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A theorem on homogeneous differential polynomials in \(R^ 2\). (Italian) Zbl 0546.35010
The author shows that a theorem of G. Bratti [ibid. 69, 169-180 (1983)] on operators \(P(D_ x,D_ y)\) with constant coefficients can be proved by a more geometric method if P is homogeneous. The main part of the paper is the proof of \(D_ x(G(B))=G(B)\) for \(D_ x\)-convex sets \(B\subset {\mathbb{R}}^ 2\), \((G(B)=\{f\in C^{\infty}({\mathbb{R}}^ 2):f|_ B=0\}).\)
Reviewer: W.Watzlawek
MSC:
35E10 Convexity properties of solutions to PDEs with constant coefficients
35G05 Linear higher-order PDEs
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References:
[1] G. Bratti , Problema di Cauchy semiglobale in due variabili , in stampa presso Rend. Sem. Mat. di Padova. [2] L. Hormander , Linear partial differential operators , Springer-Verlag , 1969 . MR 404822 | Zbl 0175.39201 · Zbl 0175.39201
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