Module 24-M-AN-RTH Regularity Theory for Partial Differential Equations

Non-official translation of the module descriptions. Only the German version is legally binding.

Students master advanced content and methods of Regularity Theory for Partial Differential Equations, in particular they can independentlycarry out very complex proofs in this area requiring a high level of mathematical expertise. They are able to analyse regularity properties of solutions of partial and integro-differential equations. Emphasis will be put on elliptic and parabolic differential equations of differentiability order two. The students can apply and compare different solution concepts. They are able to apply core ideas of the theory, such as function spaces, approximations by finite differences and bootstrapping techniques, Sobolev and Hölder spaces including the embedding theorems, in different contexts.
Students will be introduced to current research questions in the field of Regularity Theory for Partial Differential Equations. They are able to recognise and assess further development opportunities and research goals.
Furthermore, students recognise further-reaching connections to mathematical issues that have already been worked out. They can transfer and apply the knowledge and methods they have learnt so far to deeper mathematical problem areas. Students also expand their mathematical intuition as a result of more intensive study.
In combination with other in-depth modules, they will be able to write their own research papers, e.g. a master's thesis in the field of Regularity Theory for Partial Differential Equations
In the tutorials, students develop their ability to discuss mathematical topics and thus further prepare themselves for the requirements of the Master's module, in particular for the scientific discussion within the Master's seminar presentation and the defence of their Master's thesis.

The following advanced content of teaching from the field of Regularity Theory for Partial Differential Equations is compulsory:

• Difference-quotient method
• Sobolev regularity
• embedding theorems
• Calderon-Zygmund decomposition
• Schauder theory
• iteration techniques of De Giorgi and Moser

In addition, the following content of teaching can be covered, for example:

• Hölder-regularity for fully nonlinear partial differential equations
• kinetic equations
• degenerate partial differential equations
• Besov spaces

This module prepares the content of a master's thesis.

Measure and Integration, Lebesgue spaces, basics knowledge of elliptic partial differential equations

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