Functional analysis and elements of mathematical physics
Read from 7-th to 9-th semester.
7,8 semesters 4 hours of lectures per week, 9th semester, 2 hours of lectures per week
The course of functional analysis is read for 3 semesters (7-9) and contains both traditional material on the basis of functional and real analysis, and application of linear and nonlinear problems in mathematical physics. Some lectures (numbers marked with letter) are devoted to examples, as well as the development and deepening of theoretical material. For homework, students are given the task of average difficulty, directly related to the materials of each seminar. Solving problems of students protected prior to the lecturer in oral form. In 7-8 semesters studied measure theory and Lebesgue integral; properties of metric, topological, normalized (mainly Banach) Hilbert and topological vector spaces. Sufficiently detailed study of the properties of linear operators in Banach and Hilbert spaces (including. H. Elements of the spectral theory) sets out the elements of duality theory of Banach and topological vector spaces. We study the Lebesgue space, Sobolev functions of bounded variation and their applications to problems of mathematical physics. In the 9th semester continues the study of the geometrical and topological properties of Banach spaces, and presents ideas and methods of nonlinear functional analysis. First of all, we study the properties of such nonlinear mapping is Gateaux differentiable and Frechet, continuity, compactness, it is continuity and complete continuity. Introduces the important concept of Nemytskij operator and Theorem Krasnosel'skii. Then discusses the various variational methods such as the method of Lyusternik-Schnirelmann in conjunction with the principle of compactness of the Palais-Smale, then uses the global bundle Pokhozhaev method sorts the set of Krasnosel'skii in conjunction with the method based on the theorem of the mountain pass. After discusses methods such as the method of compactness, monotonicity and fixed-point theorems. At the end of the course examines the main methods of proof of destruction of solutions of initial and initial-boundary value problems for partial differential equations.