University

National research Tomsk state university

Level of English language proficiency

Intermediate

The field of study for which the graduate student will be
accepted

1.1.1. Real, complex and functional analysis (educational program)
1.1. Mathematics and mechanics (field of the educational program)

The code of the field of study for which the graduate student
will be accepted

01.06.01,
09.06.01

List of research projects of a potential supervisor
(participation / leadership)

1. Investigation of the spectral characteristics of elliptic operators in domains of Euclidean space, RSF project, leadership
2. Applications of quasiconformal analysis to spectral problems of elliptic operators, RFBR project, leadership
3. Geometric methods for the estimation of the quantum billiards' energy levels, RFS project, leadership
4. New robust statistical effective methods for signal and image processing in stochastic systems, RSF project, participation
5. Regional Scientific and Educational Mathematical Center of TSU, project of Ministry of Science and Higher Education of the Russian Federation, participation

List of possible research topics

1. Embedding theorems of Sobolev spaces
2. Development of methods for studying the spectral properties of elliptic operators
3. Investigation of the spectral characteristics of the Dirichlet problem for the quasilinear p-Laplace operator
4. Investigation of the spectral characteristics of the Neumann problem for the quasilinear p-Laplace operator
5. Lower and upper bounds for the energy levels of quantum billiards
6. Composition operators on Sobolev spaces
7. Spectral estimates of elliptic operators on Carnot groups.

Supervisor’s
main publication

In just 5 years, publications - 21,
including:
1. Gol’dshtein V., Pchelintsev V., Ukhlov A. Sobolev extension operators and Neumann eigenvalues. Journal of Spectral Theory. Vol. 10. (2020) 337-353.
2. Gol’dshtein V., Hurri-Syrjänen R., Pchelintsev V., Ukhlov A. Space quasiconformal composition operators with applications to Neumann eigenvalues. Analysis and Mathematical Physics. 2020. Vol. 10:78. 20 pp.
3. Gol’dshtein V., Pchelintsev V., Ukhlov A. Quasiconformal mappings and Neumann eigenvalues of divergent elliptic operators. Complex Variables and Elliptic Equations. Vol. 67, No 9. (2022) 2281-2302.
4. Pchelintsev V. On variations of Neumann eigenvalues of p-Laplacian generated by measure preserving quasiconformal mapping. J. Math. Sciences. Vol. 255, No 4. (2021) 503-512.
5. Pchelintsev V. Estimates for variation of the first Dirichlet eigenvalue of the Laplace operator. J. Math. Sciences. Vol. 261, No 3. (2022) 444-454.

Elliptic operators (p-Laplace operator (p>1, Schrödinger operator, etc.) arise in problems of electrostatics, heat theory, geophysics, chemical kinetics, quantum mechanics, etc. Solutions of classical equations of mathematical physics are based on the spectral expansion in eigenfunctions of the Laplace operator. Therefore, the eigenvalue problem for elliptic operators (e.g., p-Laplacian (p>1)) with various boundary conditions is very relevant. According to the minimax principle, eigenvalues are characterized in terms of the norms of embedding operators of Sobolev spaces. Previously, classical estimates of the Neumann eigenvalues for p-Laplace operator (p>1), using the method of integral representations, were obtained for convex domains in terms of the Euclidean diameter of the domain. Unfortunately, in domains of more complex geometric shape, the eigenvalues of the Neumann problem for the p-Laplace operator (p>1) cannot be characterized in terms of the Euclidean diameter of the domain. For this reason, our works proposed new methods in the theory of Sobolev spaces embedding based on the theory of composition (extension) operators on Sobolev spaces and the theory of conformal and quasi-conformal mappings, allowing us to obtain spectral estimates in convex and non-convex domains.

Upon admission to the program, it is possible to draw up a dual leadership agreement with colleagues from the Ben-Gurion University of the Negev (Israel). Upon successful completion of the program, graduate students receive financial support through inclusion in research grants.

1. Advanced knowledge in functional analysis, theory of Sobolev and Lebesgue spaces, spectral theory of elliptic operators, theory of conformal and quasiconformal mappings.
2. Ability to work in mathematical packages Maple, Mathematica.