Integration on Infinite-Dimensional Surfaces and Its Applications

Uglanov, A. V.

Rilegato

ISBN 10: 0792361334 ISBN 13: 9780792361336

Casa editrice: Kluwer Academic Pub, 2000

It seems hard to believe, but mathematicians were not interested in integration problems on infinite-dimensional nonlinear structures up to 70s of our century. At least the author is not aware of any publication concerning this theme, although as early as 1967 L. Gross mentioned that the analysis on infinite­ dimensional manifolds is a field of research with rather rich opportunities in his classical work [2. This prediction was brilliantly confirmed afterwards, but we shall return to this later on. In those days the integration theory in infinite­ dimensional linear spaces was essentially developed in the heuristic works of RP. Feynman [1], I. M. Gelfand, A. M. Yaglom [1]). The articles of J. Eells [1], J. Eells and K. D. Elworthy [1], H. -H. Kuo [1], V. Goodman [1], where the contraction of a Gaussian measure on a hypersurface, in particular, was built and the divergence theorem (the Gauss-Ostrogradskii formula) was proved, appeared only in the beginning of the 70s. In this case a Gaussian specificity was essential and it was even pointed out in a later monograph of H. -H. Kuo [3] that the surface measure for the non-Gaussian case construction problem is not simple and has not yet been solved. A. V. Skorokhod [1] and the author [6,10] offered different approaches to such a construction. Some other approaches were offered later by Yu. L. Daletskii and B. D. Maryanin [1], O. G. Smolyanov [6], N. V.

Recensione:

`This book is highly recommended for every mathematician with an interest in recent developments in functional analysis, measure and integration, differential equations, approximations, calculus of variations and stochastic processes.' Mathematical Reviews, 2001c

Contenuti:

Preface. Introduction. Basic Notations. 1. Vector Measures and Integrals. 1.1. Definitions and Elementary Properties. 1.2. Principle of Boundedness. 1.3. Passage to the Limit Under Integral Sign. 1.4. Fubini's Theorem. 1.5. Reduction of a Vector Integral to a Scalar Integral. 2. Surface Integrals. 2.1. Smooth measures. 2.2. Definition of Surface Measures. The Invariance Theorem. 2.3. Elementary Properties of Surface Measures and Integrals. 2.4. Iterated Integration Formula. 2.5. Integration by Parts Formula. 2.6. Gauss-Ostrogradskii and Green's Formulas. 2.7. Vector Surface Measures. 2.8. A Case of the Banach Surfaces. 2.9. Some Special Surface Integrals. 3. Applications. 3.1. Distributions on a Hilbert Space. 3.2. Infinite-Dimensional Differential Equations. 3.3. Integral Representation of Functions on a Banach Space. Green's Measure. 3.4. On Parabolic and Elliptic Equations in a Space of Measures. 3.5. About the Amoothness of Distributions of Stochastic Functionals. 3.6. Approximation of Functions of an Infinite-Dimensional Argument. 3.7. On a Differentiable Urysohn Function. 3.8. Calculus of Variations on a Banach Space. Comments. References. Index.