The solvability of inhomogeneous boundary-value problems in Sobolev spaces

TL;DR

This paper develops a comprehensive theory for the solvability of linear inhomogeneous boundary-value problems for differential systems in Sobolev spaces, including cases with overdetermined or fractional boundary conditions.

Contribution

It introduces the notion of a rectangular number characteristic matrix and establishes its relation to the problem's Fredholm properties, including stability and semicontinuity results.

Findings

Characteristic matrix explicitly computed in examples

Fredholm numbers are semicontinuous under certain conditions

Theory applies to boundary conditions with derivatives of fractional order

Abstract

The aim of the paper is to develop a general theory of solvability of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of arbitrary order in Sobolev spaces. Boundary conditions are allowed to be overdetermined or underdetermined. They may contain derivatives, of the unknown vector-valued function, whose integer or fractional orders exceed the order of the differential equation. Similar problems arise naturally in various applications. The theory introduces the notion of a rectangular number characteristic matrix of the problem. The index and Fredholm numbers of this matrix coincide respectively with the index and Fredholm numbers of the inhomogeneous boundary-value problem. Unlike the index, the Fredholm numbers (i.e. the dimensions of the problem kernel and co-kernel) are unstable even with respect to small (in the norm) finite-dimensional perturbations. We give examples in which the characteristic matrix can be explicitly found. We also prove a limit theorem for a sequence of characteristic matrices. Specifically, it follows from this theorem that the Fredholm numbers of the problems under investigation are semicontinuous in the strong operator topology. Such a property ceases to be valid in the general case.