Overview
- Provides a new view of traces and the divergence theorem
- Uses integrals based on finitely additive measures that were not considered before as a key tool
- Derives Gauss-Green formulas without a trace function on the boundary and treats apparently intractable singularities
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2372)
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About this book
This book provides a new approach to traces, which are viewed as linear continuous functionals on some function space. A key role in the analysis is played by integrals related to finitely additive measures, which have not previously been considered in the literature. This leads to Gauss-Green formulas on arbitrary Borel sets for vector fields having divergence measure as well as for Sobolev and BV functions. The integrals used do not require trace functions or normal fields on the boundary and they can deal with inner boundaries. For the treatment of apparently intractable degenerate cases a second boundary integral is used. The calculus developed here also allows integral representations for the precise representative of an integrable function and for the usual boundary trace of Sobolev or BV functions. The theory presented gives a new perspective on traces for beginners as well as experts interested in partial differential equations. The integral calculus might also be a stimulating tool for geometric measure theory.
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Table of contents (4 chapters)
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Front Matter
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Back Matter
Authors and Affiliations
About the authors
Friedemann Schuricht is Professor of Mathematics at TU Dresden, Germany. His main research interests are in nonlinear analysis and its applications. In particular, he has worked on problems in the calculus of variations, partial differential equations, nonsmooth analysis, geometric analysis, and related applications in continuum mechanics.
Moritz Schönherr studied mathematics and completed his doctorate at TU Dresden, Germany. He has worked on problems in partial differential equations, measure theory and the foundations of continuum mechanics. Currently he has a business position in Copenhagen, Denmark.
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Bibliographic Information
Book Title: A Theory of Traces and the Divergence Theorem
Authors: Friedemann Schuricht, Moritz Schönherr
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-3-031-86664-7
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2025
Softcover ISBN: 978-3-031-86663-0Published: 12 August 2025
eBook ISBN: 978-3-031-86664-7Published: 11 August 2025
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: XIII, 174
Number of Illustrations: 6 b/w illustrations
Topics: Measure and Integration, Analysis