Pseudodifferential Operator Methods Research Articles

Theoretical and methodological approach to the study of technological processes

The article investigates some issues of complex systems analysis and synthesis that contain local, discrete sources of temperature fields. The emphasis of the author's research lies in the calculation and optimization of the parameters of the laser action on the embryo. The biotechnological process is described by the boundary value problem of a non-stationary, multidimensional differential equation of thermal conductivity which satisfies the boundary conditions of heat flux, beginning and end of the laser action process. The author proposes an algorithm for calculation and optimization of the control parameters of laser action on multilayer microbiological material. The object of the study is the embryo under the action of a laser beam for fission. As a first approximation, at the stage of calculation and optimization of the parameters of the biotechnological process, the embryo is considered as a homogeneous body. The values of thermophysical characteristics are calculated by the method of expert evaluation of the parameters of the emitters. The correctness of the boundary value problem of the process of laser action on the embryo is proved by the author using the method of pseudodifferential operators. Seeking the solution of the boundary value problem in the form of the series, the analytical function of the temperature field distribution is obtained using the Fourier method of the separated variables in the article. Using the method of indeterminate coefficients, the author found the temperature of the laser action on the embryo. Using the approximate gradient method of finding local extrema and the method of directed search of local extrema, it is possible to obtain rational values of optimized parameters of the biotechnological process. The author outlines possible approaches for optimization of technical parameters of laser emitters. In his opinion, the results of research can be considered fundamental for the calculation and optimization of the parameters of the laser action on the embryo, taking into account the multilayer structure of the latter.

On the equivalence of the electromagnetic problem of diffraction by an inhomogeneous bounded dielectric body to a volume singular integro-differential equation

The paper is concerned with the smoothness of the solutions to the volume singular integrodifferential equations for the electric field to which the problem of electromagnetic-wave diffraction by a local inhomogeneous bounded dielectric body is reduced. The basic tool of the study is the method of pseudo-differential operators in Sobolev spaces. The theory of elliptic boundary problems and field-matching problems is also applied. It is proven that, for smooth data of the problem, the solution from the space of square-summable functions is continuous up to the boundaries and smooth inside and outside of the body. The results on the smoothness of the solutions to the volume singular integro-differential equation for the electric field make it possible to resolve the issues on the equivalence of the boundary value problem and the equation.

Volume estimates and spectral asymptotics for a class of pseudo-differential operators

Two asymptotic expansions, known as the heat and Szego expansions, are studied for a class of operators defined on $$L^2(\Omega )$$ where $$\Omega $$ is a compact region in Euclidean space with smooth boundary. Pseudo-differential operator methods are combined with a version of a theorem originally due to H. Weyl on the volume of certain tubular neighborhoods to obtain significant new information about the higher order terms in the expansions.

Yang–Mills action from minimally coupled bosons on R4 and on the four-dimensional Moyal plane

We consider bosons on (Euclidean) R4 that are minimally coupled to an external Yang–Mills field. We compute the logarithmically divergent part of the cutoff regularized quantum effective action of this system. We confirm the known result that this term is proportional to the Yang–Mills action. We use pseudodifferential operator methods throughout to prepare the ground for a generalization of our calculation to the noncommutative four-dimensional Moyal plane Rθ4. We also include a detailed comparison of our cutoff regularization to heat kernel techniques. In the case of the noncommutative space, we complement the usual technique of asymptotic expansion in the momentum variable with operator theoretic arguments in order to keep separated quantum from noncommutativity effects. We show that the result from the commutative space R4 still holds if one replaces all pointwise products by the noncommutative Moyal product.

Large-scale vortex structures in shear flows

This article considers the developments of the Hamiltonian Approach to problems of fluid dynamics, and applies the general method to strongly non-linear large-scale vortex structures in shear flows. Such vortex structures can appear in shear flows like Poiseulle and Couette flows. The Hamiltonian version of contour dynamics for two-dimensional layer models in a constant-density ideal fluid has be developed. The Hamiltonian description is formulated in terms of the dynamic variables with KdV-type Poisson brackets. The use of the method of pseudo-differential operators permits the regular application of approximation methods. Thus, governing equations can be easily derived correctly in any order of perturbation theory. We discuss also the effect of vortex structures on the average velocity profile of the flows.

Critical dimension of bosonic string theory and zeta-function regularization

A derivation of the critical dimension of the Polyakov bosonic string is presented. It is based on the use of the anholonomic formalism, a ghost-anti-ghost symmetric action, zeta-function regularization and the Seeley method of pseudo-differential operators.

Boundary Problems for the Biharmonic Operator in a Square with $L^P $-Data

Applied boundary value problems for the bi-Laplacian often actually involve domains with nonsmooth boundaries. The second fundamental problem of plane elasticity, the viscous flow and Stokes problem are representative examples of problems which can be reduced to the interior or exterior Dirichlet problem for the bi-Laplacian. In this paper we study the Dirichlet problem for $\Delta ^2 $ in a plane square with $L^p $-data, looking for a solution in potential form. By the method of pseudodifferential operators we show an existence theorem provided the data are in $L^p $, $1 < p < 3$, and satisfy a proper compatibility condition. We also solve two further boundary value problems which are related to the Dirichlet problem.