Functional Analysis for PDEs (45hours)

• Functional Analysis deals with classes of functions viewed as vectors in some functional spaces, typically infinite-dimensional. It has many applications in various fields of applied science including quantum mechanics, signal processing, etc; especially phenomena involving Partial Differential Equations. The latter theory represents powerful tools for modeling and solving concrete physical and engineering problems.
• In this course, we propose a detailed exposition of some modern functional analytical tools that avoid difficult prerequisites. The topics will revolve around Hilbertian analysis, spectral theory, sesquilinear/bilinear forms, weak derivatives, etc. The developed mathematical tools will be used to study prototype models of PDEs: Elliptic equations, Parabolic equations and Hyperbolic equations.
• Prerequisites: Lebesgue integration and 𝐿𝑝-spaces. The intended audience includes engineering and computer science students with some Lebesgue integration background.

The following topics will be covered

• Hilbertian analysis tools: Hilbert bases, spectral theory of compact operators, sesquilinear/bilinear forms, operators defined by a sesquilinear/bilinear form, and Lax Milgram theorem.
• Distributions and Sobolev spaces: test functions, density in 𝐿𝑝-spaces, weak derivatives, Sobolev spaces, Poincaré inequality.
• Elliptic PDEs: existence, uniqueness and regularity of weak solutions, variational formulation, energy estimates.
• Parabolic PDEs: the heat equation, initial value problems, generalized solutions, existence, uniqueness, smoothing effect, energy estimates, semilinear equations.
• Hyperbolic PDEs: the wave equation, energy conservation, existence, uniqueness and regularity of generalized solutions, energy estimates.
• Schrödinger equation: energy conservation, existence, uniqueness and regularity of generalized solutions, energy estimates.

Course slides and reading material will be available from the instructor.

• Optional Textbooks:
  • L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, RI, second edition, 2010.
  • H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, 2010.

1. Midterm Exam: 25%
2. Project: 25%
3. Final Exam: 50%