher : American Mathematical Society
ISBN 13 : 1470476401
Total Pages : 330 pages
Book Rating : 4.03/5 ( download)

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Download or read book A Course in Minimal Surfaces written by Tobias Holck Colding and published by American Mathematical Society. This book was released on 2024-01-18 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : Robert Osserman
Publisher : Courier Corporation
ISBN 13 : 0486167690
Total Pages : 224 pages
Book Rating : 4.95/5 ( download)

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Download or read book A Survey of Minimal Surfaces written by Robert Osserman and published by Courier Corporation. This book was released on 2013-12-10 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : Ulrich Dierkes
Publisher : Springer Science & Business Media
ISBN 13 : 3662027917
Total Pages : 528 pages
Book Rating : 4.12/5 ( download)

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Download or read book Minimal Surfaces I written by Ulrich Dierkes and published by Springer Science & Business Media. This book was released on 2013-11-27 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : Jon T. Pitts
Publisher :
ISBN 13 : 9780598051547
Total Pages : 338 pages
Book Rating : 4.46/5 ( download)

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Download or read book Existence and Regularity of Minimal Surfaces on Riemannian Manifolds written by Jon T. Pitts and published by . This book was released on 1981 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Author : Luis A. Caffarelli
Publisher : American Mathematical Soc.
ISBN 13 : 0821804375
Total Pages : 114 pages
Book Rating : 4.77/5 ( download)

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Download or read book Fully Nonlinear Elliptic Equations written by Luis A. Caffarelli and published by American Mathematical Soc.. This book was released on 1995 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
ISBN 13 : 9783642117534
Total Pages : 623 pages
Book Rating : 4.38/5 ( download)

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Download or read book Regularity of Minimal Surfaces written by Ulrich Dierkes and published by Springer. This book was released on 2010-11-05 with total page 623 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.