Nonclassic boundary value problems in the theory of irregular systems of equations with partial derivatives

TL;DR

This paper investigates nonclassic boundary value problems for irregular PDE systems with noninvertible operators, introducing skeleton chains to reduce complex problems to regular systems for better analysis.

Contribution

It introduces the concept of skeleton chains for noninvertible operators in PDEs, enabling reduction of complex boundary value problems to regular systems.

Findings

Skeleton chains facilitate problem reduction to regular systems.

The approach handles noninvertible boundary operators.

Provides a framework for irregular PDE boundary problems.

Abstract

The linear PDE ${\mathbf B} {\mathbf L} (\frac{\partial}{\partial x}) u ={\mathbf L}_1(\frac{\partial}{\partial x})u +f(x)$ with nonclassic conditions on boundary $\partial \Omega$ is considered. Here ${\mathbf B}$ is linear noninvertible bounded operator acting from linear space $E$ into $E,$ $x=(t,x_1,\dots, x_m) \in \Omega, $ $\Omega \subset {\mathbb R}^{m+1}.$ It is assumed that ${\mathbf B}$ enjoys the skeleton decomposition ${\mathbf B}={\mathbf A}_1 {\mathbf A}_2,$ ${\mathbf A}_2 \in {\mathcal L}(E\rightarrow E_1),$ ${\mathbf A}_1 \in {\mathcal L}(E_1\rightarrow E)$ where $E_1$ is linear normed space. Differential operators ${\mathbf L}, \, {\mathbf L}_1$ are partial differential operators. In the concrete cases the domains of definition of operators ${\mathbf L}, {\mathbf L}_1$ consist of linear manifolds $E_{\partial}$ of sufficiently smooth abstract functions $u(x)$ with domain in $\Omega$ and their ranges in $E,$ which satisfy certain system of homogeneous boundary conditions. The abstract function $f: \Omega \subset {\mathbb R}^{m+1} \rightarrow E $ is assumed to be given. It is requested to find the solution $u: \Omega \subset {\mathbb R}^{m+1} \rightarrow E_{\partial},$ which satisfy certain condition on boundary $\partial \Omega.$ The concept of a skeleton chains is introduced as sequence of linear operators ${\mathbf B}_i \in {\mathcal L}(E_i \rightarrow E_i), \, i=1,2,\dots, p,$ where $E_i$ are linear spaces corresponding to the skeleton decomposition of operator ${\mathbf B}.$ It is assumed that irreversible operator ${\mathbf B}$ generates skeleton chain of the finite length $p.$ The problem is reduced to a regular split system with respect to higher order derivative terms with certain initial and boundary conditions.