Science

• Burenkov V.I developed a number of original approaches and methods. His method of averaging operators with variable step and shift allowed to obtain fundamental results in the approximation of functions belonging to common functional spaces, infinitely differentiable functions, in particular, with preservation of boundary values, and especially in the problem of continuation of functions from Sobolev spaces. Constructed with this method, the operator of continuation of functions from Sobolev spaces with preservation or minimal deterioration of differential properties is mentioned as Burenkov continuation operator. With its help, Victor Ivanovich Burenkov and Alexander Lvovich Gorbunov obtained exact estimates over the order of smoothness for the minimal norm of the operator of extension. His article on the superposition of absolutely continuous functions was presented in the Reports of the USSR Academy of Sciences by Andrey Kolmogorov and the result obtained in it is cited as Burenkov theorem.
• Burenkov V.I developed a method of fractional differentiation of a priori inequalities, which allowed to obtain the necessary and sufficient terms for the conditional hypoellipticity of differential operators with partial derivatives with constant coefficients. Then this direction was developed by a group of researchers from Yerevan headed by Hayk Gegamovich Ghazaryan. In their works, the foregoing result is given as Burenkov theorem on conditional hypoellipticity, and this type of hypoellipticity is called hypoellipticity by Burenkov.
• Burenkov V.I. made a great contribution to the branches of mathematics related to mathematical analysis and differential equations. Professor Burenkov is an acknowledged world expert in the theory of functional spaces, especially Sobolev spaces and spaces with fractional order of smoothness, and its applications. He carried out important researches on the theory of partial differential equations and integral equations, in particular, on the theory of hypoelliptic equations, spectral theory of differential operators and the theory of incorrect problems.
• Professor Burenkov V.I. is the founder of the research of a new case of a finite interval. The work on inequalities for intermediate derivatives with exact constants gave an impetus to a new direction. Exact constants were also found in some inequalities of different metrics, Markov-type inequalities for polynomials, and, together with Vladimir Anatolyevich Gusakov, in some embedding theorems for Sobolev spaces. Exact constants in some Hardy-type inequalities were obtained together with Swedish mathematicians J. Bergh and Lars-Erik Persson.
• New types of theorems on multipliers of Fourier integrals for weighted Lebesgue spaces with exponential weights were proved and applications to partial differential equations were analyzed.
• A nonlinear continuation operator was constructed for the limit case of the theorem of traces for the anisotropic Nikol’skii-Besov spaces, and it was proved that in this case the linear continuation operator does not exist. Professor Burenkov V.I.  together with Professor Mikhail Lvovich Goldman studied the interaction between norms of a wide class of operators in general normalized functional spaces and norms in their periodic analogues. These results allow to transfer many of the statements proved for the non-periodic case to the periodic case and, on the contrary, from the periodic case to the non-periodic case.
• New flexible methods for constructing regularized approximate solutions of integral convolution equations related to geophysical problems were developed. The use of spaces with small fractional smoothness is the basis. These methods became more effective than traditional approaches based on the use of Sobolev spaces.
• Burenkov V.I.  collaborated with Professor Evans (William Desmond Evans) from the School of Mathematics, Cardiff University, Cardiff, the UK, that led to the publication of works on weight integral inequalities and frequently cited work on quantum mechanics: V. I. Burenkov, V. D. Evans, “On the estimate of the norm of an integral operator associated with the stability of one-electron atoms”, Proc. Roy. Soc. Edinburgh Sect. A, 128:5 (1998), 993–1005.
• Burenkov V.I.  is an expert in operator theory in general Morrie-type spaces:
• Necessary and sufficient terms for functional parameters for a wide range of numerical parameters that provide the boundedness of many classical real analysis operators (maximal operator, fractional maximal operator, Riesz potential, hardy operator, true singular integrals, Hausdorff operators) from one common local Morrie-type space to another were obtained. In the case of the maximum operator and fractional maximum operator, Burenkov V.I.  conducted a joint study was Vagif Sabirovich Guliyev, Doctor of Physics and Mathematics, Professor, corresponding member of the National Academy of Sciences of Azerbaijan (Baku, Azerbaijan).
• Burenkov V.I.  found out that the local Morrey-type spaces, in contrast to the global, are convenient for the purposes of interpolation. It was proved that the scale of the local Morrey-type spaces is closed over the real interpolation method. The study was conducted together with Erlan Dautbekovic Nursultanov, - Doctor of Physics and Mathematics, Professor (Faculty of Mechanics and Mathematics of L. N. Gumilev Eurasian national University, Astana, Kazakhstan).
• Burenkov V.I.  obtained an analogue of the Young’s inequality for convolutions of functions for global Morrey-type spaces, which has a form different from the form of the classical Young’s inequality for Lebesgue spaces, and it can be used in various applications. The results were published in collaboration with Tamara Vasilievna Tararykova, a researcher at the School of Mathematics, Cardiff University, (Cardiff University, Cardiff, the UK).
• Burenkov V.I.  is an expert in obtaining accurate spectral stability estimates for eigenvalues of self-adjoint elliptic differential operators:
• Burenkov V.I.  was the first to publish the work on the spectral stability of the Laplace operator with homogeneous boundary Neumann terms in collaboration with the President of the London Mathematical Society Edward Davis (Edward Brian Davies) - Professor of Mathematics, King's College London (King's College London, KCL - London, UK).
• Burenkov V.I.  developed the method of transition operators - together with the Italian mathematician Pier Domenico Lamberti - Professor of the Faculty of Mathematics, the University of Padova (Padova, Italy). The method made it possible to obtain exact changes in the eigenvalues under perturbation of the domain of definition through effective geometric characteristics of the proximity of the initial and perturbed domains of definition for elliptic operators of arbitrary even order given on open sets allowing arbitrarily strong degeneration both for the case of homogeneous Dirichlet boundary terms and for Neumann terms. These results are cited as Burenkov - Lamberti theorems.
• Burenkov V.I. analyzed the case of the third boundary value problem (the Robin problem) together with the Italian mathematician Massimo Lanza de Cristoforis, Professor Professor of the Faculty of Mathematics, the University of Padova (Padova, Italy).
• Burenkov V.I. applied conformal mapping to the spectral stability problem for the two-dimensional Laplacian operator. The study was conducted together with Vladimir Goldstein and Alexander Uchlov, professors of the Faculty of Mathematics, Ben-Gurion University of the Negev, BGU (Negev, Israel).
• Results on stability of singular values of non-self-adjoint elliptic operators were obtained in the joint study with Doctor of Physics and Mathematics, Professor Muharbi Otelbaevich Otelbaev, actual member of National Academy of Sciences of the Republic of Kazakhstan, Professor of the Faculty of Mechanics and Mathematics of L. N. Gumilev Eurasian National University. The results have applications to the general theory of partial differential equations and numerical methods related to the computation of eigenvalues.
• Obtaining accurate spectral stability estimates for eigenfunctions within domain perturbation is under study. Some results in this direction were obtained by V. I. Burenkov together with Gerasimos Barbatis, Professor of the Faculty of Mathematics of the National and Kapodistrian University of Athens (the national and Kapodistrian University of Athens, NKUA, (Athens, Greece), Professor Lamberti (P. D. Lamberti, Department of Mathematics, the University of Padova, Italy) and Ermal Feleqi, Professor of the Faculty of Mathematics, the University of Vlorë “Ismail Qemali”, (Vlorë, Albania).
• More than 180 articles were published. More than 100 plenary presentations at conferences and at the invitation of universities in 30 countries were made.
• Burenkov V.I. is one of the founders and an editor-in-chief of the international journal “Eurasian Mathematical Journal” (together with academician of the Russian Academy of Sciences Viktor Antonovich Sadovnichy and academician of the National Academy of Sciences of Kazakhstan Mukhtarbay Otelbaevich Otelbaev).

Scientific interests

• The theory of functions and functional analysis (Sobolev spaces, Nikol'skii-Besov spaces, total of Morrey-type spaces, interpolation theory).
• Differential equations with partial derivatives (hypoellipticity equations, spectral stability).
• Integral equations (incorrect, problems).
• Applications to geophysics, quantum mechanics, numerical methods, radar theory, acoustics.