Topics in the Calculus of Variations
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Topics in the Calculus of Variations: (Advanced Lectures in Mathematics)

Topics in the Calculus of Variations: (Advanced Lectures in Mathematics)


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About the Book

This work illustrates two basic principles in the calculus of variations - the questions of existence of solutions, and closely related, the problem of regularity of minimizers. Chapter one studies variational problems for nonquadratic energy functionals defined on suitable classes of vector-valued functions where nonlinear constraints are also incorporated. Problems of this type arise for mappings between Riemannian manifolds or in nonlinear elasticity. Using direct methods for the existence of generalized minimizers is rather easy to establish, and it is then shown that regularity holds up to a set of small measure. Chapter two contains a short introduction into geometric measure theory, which serves as a basis for developing an existence theory for (generalized) manifolds with prescribed mean curvature form and boundary in arbitrary dimensions and co-dimensions. A major aspect of the book is that it concentrates on techniques, and presents methods which turn out to be useful for applications in regularity theorems, as well as for existence problems.

Table of Contents:
1 Degenerate Variational Integrals with Nonlinear Side Conditions, p-harmonic Maps and Related Topics.- 1.1 Introduction, Notations and Results for Minimizers.- 1.2 Linearisation of the Minimum Property, Extension of Maps.- 1.3 Proofs of the Basic Theorems.- 1.4 A Survey on p-Harmonic Maps.- 1.5 Variational Inequalities and Asymptotically Regular Integrands.- 1.6 Approximations for some Model Problems in Nonlinear Elasticity.- 2 Manifolds of Prescribed Mean Curvature in the Setting of Geometric Measure Theory.- 2.1 The Mean Curvature Problem.- 2.2 Some Facts from Geometric Measure Theory.- 2.3 A First Approach to the Mean Curvature Problem.- 2.4 General Existence Theorems, Applications to Isoperimetric Problems.- 2.5 Tangent Cones, Small Solutions, Closed Hypersurfaces.


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Product Details
  • ISBN-13: 9783528066239
  • Publisher: Friedrich Vieweg & Sohn Verlagsgesellschaft mbH
  • Publisher Imprint: Friedrich Vieweg & Sohn Verlagsgesellschaft mbH
  • Height: 244 mm
  • No of Pages: 148
  • Series Title: Advanced Lectures in Mathematics
  • ISBN-10: 3528066237
  • Publisher Date: 01 Jan 1994
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Width: 158 mm


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