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On the fitting of mappings near to solutions of elliptic first order systems. (Russian) Zbl 0682.35032

The problem investigated by the author is as follows \(Dg=\sum^{n}_{i=1}A_ i(\partial g/\partial x_ i)=0,\) where \(A_ i\in R^{1\times m}\)-matrices, \(g=(g^ 1,g^ 2,...,g^ m): \Delta \to R^ m,\) \(x\in \Delta \subset R^ n\). The bundle of solutions of this system is denoted by G. The stability of G in \(W^ 1\) with respect to C norm means that the local closeness of \(f: \Delta \to R^ m\in W^ 1\) to G in C norm involves global closeness. Introducing the bundle of locally close mappings f (denoted by \(G_ p(\epsilon)\), \(\epsilon\geq 0\), \(p>1)\) it is shown that the closeness of f to G is equivalent with \(f\in G_ p(\epsilon)\) for small \(\epsilon\). The main result of this work is that for \(1=m\) there exist a sequence \(f_ j\), \(j=1,2,..\). uniformly convergence to \(f\in G_ p(\epsilon)\).
Reviewer: G.Molnárka

MSC:

35J45 Systems of elliptic equations, general (MSC2000)
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