Topics in the calculus of variations
著者
書誌事項
Topics in the calculus of variations
（Advanced lectures in mathematics）
Vieweg, c1994
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注記
Bibliography: p. 139-144
Includes index
内容説明・目次
内容説明
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.
目次
Contents: Degenerate Variational Integrals with Nonlinear Side Conditions, p-harmonic Maps and Related Topics: - Introduction, Notations and Results for Minimizers - Linearisation of the Minimum Property, Extension of Maps - Proofs of the Basic Theorems - A Survey on p-Harmonic Maps - Variational Inequalities and Asymptotically Regular Integrands - Approximations for some Model Problems in Nonlinear Elasticity - Manifolds of Prescribed Mean Curvature in the Setting of Geometric Measure Theory: -The Mean Curvature Problem - Some Facts from Geometric Measure Theory - A First Approach to the Mean Curvature Problem - General Existence Theorems, Applications to Isoperimetric Problems - Tangent Cones, Small Solutions, Closed Hypersurfaces.Kapitel 1 behandelt Variationsprobleme mit nichtlinearen Nebenbedingungen wie sie in der mathematischen Physik insbesondere der Elastizitatstheorie) gelaufig sind. Weiter Anwendungen ergeben sich im Zusammenhang mit Energiefunktionalen fur Abbildungen zwischen Riemannschen Mannigfaltigkeiten oder auch beim Studium von Variationsungleichungen.Kapitel 2 beginnt mit einer kurzen Einfuhrung in die Geometrische Masstheorie, um mit diesen Techniken einen variationellen Zugang zur Konstruktion von Mannigfaltigkeiten mit vorgeschriebenem Krummungsverhalten zu formulieren.
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