Описание: The book presents a comprehensive exposition of extension results for maps between different geometric objects and of extension-trace results for smooth functions on subsets with no a priori differential structure (Whitney problems). The account covers development of the area from the initial classical works of the first half of the 20th century to the flourishing period of the last decade. Seemingly very specific these problems have been from the very beginning a powerful source of ideas, concepts and methods that essentially influenced and in some cases even transformed considerable areas of analysis. Aside from the material linked by the aforementioned problems the book also is unified by geometric analysis approach used in the proofs of basic results. This requires a variety of geometric tools from convex and combinatorial geometry to geometry of metric space theory to Riemannian and coarse geometry and more. The necessary facts are presented mostly with detailed proofs to make the book accessible to a wide audience.
Описание: The book presents a comprehensive exposition of extension results for maps between different geometric objects and of extension-trace results for smooth functions on subsets with no a priori differential structure (Whitney problems). The account covers development of the area from the initial classical works of the first half of the 20th century to the flourishing period of the last decade. Seemingly very specific these problems have been from the very beginning a powerful source of ideas, concepts and methods that essentially influenced and in some cases even transformed considerable areas of analysis. Aside from the material linked by the aforementioned problems the book also is unified by geometric analysis approach used in the proofs of basic results. This requires a variety of geometric tools from convex and combinatorial geometry to geometry of metric space theory to Riemannian and coarse geometry and more. The necessary facts are presented mostly with detailed proofs to make the book accessible to a wide audience.
Автор: Mariano Giaquinta; Guiseppe Modica; Jiri Soucek Название: Cartesian Currents in the Calculus of Variations II ISBN: 3642083757 ISBN-13(EAN): 9783642083754 Издательство: Springer Рейтинг: Цена: 30606.00 р. Наличие на складе: Есть у поставщика Поставка под заказ.
Описание: Non-scalar variational problems appear in different fields. In geometry, for in- stance, we encounter the basic problems of harmonic maps between Riemannian manifolds and of minimal immersions; related questions appear in physics, for example in the classical theory of a-models. Non linear elasticity is another example in continuum mechanics, while Oseen-Frank theory of liquid crystals and Ginzburg-Landau theory of superconductivity require to treat variational problems in order to model quite complicated phenomena. Typically one is interested in finding energy minimizing representatives in homology or homotopy classes of maps, minimizers with prescribed topological singularities, topological charges, stable deformations i. e. minimizers in classes of diffeomorphisms or extremal fields. In the last two or three decades there has been growing interest, knowledge, and understanding of the general theory for this kind of problems, often referred to as geometric variational problems. Due to the lack of a regularity theory in the non scalar case, in contrast to the scalar one - or in other words to the occurrence of singularities in vector valued minimizers, often related with concentration phenomena for the energy density - and because of the particular relevance of those singularities for the problem being considered the question of singling out a weak formulation, or completely understanding the significance of various weak formulations becames non trivial.
Название: Introduction to the Calculus of Variations ISBN: 1783265523 ISBN-13(EAN): 9781783265527 Издательство: World Scientific Publishing Цена: 8870.00 р. Наличие на складе: Есть у поставщика Поставка под заказ.
Описание: The calculus of variations is one of the oldest subjects in mathematics, and it is very much alive and still evolving.
Описание: The purpose of this book is to give a quick and elementary, yet rigorous, presentation of the rudiments of the so-called theory of Viscosity Solutions which applies to fully nonlinear 1st and 2nd order Partial Differential Equations (PDE).
Автор: Mariano Giaquinta; Stefan Hildebrandt Название: Calculus of Variations II ISBN: 3642081924 ISBN-13(EAN): 9783642081927 Издательство: Springer Рейтинг: Цена: 18161.00 р. Наличие на складе: Есть у поставщика Поставка под заказ.
Описание: This book describes the classical aspects of the variational calculus which are of interest to analysts, geometers and physicists alike. Volume 1 deals with the for- mal apparatus of the variational calculus and with nonparametric field theory, whereas Volume 2 treats parametric variational problems as weIl as Hamilton- Jacobi theory and the classical theory of partial differential equations of first order. In a subsequent treatise we shall describe developments arising from Hilbert's 19th and 20th problems, especially direct methods and regularity theory. Of the classical variational calculus we have particularly emphasized the often neglected theory of inner variations, i. e. of variations of the independent variables, which is a source of useful information such as monotonicity for- mulas, conformality relations and conservation laws. The combined variation of dependent and independent variables leads to the general conservation laws of Emmy Noether, an important tool in exploiting symmetries. Other parts of this volume deal with Legendre-Jacobi theory and with field theories. In particular we give a detailed presentation of one-dimensional field theory for non para- metric and parametric integrals and its relations to Hamilton-Jacobi theory, geometrieal optics and point mechanics. Moreover we discuss various ways of exploiting the notion of convexity in the calculus of variations, and field theory is certainly the most subtle method to make use of convexity. We also stress the usefulness of the concept of a null Lagrangian which plays an important role in several instances.
Описание: Infinite Horizon Variational Problems.- Extremals of Nonautonomous Problems.- Extremals of Autonomous Problems.- Infinite Horizon Autonomous Problems.- Turnpike for Autonomous Problems.- Linear Periodic Control Systems.- Linear Systems with Nonperiodic Integrands.- Discrete-Time Control Systems.- Control Problems in Hilbert Spaces.- A Class of Differential Inclusions.- Convex Processes.- A Dynamic Zero-Sum Game.
Автор: Bruce van Brunt Название: The Calculus of Variations ISBN: 1441923160 ISBN-13(EAN): 9781441923165 Издательство: Springer Рейтинг: Цена: 9083.00 р. Наличие на складе: Есть у поставщика Поставка под заказ.
Описание: Thecalculusofvariationshasalonghistoryofinteractionwithotherbranches of mathematics such as geometry and di?erential equations, and with physics, particularly mechanics. More recently, the calculus of variations has found applicationsinother?eldssuchaseconomicsandelectricalengineering. Much of the mathematics underlying control theory, for instance, can be regarded as part of the calculus of variations. This book is an introduction to the calculus of variations for mathema- cians and scientists. The reader interested primarily in mathematics will ?nd results of interest in geometry and di?erential equations. I have paused at times to develop the proofs of some of these results, and discuss brie?y v- ious topics not normally found in an introductory book on this subject such as the existence and uniqueness of solutions to boundary-value problems, the inverse problem, and Morse theory. I have made "passive use" of functional analysis (in particular normed vector spaces) to place certain results in c- text and reassure the mathematician that a suitable framework is available for a more rigorous study. For the reader interested mainly in techniques and applications of the calculus of variations, I leavened the book with num- ous examples mostly from physics. In addition, topics such as Hamilton's Principle, eigenvalue approximations, conservation laws, and nonholonomic constraints in mechanics are discussed. More importantly, the book is written on two levels. The technical details for many of the results can be skipped on the initial reading. The student can thus learn the main results in each chapter and return as needed to the proofs for a deeper understanding.
Описание: Proceedings of the IUTAM Symposium held in Paris, France, 22-25 April 1997
Автор: P.A. Griffiths Название: Exterior Differential Systems and the Calculus of Variations ISBN: 0817631038 ISBN-13(EAN): 9780817631031 Издательство: Springer Рейтинг: Цена: 11173.00 р. Наличие на складе: Есть у поставщика Поставка под заказ.
Описание: Ideals e) Exterior Differential Systems EULER-LAGRANGE EQUATIONS FOR DIFFERENTIAL SYSTEMS ~liTH ONE I. Classical Examples b) Variational Equations for Integral Manifolds of Differential Systems c) Differential Systems in Good Form; iii) the Genera 1 Euler Equations 2 K /2 ds b) Examples: i) the Euler Equations Associated to f for lEn;
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