Distributions and Differential Operators SS 18

The aim of this lecture is to present the theory of distributions of Laurent Schwartz in a rigorous, accessible way, together with applications to different areas of mathematics, like (systems of) linear partial differential equations and Fourier analysis, as well as to problems from physics. The need for distributions arises from the unpleasant fact that not every function is differentiable. The purpose of distribution theory is to remedy this flaw; indeed the space of distributions is essentially the smallest extension of the space of continuous functions where differentiation is always well defined. In a certain sense, distributions are to functions what the real numbers are to the rational numbers.

In the course of the lecture we will be dealing with basic properties of distributions, elementary operations with distributions, convolution of distributions, Fourier transform of temperate distributions, fundamental solutions for linear partial differential operators, and if time permits the Malgrange-Ehrenpreis Theorem which assures the existence of a fundamental solutions for every non-zero linear partial differential operator with constant coefficients.

Prerequisites for attending the lecture are kept to a minimum. Students are only assumed to have a sound knowledge of linear algebra and analysis of several variables.