Advanced Functional Analysis, 5 op

The course focuses on the theory of unbounded linear operators, especially differential operators, and on analysis of elliptic partial differential equations. In particular, the course introduces the functional analytic tools for the study of existence, regularity properties, and approximation of solutions of elliptic equations.

List of the main topics:

• Closed operators on Banach spaces, definition and characterizations
• Closed-Graph Theorem, Open Mapping Theorem, Uniform Boundedness Principle
• Operator representation of elliptic partial differential equations
• Theory of Sobolev spaces, Sobolev embedding theorems
  
• Definitions of classical, strong, and weak solutions of elliptic equations
• Existence and regularity of solutions of elliptic equations
• Compact embeddings and spectral representation of differential operators
• The Fourier transform and Fourier multipliers in the analysis of elliptic equations