Compact Course on Linear PDEs

​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.

• 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.