Topics in the Calculus of Variations

Dieses Buch richtet sich an Mathematiker und Physiker mit Interesse an neuen Entwicklungen auf einem Teilgebiet der nichtlinearen Analysis, wo verschiedene Methoden zur Lösung variationeller Probleme zum Einsatz kommen. Einige Grundkenntnisse sind erforderlich, neue Begriffsbildungen werden am konkreten Problem entwickelt.

This report grew out of a series of lectures given at the East China Institute of Technology, Nanjing, during September 1992. The purpose of this book is to make beginning research students familiar with some problems in varia tional calculus which have been chosen following my personal taste but with the attempt to illustrate two basic principles in the calculus of variations which are the fundamental question of existence of (generalized) solutions and closely related the question of regularity. Chapter one is devoted to the study of variational integrals for vectorvalued functions which began with the pioneering work of Morrey [68] in the thirties. We concentrate on problems where also nonlinear side conditions are imposed on the classes of admissi ble comparison functions. As special cases we include mappings whose range is forced to lie in some Riemannian manifold possibly with boundary or functions whose Jacobian is required to be strictly positive. The variational integrals under consideration are typically nonquadratic with respect to the gradient which immediately leads us to classes of degenerate elliptic systems. Let us mention some of the most important applications: • p-harmonic maps between Riemannian manifolds • systems of degenerate variational inequalities • model problems in nonlinear elasticity. Usually by working in appropriate Sobolev spaces, the existence of gener alized solutions is rather easy to establish (see [7]) but leads to apriori dis continuous functions.

Autorentext

Prof. Dr. Martin Fuchs ist an der Universität des Saarlandes im Bereich Variationsrechnung und partielle Differentialgleichungen mit Bezügen zur mathematischen Physik und Differentialgeometrie tätig.

Klappentext

This book illustrates two basic principles in the calculus of variations which are the question 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 vectorvalued functions where also nonlinear constraints are incorporated. Problems of this type arise for mappings between Riemannian manifolds or in nonlinear elasticity. Using direct methods 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 codimensions. One major aspect of the book is to concentrate on techniques and to present methods which turn out to be useful for applications in regularity theorems as well as for existence problems.

Inhalt
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.