On separation of a class of degenerate differential operators in the Lebesgue space

M. G. Gadoev ; S. A. Iskhokov ; F. S. Iskhokov

Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 86-107

Résumé

Let $\Omega$ be an arbitrary open set in $n$-dimensional Euclidian space $R_{n}$ and let $\Pi(0)$ be the unit cube centered at the origin. For each point $\xi=(\xi_{1},\xi_{2},\ldots,\xi_{n})\in R_{n}$ and each vector $\vec{t}=(t_{1},t_{2},\ldots,t_{n})$ with positive components we define a parallelepiped $\Pi_{\overrightarrow{t}}(\xi)$ by the identity $$ \Pi_{\overrightarrow{t}}(\xi)=\left\{x\in R_{n} :\left(\left(x_{1}-\xi_{1}\right)/t_{1}, \left(x_{2}-\xi_{2}\right)/t_{2},\ldots, \left(x_{n}-\xi_{n}\right)/t_{n}\right)\in \Pi(0)\right\}. $$ Let $g_{j}(x) (j=\overline{1,n})$ be positive functions defined in $\Omega$. We let $\Pi_{\varepsilon,\overrightarrow{g}}(\xi)=\Pi_{\varepsilon\overrightarrow{g}(\xi)}(\xi)$, where $\varepsilon>0$ and $\overrightarrow{g}(\xi)=(g_{1}(\xi),g_{2}(\xi),\ldots,g_{n}(\xi))$. It is assumed that the set $\Omega$ and functions $g_{j}(x), j=\overline{1,n},$ are related by condition: (A) There exists a number $\varepsilon_{0}>0$ such that for each $\xi\in\Omega$ and any $\varepsilon\in (0, \varepsilon_{0})$ the parallelepiped $\Pi_{\varepsilon,\overrightarrow{g}}(\xi)$ is contained in $\Omega$. The condition (A) is an analogue of the immersion condition introduced by P.I. Lizorkin in 1980. In the paper we investigate separation of a differential expression \begin{equation}\label{*} L(x,D_{x})=\sum_{|k|\leq 2r}a_{k}(x)D_{x}^{k} (x\in \Omega), \end{equation} where $r$ – a natural number, $k=(k_{1}, k_{2}, \ldots , k_{n})$ is a multi-index, $|k|=k_{1}+k_{2}+\ldots+k_{n}$ is length of the multi-index, in the Lebesgue space $L_{p}(\Omega), 1$. We denote by $\mathcal{K}$ the set of all multi-indexes $k$ such that $a_k(x)\not\equiv 0$. Let $O_\mathcal{K}$ be the set of all functions $u(x)\in L_{1, loc}(\Omega)$, that have Sobolev generalized derivatives $D_x^ku(x)$ for all $k\in\mathcal{K}$. The differential expression (*) is said to be $L_p$-separated if for all $u(x)\in O_\mathcal{K}$ such that $u(x)\in L_{p}(\Omega)$, $L(x, D_x)u(x)\in L_{p}(\Omega)$ the inclusion $a_k(x)D^k_x u(x)\in L_{p}(\Omega)$ holds for all multi-indexes $k\in \mathcal{K}$. The work consists of five sections. The first section contains the statement of the main results, the right regularizer for the studied class of differential expressions is constructed in the second section, and sections 3-5 provide proofs of the main theorems of the paper.

Détail

Keywords: separation, partial differential operator, non-power degeneration, right-hand regularizing operator, inverse operator.