This book presents a simple and novel theory of integration, both real and vectorial, particularly suitable for the study of PDEs. This theory allows for integration with values in a Neumann space E, i.e. in which all Cauchy sequences converge, encompassing Neumann and Frechet spaces, as well as "weak" spaces and distribution spaces.

We integrate "integrable measures", which are equivalent to "classes of integrable functions which are a.e. equals" when E is a Frechet space. More precisely, we associate the measure f with a class f, where f(u) is the integral of fu for any test function u. The classic space Lp( ;E) is the set of f, and ours is the set of f; these two spaces are isomorphic.

Integration studies, in detail, for any Neumann space E, the properties of the integral and of Lp( ;E): regularization, image by a linear or multilinear application, change of variable, separation of multiple variables, compacts and duals. When E is a Frechet space, we study the equivalence of the two definitions and the properties related to dominated convergence.

About the Author

Jacques Simon is Director Emeritus of Research at the CNRS, France. His research focuses on partial differential equations, particularly on the spaces used by these equations and on shape optimization.

Book Information
ISBN 9781786300133
Author Jacques Simon
Format Hardback
Page Count 448
Imprint ISTE Ltd and John Wiley & Sons Inc
Publisher ISTE Ltd and John Wiley & Sons Inc