Pavel Vilkov | Searcharxiv

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↗

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↗