A Compact Course on Linear PDEs

by Alberto Valli

Overview

This textbook is devoted to second order linear partial differential equations. The focus is on variational formulations in Hilbert spaces. It contains elliptic equations, including the biharmonic problem, some useful notes on functional analysis, a brief presentation of Sobolev spaces and their properties, some basic results on Fredholm alternative and spectral theory, saddle point problems, parabolic and linear Navier-Stokes equations, and hyperbolic and Maxwell equations. Almost 80 exercises are added, and the complete solution of all of them is included. The work is mainly addressed to students in Mathematics, but also students in Engineering with a good mathematical background should be able to follow the theory presented here. This second edition has been enriched by some new sections and new exercises; in particular, three important equations are now included: the biharmonic equation, the linear Navier-Stokes equations and the Maxwell equations. 

Product Details

ISBN-13:	9783031359767
Publisher:	Springer-Verlag New York, LLC
Publication date:	08/29/2023
Series:	UNITEXT , #154
Sold by:	Barnes & Noble
Format:	eBook
File size:	28 MB
Note:	This product may take a few minutes to download.

About the Author

​Alberto Valli is a Professor of Mathematical Analysis at University of Trento. His research activity has focused on the mathematical analysis of linear and nonlinear partial differential equations, in particular in fluid dynamics and electromagnetism, and on their numerical approximation by means of the finite element method. He published more than 80 papers in prestigious international journals and 4 books on partial differential equations and their numerical approximation.

Table of Contents

1. Introduction.- 2. Second order linear elliptic equations.- 3. A bit of functional analysis.- 4. Weak derivatives and Sobolev spaces.- 5. Weak formulation of elliptic PDEs.- 6. Technical results.- 7. Additional results.- 8. Saddle points problems.- 9. Parabolic PDEs.- 10. Hyperbolic PDEs.- Appendix A: Partition of unity.- Appendix B: Lipschitz continuous and smooth domains.- Appendix C: Integration by parts for smooth functions and vector fields.- Appendix D: Reynolds transport theorem.- Appendix E: Gronwall lemma.- Appendix F: Necessary and sufficient conditions for the well-posedness of the variational problem.

Editorial Reviews

The book is designed to be compact, and it is – very. The author treats many topics very quickly, and the connecting links are also much abbreviated. A lot is demanded of the reader, even one starting with a background of classical PDEs at the undergraduate level.” (Bill Satzer, MAA Reviews, December 27, 2021)

From the Publisher