Department of Functional Analysis and its Applications

The Department was founded in 2008. Main topics are: boundary value and spectral problems for PDEs of elliptic and mixed type, spectral analysis, boundary control problems for oscillating processes, linear integral equations, completeness and basis properties of functional systems.

Students of the Department attend a special pro-seminar on functional analysis.

Staff members:

• Kapustin Nikolay, Professor, Dr. Sc.
• Polosin Alexey, Associated Professor, PhD, Scientific Secretary of the Department
• Kholomeeva Anna, Assistant Professor, PhD
• Gulyaev Denis, Assistant Professor, PhD

Regular courses:

• Functional Analysis by Prof. Moiseev, 64 hours, 5th semester.
• Functional Analysis by Prof. Moiseev, 32 hours, 6th semester.
• Functional Analysis by Assoc. Prof. Kapustin, 32 hours, 5th semester.
• Functional Analysis by Assoc. Prof. Kapustin, 32 hours, 7th semester.
• Mathematical Analysis by Assoc. Prof. Polosin, 64 hours, 2nd semester.
• Mathematical Analysis by Assoc. Prof. Polosin, 64 hours, 3rd semester.

Special courses:

• Methods for Solving Integral Equations by Assoc. Prof. Polosin, 32 hours, 5th semester.

Special scientific seminars:

• Spectral Theory of Differential Operators and Actual Problems of Mathematical Physics by Prof. Moiseev and Prof. Il'in.
• Boundary Value and Spectral Problems by Prof. Moiseev.

Main Scientific Directions

Boundary value problems for mixed type equations

(Prof. E.I. Moiseev)

Fundamental results on representing solutions to boundary value problems in the form of biorthogonal series have been obtained in the spectral theory. To justify the obtained expansions, important theorems on completeness and basis properties of appropriate systems of root functions have been proved.

The method of integral equations (especially singular ones) is another powerful tool to explore boundary value problems. Using this method, many non-classical problems have been successfully investigated.

Location of the spectrum for non-self-adjoint problems, in particular, problems with non-local boundary conditions or oblique derivative problems, is of special interest. In this sphere, important results on spectrum's location, on its belonging to the Carleman parabola and on various properties of root functions have been obtained.

Boundary control problems for oscillating processes

(Prof. E.I. Moiseev)

This field, which is being developed in the close cooperation with the team of the Department of General Mathematics, dwells upon boundary value problems for optimal control of oscillating processes. In particular, for a large time interval, which is a multiple of the string length, the optimal boundary control of string oscillations has been found for various problems of the first, second and third kind, in terms of generalized solutions to the wave equation which have a uniformly finite energy.
Recent papers:

1. E.I.Moiseev and V.E.Ambartsumyan, On the basis property of eigenfunctions of the Frankl problem with a nonlocal oddness condition of the second kind // Integral Transforms and Special Functions, vol. 21, no. 10, pp. 745-754, 2010.
2. E.I.Moiseev and V.E.Ambartsumyan, On the solvability of nonlocal boundary value problem for the Helmholtz equation with the equality of flows at the part of the boundary and its conjugated problem // Integral Transforms and Special Functions, vol. 21, no. 12, 2010.

Recent publications:

• 2013

1. Moiseev E.I., Nefedov P.V. Frankl problem for the Lavrent'ev-Bitsadze equation in a 3D-domain // Integral Transforms and Special Functions. 2013. 24. N 7. P. 554-560.

• 2012

1. Kapustin N.Yu. On the spectral problem arising in the solution of a mixed problem for the heat equation with a mixed derivative in the boundary condition // Differential Equations. 2012. 48. N 5. P. 701-706.
2. Moiseev E.I., Kholomeeva A.A. Optimal boundary control by displacement at one end of a string under a given elastic force at the other end // Proc. of the Steklov Inst. of Math. 2012. 276. N 1. P. 153-160.
3. Moiseev E.I., Kholomeeva A.A. Optimization of the boundary control of string vibrations at one string end with a given mode at the other end // Doklady Mathematics. 2012. 86. N 1. P. 454-456.
4. Moiseev E.I., Likhomanenko T.N. A nonlocal boundary value problem for the Lavrent’ev-Bitsadze equation // Doklady Mathematics. 2012. 86. N 2. P. 635-637.