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Fredholm theory connected with a Douglis-Nirenberg system of differential equations over \(\mathbb R^n\). (English) Zbl 1374.35154

The author studies the spectral problem \(A(x,D)u(x)-\lambda u(x)=f(x)\), \(x\in \mathbb R^n\), where \(A\) is a Douglis-Nirenberg system of linear differential operators satisfying the parameter-ellipticity condition in a closed sector \(\mathcal L\) of the complex plane with vertex in the origin. Under minimal smoothness assumptions, the spectral problem is considered in the \(L_p\) Sobolev-Bessel potentials space setting, \(1<p<\infty\). The main result is the Fredholm theory for the corresponding operator \(A_p\), for values of the spectral parameter lying in \(\mathcal L\). Under some assumptions, sufficiently large (in the absolute value) eigenvalues from \(\mathcal L\) are shown to be common, with the same multiplicities, for different values of \(p\).

MSC:

35J48 Higher-order elliptic systems
47A53 (Semi-) Fredholm operators; index theories
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