Elliptic functional differential equations in bounded and unbounded domains
Elliptic functional differential equations in bounded and unbounded domains

Area of science: 

Mathematics, mechanics and informatics.

Scientific direction: 

Partial differential equations.

The project analyzes boundary value problems for elliptic functional-differential equations in bounded domains and half-space, as well as elliptic functional-differential equations in the entire space R^n. The Dirichlet problem in the half-space for strongly elliptic differential-difference equations will be studied. Both necessary and sufficient conditions for coercivity of differential-difference equations with incommensurable shifts of arguments in bounded domains were obtained, and applications of these problems to the Kato problem of the square root of the operator were analyzed. The Kato hypothesis for elliptic degenerate differential-difference operators will be proved and the solvability of parabolic problems for differential-difference equations in weight spaces will be studied. Elliptic functional-differential equations with different compressions whose logarithms are incommensurable will be analyzed, and the terms for the unique and Fredholm solvability of the boundary value problem and the continuous dependence of the solutions on the compression coefficients will be found. The solvability of functional-differential equations in the weight spaces containing one-variable compression and stretching in the higher part of the compression (such transformations are called orthotropic compressions) will be also studied.

Project goals
  • Proof of solvability, determination of uniqueness classes, obtaining an integral representation of solutions to the Dirichlet problem in half-space for elliptic differential-difference equations and studying their qualitative properties. Obtaining explicitly necessary and sufficient terms for the implementation of Garding inequality for differential-difference equations with incommensurable shifts in a bounded domain.
  • Proof of the Kato hypothesis for elliptic differential-difference operators of the second order with degeneracy in some bounded domains, as well as a study of the solvability of parabolic problems for differential-difference equations in weight spaces. Obtaining conditions for solvability of boundary value problems for different types of functional differential equations containing compression and stretching of arguments of higher derivatives of an unknown function, including equations with incommensurable compression and orthotropic compression.
Project leader All participants
Alexander Skubachevskii

Alexander Skubachevskii

Doctor of Sciences, Professor, Director of the S.M. Nikol’skii Mathematical Institute, the leader of the project
Project results
Elliptic differential-difference operators with degeneration in the cylinder were studied. It was proved that these operators are regularly accretive and satisfy the Kato hypothesis on the square root of the regularly accretive operator. Thus, a new class of operators satisfying the Kato hypothesis was found. The difficulties of studying this problem for elliptic differential-difference operators with degeneration are due to the fact that the smoothness of generalized solutions of the corresponding boundary value problems is violated. Moreover, generalized solutions of boundary value problems for elliptic differential-difference equations with degeneration do not even belong to the Sobolev space of the first order. Therefore, it is impossible to write out the scope of the corresponding operators in explicit form. And, therefore, it is not possible to write the interpolation space between this domain of definition and the space of Lebesgue functions integrated with the square. The interpolation between the image of the domain of the elliptic differential-difference operator, which is generated by the difference operator, is used to describe the domain of the square root of the operator and the root of the square of the conjugate operator. The description of a new class of regularly accretive operators satisfying the Kato hypothesis and the above proof method are fundamentally new and have no analogues in the works of other authors.
For differential-difference equations with incommensurable shifts of arguments in bounded domains, sufficient terms for the implementation of the inequality of the Garding type in algebraic form were obtained. These terms take into account the properties and size of the domain and are uniform over small shear perturbations. The method of constructing a system of specially formed and interconnected differential-difference equations with commensurate shifts was used. In the study of terms for the implementation of the Garding type inequality for the resulting system, the method of splitting the initial region into subdomains and reducing the differential-difference equations to systems of strongly elliptic differential equations in subdomains was applied.
For perturbed differential-difference equations (with a small parameter in shifts) in bounded domains, the necessary and sufficient terms for the implementation of the Garding type inequality were obtained and the solvability of boundary value problems for these equations was studied, as well as the asymptotic behavior of generalized solutions when the small parameter tends to zero. A symbol based on the Fourier transform was used to solve the coercivity problem.
The Dirichlet problem in the half-plane was studied for an elliptic differential-difference equation of general form, i.e. for an equation in which an arbitrary number of superpositions of the second derivative and shift operators, unrelated to each other by any terms of commensurability, act on a variable parallel to the boundary axis. Integral representation of its solution in the sense of generalized functions (Gelfand - Shilov) was built and its smoothness beyond the boundary straight was proved [A. Muravnik, On the half-plane Dirichlet problem for differential-difference several elliptic equations with several nonlocal terms, Mathematical Modelling of Natural Phenomena, vol. 12, issue. 6, 2017 (in print)]. The theorem on the asymptotic proximity of the constructed solution and the solution of the Dirichlet problem in the same half-plane for the elliptic differential equation was proved. It was obtained from the initial (nonlocal) problem as follows: the values of all shifts are assumed to be zero; as a result, the main result of stabilizing the solution of the original (nonlocal) problem is derived – to stabilize its solution, the classical Repnikov-Eidelman criterion [A. B. Muravnik, Asymptotic properties of solutions of two-dimensional differential-difference elliptic problems, Contemporary Mathematics. Fundamental Directions, vol. 63, vol. 4, 2017 (in print)].
These results are fundamentally new: previously, only the case of a single nonlocal term (superposition of the second derivative and shift operator) was studied, which did not allow to conclude whether different terms with incommensurable shifts were permissible, or that the constraints on the coefficients of the equation guaranteeing validity were the requirement of positive certainty of the differential difference operator (i.e., one of the key requirements of the general nonlocal elliptic theory). The classical operating scheme based on the fact that shift operators are multipliers in Fourier images (like differential operators) was used to obtain the results, but qualitative modifications of this scheme are required in comparison with the case of a single nonlocal term: in particular, if in the case of a single nonlocal term, the kernel of the integral Poisson operator, by which the solution of the problem is represented, has a unique feature (and it is concentrated at the origin), for equations of the general form of such a feature is not the only, and the location of features on the real axis depends on the values of the shifts and the coefficients of the nonlocal members.
The decisive assumption in the study of solvability and regularity of solutions of elliptic functional differential equations often is the “commensurability of transformations” of arguments, while the equations with “incommensurable transformations” were studied much less. The project was the first to analyze strongly elliptic functional-differential equations of the second order, containing in the older derivatives of the compression transformation of arguments with several compression parameters, logarithms of which are incommensurable. Necessary and sufficient terms of strong ellipticity were obtained for them in algebraic form. It was proved that the Dirichlet problem for a strongly elliptic equation has a unique solution that depends continuously on the compression parameters, and the unbounded operator corresponding to the problem is the generator of the analytic semigroup. The analysis of the obtained conditions shows that, being strongly elliptic at fixed incommensurable values of compression parameters, the functional differential equation will be the same in the vicinity of these values. This is fundamentally different from the case when there are only comparable compressions in the equation - here the property of strong ellipticity is not stable over small independent perturbations of the compression parameters. In addition, functional-differential equations with incommensurable compressions that are not strongly elliptic were studied. The Dirichlet problem for such equations can have an infinite-dimensional kernel.
An example showing that an arbitrarily small perturbation of the compression parameter can lead to a qualitative change in the properties of the problem, even if the coefficient for the corresponding term of the equation is arbitrarily small was set. The study is based on the elucidation of the structure corresponding to the commutative B*-algebra generated by functional compression operators and multiplication operators by homogeneous functions of zero degree. Additionally, the absence of stability of spectral properties of functional operators with compressions with respect to arbitrarily small perturbations of these parameters was demonstrated.
In the scale of weight spaces, proposed by V. A. Kondratiev for the study of elliptic equations in areas with conical points and having as a weight function a certain degree of polar radius, the solvability of the linear functional differential equation of the second order was studied, containing in the upper part of the orthotropic compression transformation, where the compression parameters are positive real numbers, different from each other. Sufficient conditions for the solvability of the functional-differential equation with orthotropic contractions in the form of conditions on the coefficients of the equation were obtained and it was shown that by choosing appropriate weight parameter it is possible to achieve unique solvability [Rossovsky, L. E., Tesevich A. L., On the unique solvability of the functional-differential equation with orthotropic compressions in weight spaces // Different. equations, 53:12 (2017), 1679-1692]. In addition, it was proved that the difference equation with infinitely smooth coefficients stabilizing at infinity is uniquely solvable in the Lebesgue space of squared integrable functions on the line. A priori estimation of generalized solutions in weight spaces was also obtained [Tasevich A., Analysis of Functional-Differential Equation with Orthotropic Constrictions // Math. Model. Phenomenon.,Vol. 12, No. 6 2017. - P. 62-69.].
In the study of strong ellipticity and solvability in the Sobolev spaces and in the Kondratiev weight spaces of the functional-differential equation containing orthotropic compression of the arguments of the desired function, the main point is to find a special replacement of variables dictated by the structure of the orbits of the points of the region under the action of the group generated by the compression transformations included in the equation. After this replacement and a number of others, the original equation is reduced to the difference equation on the line, which has smooth coefficients stabilizing at infinity. This problem has a simpler formulation, however, explicit results on the solvability of the difference equation with variable coefficients with at least three terms corresponding to the identical operator, the shift operator and the inverse to it, were not obtained by other authors before this study.
For parabolic equations with operator coefficients analyzed in the Lipschitz cylinder, a unique classical solvability in generalized Sobolev spaces $H_p^s$ was proved, for p and s close to 2 and 1, respectively, provided that the operator coefficients of the elliptic part are multipliers in Sobolev - Slobodetskii spaces of small order of smoothness. In the study of such equations, the methods of the theory of semigroups of linear operators in Banach spaces were used. The proof that the operator corresponding to the elliptic part of the equation is the generator of the analytical semigroup was based on the theorem of extrapolation of the reversibility of the bounded operator acting in some interpolation scale, as well as the fact that the analyzed generalized Sobolev spaces form an interpolation scale.
Application area
  • The theory of boundary value problems for elliptic differential-difference equations with argument shifts in the senior terms has important applications in the theory of multilayer plates and shells, the theory of multidimensional laser systems, and control theory.