1 
Name of course 
Sobolev spaces 
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Year 
1, speciality Mathematics 
3 
Semester 
2 
4 
Credits 
4 
5 
Lecturer’s scientific and academic degree, First Name, Patronic Name, Surname 
Doctor of Sciences (habilitated), Professor, Veniyamin Ruhoravič Krotaū 
6 
Goals of course 
Progress in a professional competence of students, forming of a concept about technical potential of the discipline and role of Sobolev spaces in scientific problems. At the end of the course a student will be able:

7 
Background 
Real and complex analysis, functional analysis, equations in partial derivatives. 
8 
Contents 
Dense classes in . Linear functional representation theorem in space. Weak compactness theorem for . Absolutely continuous functions and Sobolev classes. Classes of functions of bounded variation due to Riss. Moduli of continuity in and their properties. Hölders classes. Comparation of classes and its description in terms of a global derivative. Weak derivatives and their existence. Coverings lemma. HardyLittlewood maximal function and its properties. Riss potentials and their properties on the basis of HardyLittlewood maximal function. Necessary conditions of boundedness. HardyLittlewoodSobolev theorem of fractional integration. General theorem of the convergence for approximative identities. Applications of approximative identities to differentiation of multiple integrals to harmonic functions and to temperatures. General derivatives and its properties. Sobolev classes. Description of Sobolev spaces without using of generalized derivatives. Embedding theorems for Sobolev spaces: case . GagliardoNirenberg theorem. Survey of other properties of Sobolev spaces. 
9 
Bibliography 

10 
Methods of teaching 
Comparative method, problematic method, dialog and heuristic method, method of forming knowledge of a personal significance. 
11 
Language 
English 
12 
Requirements, monitoring 
Final grade on the discipline is a cumulative characteristic which is formed on a base of an undergraduate student’s knowledge during the total period of monitoring and on the semester grade(s). Monitoring grade component can attain 50% of a final grade. 
13 
Form of assessment 
Exam 