A boundary value problem for a system of first order equations. (Russian. English summary) Zbl 0753.35016

Kurbatova, N.

Differ. Uravn. Primen. 45, 51-58 (1990).

The following system is investigated: \(\sum_{i=1}^ n(u_{x_ i}^ i+a_ i u_ i)=0\), \(u_{x_ j}^ 1-u_{x_ 1}^ j+c_ ju_ 1+bu_ j=0\), \(j=2,\dots,n\), where \(a_ i\), \(c_ j\) and \(b\) are either constant or continuously differentiable functions and \(x\in D=\{y\in R^ n;\;0<x_ 1<h(x_ 2,\dots,x_ n)\), \((x_ 2,\dots,x_ n)\in E\}\), with a given bounded domain \(E\) in \(\{x_ 1=0\}\) and a given positive function \(h\). Moreover, the special boundary conditions are prescribed. It is shown that the problem is either Fredholm or uniquely solvable according to specific assumptions on coefficients which differ in the case of constant coefficients from those for nonconstant case.

Reviewer: I.Straškraba

MSC:

35F15

Boundary value problems for linear first-order PDEs

Keywords:

Fredholm problem; uniquely solvable