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Lecture Note

Lectures on elliptic partial differential equations (Pseudodifferential operator approach)

Alexander Komech


Main goal of these lectures is a concise introduction to classical theory of elliptic partial differential equations. The lecture course was delivered at Mathematical Department of Vienna University in the fall of 2006.

We give detailed explanations of basic operations on distributions including the Fourier transform, the Sobolev spaces and embedding theorems, methods of pseudodifferential operators and their applications to obtain fundamental properties of elliptic partial differential equations: the Fredholm properties and smoothness of solutions.

This one-semester course is aimed at the third year students specialised in Mathematics, Physics and Engineering. We assume that the reader is familiar with the standard Calculus, the classical Fourier transform, and basic facts from the Hilbert space such as completeness of $L^2(R^n)$, and the Fredholm theory.

MSC Codes:
35G05, 35S05, 46F10
distribution, Fourier transform, elliptic operator