/ October 26, 2021

Lecture notes: A primer on functional Analytic methods for PDEs

FAU DCN-AvH. Friedrich-Alexander Universität Erlangen-Nürnberg (Germany)
Period: Summer semester 2021

These lecture notes cover topics suitable for an introductory 5 ECTS (2+1 weekly hours) primer on functional analytic methods in partial differential equations.

The goal of the course is to provide a solid foundation in the aspects and tools of functional analysis used in modern abstract PDE analysis. Of course such a course cannot comprehensively consider all aspects of either functional analysis or PDE applications within a 5 ECTS scope, so there will be shortcuts and several results without proofs. The proofs will be part of the exercises or can be found easily in the literature mentioned.

The overarching idea is to establish essentially three fundamental ideas which are prevalent in modern PDE theory basing on functional analysis; these being:
• Positivity (ellipticity)
• Fredholm theory
• Diagonalization

The theoretical foundation will therefor lead to bilinear forms, compactness, and spectral theory, particularly in Hilbert spaces.
Due to their important compactness properties, weak topologies will also be considered. Sobolev spaces are then the natural environment to consider elliptic boundary value problems in.
Due to time constraints, time-dependent problems (evolution equations) will merely be mentioned; however, the topics taught will transfer very well to this more advanced topic.

These lecture notes are written during the summer term 2021 at Friedrich-Alexander-Universität Erlangen-Nürnberg. There will certainly be mistakes and inconsistencies, and maybe I will want to restructure some parts later on. In this sense, these notes are a work in progress.

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1 Introduction    3

1.1 Physical motivation………………………… 3
1.2 Linear algebra inspirations……………………5

 

2 Fundamentals    7

2.1 Normed vector spaces ………………………. 7
2.2 Linear operators………………………….. 10

 

3 Dual space, linear functionals and weak topology    13

3.1 Dual space and linear functionals…………………. 13
3.2 Weak convergence ………………………… 15
3.3 Examples……………………………… 18

 

4 Linear operators in Banach spaces  18

4.1 Main theorems about linear operators in Banach spaces ……….. 18
4.2 Adjoint operators …………………………. 23
4.3 Compact operators ………………………… 27
4.4 Examples……………………………… 32

 

5 Hilbert spaces    34

5.1 Dual space of a Hilbert space …………………… 38
5.2 The Lax-Milgram lemma……………………… 41
5.3 Orthonormal basis ………………………… 43

 

6 Spectral theory    45

6.1 Spectrum of compact operators ………………….. 49
6.2 Spectral theorem for normal operators………………. 50

 

7 Sobolev spaces    53

7.1 Weak derivative………………………….. 54
7.2 Basics in Sobolev spaces ……………………… 56
7.3 Extension, embedding and compactness……………… 61

 

8 Linear elliptic partial differential equations    66

8.1 Weak formulation…………………………. 66
8.2 Existence and uniqueness for uniformly elliptic operators . . . . . . . . 69
8.3 General linear elliptic operators………………….. 72

 

Comments?

For any comments and suggestions and of course particularly so for any notification of errors, please contact Hannes Meinlschmidt

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