Single-Course English 5 ECTS

Pseudodifferential operators for boundary value problems

Overall Course Objectives

Modern analysis of boundary value problems for partial differential equations (PDE) relies on the theory of pseudodifferential operators. In addition to establishing locally the existence, uniqueness and regularity of solutions of rather general elliptic PDE problems, this analysis can be linked with well-developed tools from topology, differential geometry and microlocal analysis to allow additional fundamental insight into the properties and behavior of solutions. In certain cases this additional insight can improve the mathematical modelling, the physical understanding, and the numerical solution efficiency and stability for the given boundary value problem. The purpose of this course is to teach the theory of standard pseudodifferential operators with smooth symbols, including the rigorous definition, continuity properties, symbol calculus and parametrix construction. This is then applied in the analysis of selected boundary value problems.

Explain the relevance and the fundamental properties of symbols of standard pseudodifferential operators.
Prove the properties of and use the algebra of symbols of standard pseudodifferential operators.
Construct symbols of compounds and adjoints of standard pseudodifferential operators.
Prove mapping properties of standard ps.d.o.
Construct symbols of parametrices of elliptic ps.d.o.
Construct and prove results regarding symbols in localization of boundary problems via diffeomorphic charts.
Use ps.d.o. for the analysis of local elliptic boundary problems.
Prove results relating Essential Support of ps.d.o. and Wavefront Sets of distributions.

Smooth symbol classes: definition, results regarding compounds and adjoints. Fourier integral representation and Schwartz kernels of ps.d.o. Mapping properties of standard pseudodifferential operators. Parametrices of elliptic ps.d.o. Essential support, wavefront sets and microlocal regularity. Layer potentials. Transformation using local diffeomorphic charts and extension by zero.

01715

Lectures presenting, proving and contextualizing the main results.