Advanced Methods in Mathematical Analysis (FSI-SDR-A)

The aim of the course is to provide students with an overview of modern and advanced methods (based mainly on functional analysis) suitable, in particular, for the qualitative analysis of linear as well as nonlinear problems for differential equations. Students will became familiar with the generalized formulations (weak and variational) of the problems.

Differential calculus, integral calculus, linear algebra, ordinary and partial differential equations, functional analysis.

The course is devoted to two basic areas, which partially overlap. The first part is an introduction to the so-called modern theory of (partial) differential equations and related concepts such as generalized functions, Sobolev spaces, embedding theorems, weak and variational formulation of problems. The second part is devoted to selected methods of nonlinear analysis. These are mainly topological methods, monotonicity methods and variational methods. Applications of these methods to different types of problems are also discussed. Elements of differential calculus in normed linear spaces are mentioned.

Course-unit credit is awarded on condition of having attended the seminars actively (the attendance is compulsory) and passed a control test during the semester.
Examination: It has an oral form. Theory as well as examples will be discussed. Students should show they are familiar with basic topics and principles and how they are able to illustrate the theory in particular situations.

Motivation.
Overview of selected basic concepts of functional analysis and theory of differential equations.
Generalized functions and generalized derivatives.
Sobolev spaces.
Embedding theorems.
Trace of a function.
Weak and variational formulation of linear problems.
Lax-Milgram lemma.
Differential calculus in normed linear spaces.
Topological methods (Brouwer theorem, Schauder theorem).
Applications of fixed point theorems.
Theory of monotone operators.
Applications of monotonicity methods.
Variational methods.
Applications of variational methods.

Illustration of the concepts presented at the lectures on examples. Application of theoretical results in particular cases and in selected equations.

K. Deimling, Nonlinear functional analysis, Springer 1985.

P. Drábek, J. Milota, Methods of nonlinear analysis, Birkhauser 2013.

L. C. Evans, Partial differential equations, American Mathematical Society 2010.

J. Franců, Moderní metody řešení parciálních diferenciálních rovnic, FSI VUT Brno 2019.

P. Řehák, Advanced methods in mathematical analysis, FME BUT Brno 2025.