A Compact Course on Linear PDEs

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.

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.- A Partition of unity.- B Lipschitz continuous and smooth domains.- C Integration by parts for smooth functions and vector fields.- D Reynolds transport theorem.- E Gronwall lemma.- F Necessary and su cient conditions for the well-posedness of the variationalproblem.

"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)

“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)