arXiv subjects

Alexander Shlapunov

Publications and source records attributed to Alexander Shlapunov.

At least 19 recordsLinked to original sources

arXiv2026

On a homotopy formula for generalized steady Stokes' operators, associated with the de Rham complex

We construct left, right and bilateral fundamental solutions for generalized steady Stokes' operators $S$ with smooth coefficients coefficients, associated with the de Rham complex of differentials on differential forms over a domain $X$ in ${\mathbb R}^n$. The investigated operators are Douglis-Nirenberg elliptic under reasonable assumptions. As an immediate corollary we produce a homotopy formula for regular solutions to this operator.

math.AP↗

arXiv2025

On approximation theorems for solutions to strongly parabolic systems in anisotropic Sobolev spaces

We investigate the problem on Runge pairs for Sobolev solutions of strongly uniformly parabolic systems in non-cylindrical domains of a special kind. We prove that if the coefficients of a parabolic operator are constant, then two domains with sufficiently smooth boundaries, no parts of which are parallel to the plane $t=0$, form a Runge pair if and only if the complements of any section of the larger domain to the section of the smaller domain by planes $t = const$, have no compact components in the larger section.

math.AP↗

arXiv2025

The Grothendieck duality and sparse minimizing in spaces of Sobolev solutions to elliptic systems

We present an instructive example of using Banach spaces of solutions to (linear, generally, non-scalar) elliptic operator $A$ to investigate variational inverse problems related to neural networks and/or to regularization of solutions to boundary value problems. More precisely, inspired by kernel's method for optimization problems in locally convex spaces, we prove the existence of the so-called sparse minimizers for the related variational problem and produce a representer theorem where a suitable fundamental solution of the operator $A$ is used as a reproducing kernel. The Grothendieck type duality for the Sobolev spaces of solutions to elliptic operator $A$ plays an essential role in the considerations. The case where the number of data passes to infinity is also discussed. Some typical situations related to the standard elliptic operators, the corresponding function spaces and fundamental solutions are considered.

math.AP↗

arXiv2024

On the Grothendieck duality for the space of holomorphic Sobolev functions

We describe the strong dual space $({\mathcal O}^s (D))^*$ for the space ${\mathcal O}^s (D) = H^s (D) \cap {\mathcal O} (D)$ of holomorphic functions from the Sobolev space $H^s(D)$, $s \in \mathbb Z$, over a bounded simply connected plane domain $D$ with infinitely differential boundary $\partial D$. We identify the dual space with the space of holomorhic functions on ${\mathbb C}^n\setminus \overline D$ that belong to $H^{1-s} (G\setminus \overline D)$ for any bounded domain $G$, containing the compact $\overline D$, and vanish at the infinity. As a corollary, we obtain a description of the strong dual space $({\mathcal O}_F (D))^*$ for the space ${\mathcal O}_F (D)$ of holomorphic functions of finite order of growth in $D$ (here, ${\mathcal O}_F (D)$ is endowed with the inductive limit topology with respect to the family of spaces ${\mathcal O}^s (D)$, $s \in \mathbb Z$). In this way we extend the classical Grothendieck-K{ö}the-Sebastião e Silva duality for the space of holomorphic functions.

math.CV↗

arXiv2024

Maxwell's and Stokes' operators associated with elliptic differential complexes

We propose a new technique to generate reasonable systems of partial differential equations (PDE) that could be potential candidates for depicting models in natural sciences related to quasi-linear equations. Such systems appear within typical constructions of the Homological Algebra as complexes of differential operators describing compatibility conditions for overdetermined systems of PDE's. The related models can be both steady and evolutionary. Additional assumptions on the ellipticity of the differential complex provide a wide class of elliptic, parabolic and hyperbolic operators that could be generated in this way. In particular, it appears that an essentially large amount of equations related to the modern Mathematical Physics is generated by the de Rham complex of differentials on the exterior differential forms. These includes the elliptic Laplace and Lamé type operators; the parabolic heat transfer equation; the Euler type and Navier-Stokes type equations in Hydrodynamics; the hyperbolic wave equation and the Maxwell equations in Electrodynamics; the Klein-Gordon equation in Relativistic Quantum Mechanics; and so on. Our model generation method covers a broad class of generating systems, especially in higher spatial dimensions, due to different basic algebraic structures at play.

math-ph↗

arXiv2023

Mixed Problems with a Parameter

Let $X$ be a smooth $n\,$-dimensional manifold and $D$ be an open connected set in $X$ with smooth boundary $\partial D$. Perturbing the Cauchy problem for an elliptic system $Au = f$ in $D$ with data on a closed set $\iG \subset \partial D$ we obtain a family of mixed problems depending on a small parameter $\varepsilon > 0$. Although the mixed problems are subject to a non-coercive boundary condition on $\partial D \setminus \iG$ in general, each of them is uniquely solvable in an appropriate Hilbert space $\cD_{T}$ and the corresponding family $\{ u_{\varepsilon} \}$ of solutions approximates the solution of the Cauchy problem in $\cD_{T}$ whenever the solution exists. We also prove that the existence of a solution to the Cauchy problem in $\cD_{T}$ is equivalent to the boundedness of the family $\{ u_{\varepsilon} \}$. We thus derive a solvability condition for the Cauchy problem and an effective method of constructing its solution. Examples for Dirac operators in the Euclidean space $\R^n$ are considered. In the latter case we obtain a family of mixed boundary problems for the Helmholtz equation.

math.AP↗

arXiv2023

On the Cauchy Problem for Elliptic Complexes in Spaces of Distributions

Let D be a bounded domain in n-dimensional Eucledian space with a smooth boundary. We indicate appropriate Sobolev spaces of negative smoothness to study the non-homogeneous Cauchy problem for an elliptic differential complex {A_i} of first order operators. In particular, we describe traces on the boundary of tangential part t_i (u) and normal part n_i(u) of a (vector)-function u from the corresponding Sobolev space and give an adequate formulation of the problem. If the Laplacians of the complex satisfy the uniqueness condition in the small then we obtain necessary and sufficient solvability conditions of the problem and produce formulae for its exact and approximate solutions. For the Cauchy problem in the Lebesgue spaces L^2(D) we construct the approximate and exact solutions to the Cauchy problem with maximal possible regularity. Moreover, using Hilbert space methods, we construct Carleman's formulae for a (vector-) function u from the Sobolev space H^1(D) by its Cauchy data t_i (u) on a subset S on the boundary of the domain D and the values of A_i u in D modulo the null-space of the Cauchy problem. Some instructive examples for elliptic complexes of operators with constant coefficients are considered.

math.AP↗

arXiv2023

On Grothendieck type duality for the space of holomorphic functions of several variables

We describe the strong dual space $({\mathcal O} (D))^*$ for the space ${\mathcal O} (D)$ of holomorphic functions of several complex variables over a bounded Lipschitz domain $D$ with connected boundary $\partial D$ (as usual, ${\mathcal O} (D)$ is endowed with the topology of the uniform convergence on the compact subsets of $D$). We identify the dual space with a closed subspace of the space of harmonic functions on the closed set ${\mathbb C}^n\setminus D$, $n>1$, with elements vanishing at the infinity and satisfying the tangential Cauchy-Riemann equations on $\partial D$. In particular, we extend in a way the classical Grothendieck-K{ö}the-Sebastião e Silva duality for the space of holomorphic functions of one complex variable to the multi-dimensional situation. We use the Bochner-Martinelli kernel ${\mathfrak U}_n$ in ${\mathbb C}^n$, $n>1$, instead of the Cauchy kernel over the complex plane ${\mathbb C}$ and we prove that the duality holds true if and only if the space ${\mathcal O} (D)\cap H^1 (D)$ of the Sobolev holomorphic functions over $D$ is dense in ${\mathcal O} (D)$.

math.CV↗

arXiv2022

On the ill-posed Cauchy problem for the polyharmonic heat equation

We consider the ill-posed Cauchy problem for the polyharmonic heat equation on recovering a function, satisfying the equation $(\partial _t + (- Δ)^m) u=0$ in a cylindrical domain in the half-space ${\mathbb R}^n \times [0,+\infty)$, where $n\geq 1$, $m\geq 1$ and $Δ$ is the Laplace operator, via its values and the values of its normal derivatives up to order $(2m-1)$ on a given part of the lateral surface of the cylinder. We obtain a Uniqueness Theorem for the problem and a criterion of its solvability in terms of the real-analytic continuation of parabolic potentials, associated with the Cauchy data.

math.AP↗

arXiv2022

On uniqueness theorems for the inverse problem of Electrocardiography in the Sobolev spaces

We consider a mathematical model related to reconstruction of cardiac electrical activity from ECG measurements on the body surface. An application of recent developments in solving boundary value problems for elliptic and parabolic equations in Sobolev type spaces allows us to obtain uniqueness theorems for the model. The obtained results can be used as a sound basis for creating numerical methods for non-invasive mapping of the heart.

math.AP↗

arXiv2022

Exterior extension problems for strongly elliptic operators: solvability and approximation using fundamental solutions

In this work we study three exterior extension problems for strongly elliptic partial equations: the Cauchy problem (in a special statement), the "analytical" continuation problem and the so called "inner" Dirichlet problem in the scale of the Sobolev spaces over a domain with relatively smooth boundaries. We consider the existence of solutions to these problems, the dense solvability and conditional well-posedness of these problems for a wide class of strongly elliptic systems. We also consider the approximation of solutions to these problems by a single layer potential and by a linear combination of "discrete" fundamental solutions in relation to a narrower class of strongly elliptic operators of the second order. The obtained results justify the applicability of the indirect method of boundary integral equations and for numerical solving the exterior extension problems.

math.AP↗

arXiv2022

Existence theorems for regular spatially periodic solutions to ersatz Navier-Stokes equations

The initial problem for the Navier-Stokes type equations over ${\mathbb R}^n \times [0,T]$, $n\geq 2$, with a positive time $T$ in the spatially periodic setting is considered. First, we prove that the problem induces an open injective continuous mapping on scales of specially constructed function spaces of Bo\-chner-Sobolev type over the $n\,$-dimensional torus ${\mathbb T}^n$. Next, rejecting the idea of proving a universal a priori estimate for high-order derivatives, we obtain a surjectivity criterion for the non-linear mapping under the considerations in terms of boundedness for its inverse images of precompact sets. Finally, we prove that the mapping is surjective if we consider the versions of the Navier-Stokes type equations containing no `pressure'{}. This gives a uniqueness and existence theorem for regular solutions to this particular ersatz of the Navier-Stokes type equations. The used techniques consist in proving the closedness of the image by estimating all possible divergent sequences in the preimage and matching the asymptotics. The following facts are essential: i) the torus is a compact closed manifold, ii) the corresponding system is `local'.

math.AP↗

arXiv2022

Approximation of solutions to parabolic Lamé type operators in cylinder domains and Carleman's formulas for them

Let $s \in {\mathbb N}$, $T_1,T_2 \in {\mathbb R}$, $T_1<T_2$, and let $Ω, ω$ be bounded domains in ${\mathbb R}^n$, $n \geq 1$ such that $ω\subset Ω$ and the complement $Ω\setminus ω$ have no non-empty compact components in $Ω$. We investigate the problem of approximation of solutions to parabolic Lamé type system from the Lebesgue class $L^2(ω\times (T_1,T_2))$ in a cylinder domain $ω\times (T_1,T_2) \subset {\mathbb R}^{n+1}$ by more regular solutions in a bigger domain $Ω\times (T_1,T_2)$. As an application of the obtained approximation theorems we construct Carleman's formulas for recovering solutions to these parabolic operators from the Sobolev class $H^{2s,s}(Ω\times (T_1,T_2))$ via values the solutions on a part of the lateral surface of the cylinder and the corresponding them stress tensors.

math.AP↗

arXiv2022

The Fredholm Navier-Stokes type equations for the de Rham complex over weighted Hölder spaces

We consider a family of initial problems for the Navier-Stokes type equations generated by the de Rham complex in ${\mathbb R}^n \times [0,T]$, $n\geq 2$, with a positive time $T$ over a scale weighted anisotropic Hölder spaces. As the weights control the order of zero at the infinity with respect to the space variables for vectors fields under the consideration, this actually leads to initial problems over a compact manifold with the singular conic point at the infinity. We prove that each problem from the family induces Fredholm open injective mappings on elements of the scales. At the step $1$ of the complex we may apply the results to the classical Navier-Stokes equations for incompressible viscous fluid.

math.AP↗

arXiv2022

On approximation of solutions to the heat equation from Lebesgue class $L^2$ by more regular solutions

Let $s \in {\mathbb N}$, $T_1,T_2 \in {\mathbb R}$, $T_1<T_2$, and $Ω, ω$ be bounded domains in ${\mathbb R}^n$, $n \geq 1$, such that $ω\subset Ω$ and the complement $Ω\setminus ω$ has no (non-empty) compact components in $Ω$. We prove that this is the necessary and sufficient condition for the space $H^{2s,s} _{\mathcal H} (Ω\times (T_1,T_2))$ of solutions to the heat operator ${\mathcal H} $ in a cylinder domain $Ω\times (T_1,T_2)$ from the anisotropic Sobolev space $H^{2s,s} (Ω\times (T_1,T_2))$ to be dense in the space $L^{2} _{\mathcal H}(ω\times (T_1,T_2))$, consisting of solutions in the domain $ω\times (T_1,T_2)$ from the Lebesgue class $L^{2} (ω\times (T_1,T_2))$. As an important corollary we obtain the theorem on the existence of a basis with the double orthogonality property for the pair of the Hilbert spaces $H^{2s,s} _{\mathcal H} (Ω\times (T_1,T_2))$ and $L^{2} _{\mathcal H}(ω\times (T_1,T_2))$ .

math.AP↗

arXiv2021

Existence theorems for regular solutions to the Cauchy problem for the Navier-Stokes equations in ${\mathbb R}^3$

We consider the initial problem for the Navier-Stokes equations over ${\mathbb R}^3 \times [0,T]$ with a positive time $T$ over specially constructed scale of function spaces of Bochner-Sobolev type. We prove that the problem induces an open both injective and surjective mapping of each space of the scale. In particular, intersection of these classes gives a uniqueness and existence theorem for smooth solutions to the Navier-Stokes equations for smooth data with a prescribed asymptotic behaviour at the infinity with respect to the time and the space variables.

math.AP↗

arXiv2021

Inverse image of precompact sets and existence theorems for the Navier-Stokes equations in spatially periodic setting

We consider the initial problem for the Navier-Stokes equations over ${\mathbb R}^3 \times [0,T]$ with a positive time $T$ in the spatially periodic setting. Identifying periodic vector-valued functions on ${\mathbb R}^3$ with functions on the $3\,$-dimensional torus ${\mathbb T}^3$, we prove that the problem induces an open injective mapping ${\mathcal A} _s: B^{s}_1 \to B^{s-1}_2$ where $B^{s}_1$, $B^{s-1}_2$ are elements from scales of specially constructed function spaces of Bochner-Sobolev type parametrized with the smoothness index $s \in \mathbb N$. Finally, we prove rather expectable statement that a map ${\mathcal A} _s$ is surjective if and only if the inverse image ${\mathcal A} _s ^{-1}(K)$ of any precompact set $K$ from the range of the map ${\mathcal A} _s $ is bounded in the Bochner space $L^{\mathfrak s} ([0,T], L ^{\mathfrak s} ({\mathbb T}^3))$ with the Ladyzhenskaya-Prodi-Serrin numbers ${\mathfrak s}$, ${\mathfrak r}$.

math.AP↗

arXiv2021

Existence Theorems for Regular Spatially Periodic Solutions to the Navier-Stokes Equations

We consider the initial value problem for the Navier-Stokes equations over $R^{3} \times [0,T]$ with a positive time $T$ in the spatially periodic setting. Identifying periodic vector-valued functions on $R^{3}$ with functions on the three-dimensional torus $T^{3}$, we prove that the problem induces an open both injective and surjective mapping of specially constructed function spaces of Bochner-Sobolev type. This gives a uniqueness and existence theorem for regular solutions to the Navier-Stokes equations. Our techniques consist in proving the closedness of the image by estimating all possible divergent sequences in the preimage and matching the asymptotics.

math.AP↗