# Topics in the Calculus of Variations

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*Martin Fuchs*

A publication of Vieweg+Teubner

This book illustrates two basic principles in the calculus of variations which are the question of existence of solutions and the closely related 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 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 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.

A publication of Vieweg+Teubner. The AMS is exclusive distributor in North America. Vieweg+Teubner Publications are available worldwide from the AMS outside of Germany, Switzerland, Austria, and Japan.