A Compact Course on Linear PDEs

1. Front Matter

   Pages i-xiii

   PDF

2. Introduction

   • Alberto Valli

   Pages 1-8

3. Second Order Linear Elliptic Equations

   • Alberto Valli

   Pages 9-29

4. A Bit of Functional Analysis

   • Alberto Valli

   Pages 31-38

5. Weak Derivatives and Sobolev Spaces

   • Alberto Valli

   Pages 39-51

6. Weak Formulation of Elliptic PDEs

   • Alberto Valli

   Pages 53-81

7. Technical Results

   • Alberto Valli

   Pages 83-107

8. Additional Results

   • Alberto Valli

   Pages 109-149

9. Saddle Points Problems

   • Alberto Valli

   Pages 151-175

10. Parabolic PDEs

    • Alberto Valli

    Pages 177-193

11. Hyperbolic PDEs

    • Alberto Valli

    Pages 195-211

12. Back Matter

    Pages 213-235

    PDF

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 some basic results on Fredholm alternative and spectral theory, some useful notes on functional analysis, a brief presentation of Sobolev spaces and their properties, saddle point problems, parabolic equations and hyperbolic equations. Many 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.

• Elliptic equations
• Saddle-point formulation
• Parabolic equations
• Hyperbolic equations
• Analysis
• Hilbert spaces
• partial differential equations

“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)

• Department of Mathematics, University of Trento, Trento, Italy

  Alberto Valli

Alberto Valli is professor of Mathematical Analysis at the Department of Mathematics of the University of Trento. His research activity has concerned the analysis of partial differential equations in fluid dynamics and electromagnetism and of their numerical approximation by the finite element method. He also studied domain decomposition methods and their use in the discretization of partial differential equations. On these topics he wrote three books.

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