2 semester

1

Name of course

Sobolev spaces

2

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:

  • to use concepts of generalized derivatives and Sobolev spaces in mathematical problems;
  • to prove main theorems of Sobolev spaces theory;
  • to construct a resolvent  for Fredholm and Volterra equations;
  • to use main results of the integral theory in practice;
  • to use theoretical and practical basic principles of the theory in mathematics.

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Background

Real and complex analysis, functional analysis, equations in partial derivatives.

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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. Hardy-Littlewood maximal function and its properties.

Riss potentials and their properties on the basis of Hardy-Littlewood maximal function. Necessary conditions of boundedness. Hardy-Littlewood-Sobolev 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 . Gagliardo-Nirenberg theorem. Survey of other properties of Sobolev spaces.

9

Bibliography

  • E.Stein, Singular integrals and differentiability properties of functions, Princeton, New Jersey, Princeton University Press, 1970.
  • E.Stein, G.Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton, New Jersey, Princeton University Press, 1971.
  • V. Maz’ja, Sobolev spaces, Berlin, Heidelberg, New-York, Tokyo, Springer-Verlag, 1985.

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Methods of teaching

Comparative method, problematic method, dialog and heuristic method, method of forming knowledge of a personal significance.

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Language

English

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Requirements, monitoring

  • evaluation of an exercise.

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.

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Form of assessment

Exam