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A remark on certain overdetermined systems of partial differential equations. (English) Zbl 0551.35062

P(D) sei ein linearer Differentialoperator mit konstanten Koeffizienten und \(d_ j\geq 1\) \((j=1,...,n)\) rationale Zahlen, \(d=(d_ 1,...,d_ n)\). \(\Omega \subset {\mathbb{R}}^ n\) sei offen und \(\Gamma^ d(\Omega)\) ein Gevrey-Raum. \[ Q(D,D_ t)=\sum_{<d,\alpha >+h\leq m}c_{\alpha,h}D^{\alpha}D^ h_ t,\quad D_ t=-i\partial_ t \] sei ein (d,1) hypoelliptischer Differentialoperator mit konstanten Koeffizienten. Dann sind folgende Aussagen äquivalent: \((i)\quad P(D)\Gamma^ d(\Omega)=\Gamma^ d(\Omega),\) (ii) für jedes w mit \(Q(D,D_ t)w=0\) in V existiert eine Lösung des Systems \(P(D)v=w,\quad Q(D,D_ t)v=0\) in einer offenen Umgebung von \(\Omega\) im \({\mathbb{R}}^{n+1}\). Eine solche Beziehung zwischen der Surjektivität von P(D) und der Lösbarkeit des Systems für den Spezialfall \(Q(D,D_ t)=\sum^{n}_{j=1}(D^ 2_ j+D^ 2_ t)\) gab Verf. im Fall \(d=1\) bereits in einer früheren Arbeit.
Reviewer: J.Jaenicke

MSC:

35N05 Overdetermined systems of PDEs with constant coefficients
65H10 Numerical computation of solutions to systems of equations
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References:

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