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Geometric Measure Theory and the Calculus of Variations
 
Front Cover for Geometric Measure Theory and the Calculus of Variations
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Hardcover ISBN: 978-0-8218-1470-3
Product Code: PSPUM/44
List Price: $102.00
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AMS Member Price: $81.60
Electronic ISBN: 978-0-8218-9336-4
Product Code: PSPUM/44.E
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MAA Member Price: $86.40
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Front Cover for Geometric Measure Theory and the Calculus of Variations
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Geometric Measure Theory and the Calculus of Variations
Available Formats:
Hardcover ISBN:  978-0-8218-1470-3
Product Code:  PSPUM/44
List Price: $102.00
MAA Member Price: $91.80
AMS Member Price: $81.60
Electronic ISBN:  978-0-8218-9336-4
Product Code:  PSPUM/44.E
List Price: $96.00
MAA Member Price: $86.40
AMS Member Price: $76.80
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $153.00
MAA Member Price: $137.70
AMS Member Price: $122.40
  • Book Details
     
     
    Proceedings of Symposia in Pure Mathematics
    Volume: 441986; 464 pp
    MSC: Primary 00; Secondary 28; 49; 53;

    These twenty-six papers survey a cross section of current work in modern geometric measure theory and its applications in the calculus of variations. Presently the field consists of a jumble of new ideas, techniques and intuitive hunches; an exchange of information has been hindered, however, by the characteristic length and complexity of formal research papers in higher-dimensional geometric analysis. This volume provides an easier access to the material, including introductions and summaries of many of the authors' much longer works and a section containing 80 open problems in the field. The papers are aimed at analysts and geometers who may use geometric measure-theoretic techniques, and they require a mathematical sophistication at the level of a second year graduate student.

    The papers included were presented at the 1984 AMS Summer Research Institute held at Humboldt State University. A major theme of this institute was the introduction and application of multiple-valued function techniques as a basic new tool in geometric analysis, highlighted by Almgren's fundamental paper Deformations and multiple-valued functions. Major new results discussed at the conference included the following: Allard's integrality and regularity theorems for surfaces stationary with respect to general elliptic integrands; Scheffer's first example of a singular solution to the Navier-Stokes equations for a fluid flow with opposing force; and Hutchinson's new definition of the second fundamental form of a general varifold.

    Readership

  • Table of Contents
     
     
    • Articles
    • William K. Allard - An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled [ MR 840267 ]
    • F. Almgren - Deformations and multiple-valued functions [ MR 840268 ]
    • Michael T. Anderson - Local estimates for minimal submanifolds in dimensions greater than two [ MR 840269 ]
    • John E. Brothers - Second variation estimates for minimal orbits [ MR 840270 ]
    • Richard W. Carey and Joel D. Pincus - Index theory for operator ranges and geometric measure theory [ MR 840271 ]
    • Paul Concus and Mario Miranda - MACSYMA and minimal surfaces [ MR 840272 ]
    • Pierre Dolbeault - Sur les chaînes maximalement complexes de bord donné [ MR 840273 ]
    • Robert Gulliver - Index and total curvature of complete minimal surfaces [ MR 840274 ]
    • Robert Gulliver and H. Blaine Lawson, Jr. - The structure of stable minimal hypersurfaces near a singularity [ MR 840275 ]
    • Robert M. Hardt and David Kinderlehrer - Some regularity results in plasticity [ MR 840276 ]
    • Robert M. Hardt and Fang-Hua Lin - Tangential regularity near the $\mathcal {C}^1$-boundary [ MR 840277 ]
    • Robert M. Hardt and Jon T. Pitts - Solving Plateau’s problem for hypersurfaces without the compactness theorem for integral currents [ MR 840278 ]
    • F. Reese Harvey and H. Blaine Lawson, Jr. - Complex analytic geometry and measure theory [ MR 840279 ]
    • Gerhard Huisken - Mean curvature contraction of convex hypersurfaces [ MR 840280 ]
    • John E. Hutchinson - $C^{1,\alpha }$ multiple function regularity and tangent cone behaviour for varifolds with second fundamental form in $L^p$ [ MR 840281 ]
    • Christophe Margerin - Pointwise pinched manifolds are space forms [ MR 840282 ]
    • Dana Nance - The multiplicity of generic projections of $n$-dimensional surfaces in $\mathbf {R}^{n+k}$ $(n+k\leq 4)$ [ MR 840283 ]
    • Seiki Nishikawa - Deformation of Riemannian metrics and manifolds with bounded curvature ratios [ MR 840284 ]
    • George Paulik - A regularity condition at the boundary for weak solutions of some nonlinear elliptic systems [ MR 840285 ]
    • Vladimir Scheffer - Solutions to the Navier-Stokes inequality with singularities on a Cantor set [ MR 840286 ]
    • Leon Simon - Asymptotic behaviour of minimal submanifolds and harmonic maps [ MR 840287 ]
    • Jean E. Taylor - Complete catalog of minimizing embedded crystalline cones [ MR 840288 ]
    • S. Walter Wei - Liouville theorems for stable harmonic maps into either strongly unstable, or $\delta $-pinched, manifolds [ MR 840289 ]
    • Brian White - A regularity theorem for minimizing hypersurfaces modulo $p$ [ MR 840290 ]
    • William P. Ziemer - Regularity of quasiminima and obstacle problems [ MR 840291 ]
    • Edited by John E. Brothers - Some open problems in geometric measure theory and its applications suggested by participants of the 1984 AMS summer institute [ MR 840292 ]
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Volume: 441986; 464 pp
MSC: Primary 00; Secondary 28; 49; 53;

These twenty-six papers survey a cross section of current work in modern geometric measure theory and its applications in the calculus of variations. Presently the field consists of a jumble of new ideas, techniques and intuitive hunches; an exchange of information has been hindered, however, by the characteristic length and complexity of formal research papers in higher-dimensional geometric analysis. This volume provides an easier access to the material, including introductions and summaries of many of the authors' much longer works and a section containing 80 open problems in the field. The papers are aimed at analysts and geometers who may use geometric measure-theoretic techniques, and they require a mathematical sophistication at the level of a second year graduate student.

The papers included were presented at the 1984 AMS Summer Research Institute held at Humboldt State University. A major theme of this institute was the introduction and application of multiple-valued function techniques as a basic new tool in geometric analysis, highlighted by Almgren's fundamental paper Deformations and multiple-valued functions. Major new results discussed at the conference included the following: Allard's integrality and regularity theorems for surfaces stationary with respect to general elliptic integrands; Scheffer's first example of a singular solution to the Navier-Stokes equations for a fluid flow with opposing force; and Hutchinson's new definition of the second fundamental form of a general varifold.

Readership

  • Articles
  • William K. Allard - An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled [ MR 840267 ]
  • F. Almgren - Deformations and multiple-valued functions [ MR 840268 ]
  • Michael T. Anderson - Local estimates for minimal submanifolds in dimensions greater than two [ MR 840269 ]
  • John E. Brothers - Second variation estimates for minimal orbits [ MR 840270 ]
  • Richard W. Carey and Joel D. Pincus - Index theory for operator ranges and geometric measure theory [ MR 840271 ]
  • Paul Concus and Mario Miranda - MACSYMA and minimal surfaces [ MR 840272 ]
  • Pierre Dolbeault - Sur les chaînes maximalement complexes de bord donné [ MR 840273 ]
  • Robert Gulliver - Index and total curvature of complete minimal surfaces [ MR 840274 ]
  • Robert Gulliver and H. Blaine Lawson, Jr. - The structure of stable minimal hypersurfaces near a singularity [ MR 840275 ]
  • Robert M. Hardt and David Kinderlehrer - Some regularity results in plasticity [ MR 840276 ]
  • Robert M. Hardt and Fang-Hua Lin - Tangential regularity near the $\mathcal {C}^1$-boundary [ MR 840277 ]
  • Robert M. Hardt and Jon T. Pitts - Solving Plateau’s problem for hypersurfaces without the compactness theorem for integral currents [ MR 840278 ]
  • F. Reese Harvey and H. Blaine Lawson, Jr. - Complex analytic geometry and measure theory [ MR 840279 ]
  • Gerhard Huisken - Mean curvature contraction of convex hypersurfaces [ MR 840280 ]
  • John E. Hutchinson - $C^{1,\alpha }$ multiple function regularity and tangent cone behaviour for varifolds with second fundamental form in $L^p$ [ MR 840281 ]
  • Christophe Margerin - Pointwise pinched manifolds are space forms [ MR 840282 ]
  • Dana Nance - The multiplicity of generic projections of $n$-dimensional surfaces in $\mathbf {R}^{n+k}$ $(n+k\leq 4)$ [ MR 840283 ]
  • Seiki Nishikawa - Deformation of Riemannian metrics and manifolds with bounded curvature ratios [ MR 840284 ]
  • George Paulik - A regularity condition at the boundary for weak solutions of some nonlinear elliptic systems [ MR 840285 ]
  • Vladimir Scheffer - Solutions to the Navier-Stokes inequality with singularities on a Cantor set [ MR 840286 ]
  • Leon Simon - Asymptotic behaviour of minimal submanifolds and harmonic maps [ MR 840287 ]
  • Jean E. Taylor - Complete catalog of minimizing embedded crystalline cones [ MR 840288 ]
  • S. Walter Wei - Liouville theorems for stable harmonic maps into either strongly unstable, or $\delta $-pinched, manifolds [ MR 840289 ]
  • Brian White - A regularity theorem for minimizing hypersurfaces modulo $p$ [ MR 840290 ]
  • William P. Ziemer - Regularity of quasiminima and obstacle problems [ MR 840291 ]
  • Edited by John E. Brothers - Some open problems in geometric measure theory and its applications suggested by participants of the 1984 AMS summer institute [ MR 840292 ]
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