Vorlesung: Elliptic PDEs: Schauder theory (SoSe 2025)

!!! TO BE UPDATED - PLEASE CHECK BACK !!!

Lecture (Vorlesung):
Mon 10-12 in A 027.   LSF

First time (Erstes Mal): 28 April 2025.

Exercises (Übungen):
There are NO exercises!

Synopsis (Kurzbeschreibung):
The modern theory of Partial Differential Equations (PDEs) is (mainly) about proving estimates (inequalities).
These are used to study the main questions of wellposedness: existence (solvability), uniqueness, and stability (continuity in the data).
This course illustrates this on 2nd order linear uniformly elliptic PDEs (generalising the Laplace and Poisson equations).
For the study of classical solutions to boundary value problems (BVPs) for such equations with Hölder continuous coefficients, this is Schauder theory, and the estimates are called Schauder estimates.
The developed results and techniques are important building blocks for the further study of quasi-linear and fully nonlinear (2nd order elliptic) equations.
See also this description.

Audience (Hörerkreis):
Master students of Mathematics (PO 2011: WP 17.2, 18.1, 18.2, 44.3, 45.2, 45.3 - PO 2021: WP 13, 16, 40, 41), TMP-Master.
Motivated Bachelor students of Mathematics are welcome; they will get "Schein" if pass the course (contact the Lecturer).

Credits:
3 ECTS.

Prerequisites (Vorkenntnisse):
Analysis I-III. Basic knowledge of Functional Analysis and/or Partial Differential Equations is helpful, but not required.

Language (Sprache):
English. (Die mündliche Prüfung kan auch auf Deutsch gemacht werden).

Exam (Prüfung):
There will be an oral exam (30 min; dates to be announced) (Es wird eine mündliche Prüfung geben).
See separate webpage (Moodle).

Content (Inhalt) (PRELIMINARY!):

0. Introduction and motivation
1. Laplace & Poisson equation & Harmonic functions
2. Maximum Principle & a priori estimates
3. Solvability Poisson equation in R^d and in domains
4. Hölder spaces, characterisations, and interpolation
5. A priori estimate in R^d for Laplace operator
6. Schauder estimates & solvability in R^d
7. Interior Schauder estimates
8. Boundary Schauder estimates
9. Solvability of Dirichlet BVP in domains
10. Regularity

Literature:
In Moodle you will find a copy of the notes from the lecture.
Above you will find a short description of the content of the lecture.

Supplementary literature:

• [GT] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, 1997)
• [FR] X. Fernández-Real and X. Ros-Oton, Regularity Theory for Elliptic PDE (EMS, 2022)
• [J] J. Jost, Partial Differential Equations (Springer, 2013)
• [K] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces (AMS, 1996)
• [HL] Q. Han and F. Lin, Elliptic Partial Differential Equations (AMS, 2000)
• [H] Q. Han, Nonlinear Elliptic Equations of the Second Order (AMS, 2016)
• [CW] Y.-Z. Chen and L.-C. Wu, Second order elliptic equations and elliptic systems (AMS, 1998)
• [WKK] E. Wienholtz, H. Kalf, T. Kriecherbauer, Elliptische Differentialgleichungen zweiter Ordnung (Springer, 2009)

Office hours:
See Moodle.

To access the course material, you need to sign up (opens 14. April 2025) in Moodle here (Psword: Schauder).

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Letzte Änderung: 07 April 2025.

Thomas Østergaard Sørensen