Pseudodifferential-Operatoren und Anwendungen

The class of pseudo-differential operators is a natural extension of the class differential operators, and contains many more important operators. For example, if an elliptic differential operator has an inverse, then this inverse is a pseudo-differential operator as well. Inside the class of pseudo-differential operators one can also take complex powers, and expression like $\sqrt{1-\Delta}$ are well-defined pseudo-differential operators.

The lecture will start with an introduction of pseudo-differential operators on euclidean space. Here Fourier transformation is an important technique, and we will see how one can use the symbol calculus to invert many pseudo-differential operators up to smoothing operators. Applications to partial differential equations will be developed, we discuss adjoints and composition of operators, and we discuss regularity theory.

The definition of the operators on euclidean spaces is then used to define pseudo-differential operators on manifolds. We study the behavior of the wave front set of solutions of hyperbolic equations and study the relation to geodesics on an associated Lorentzian manifold. We get a very conceptual description of the propagation of waves in space-time.

In the last part of the lecture we intend to give an overview over the Boutet-de-Monvel calculus, which extends the class of usual pseudodifferential operators to model boundary value problems and their solution operators.

For related literature we refer to the website of the lecture series.