On hypoellipticity in \(\mathcal G\). (English) Zbl 1093.35019

Nedeljkov, M.; Pilipović, S.

Bull., Cl. Sci. Math. Nat., Sci. Math. 123, No. 27, 47-56 (2002).

Let \(G\) denote Colombeau’s algebra of generalized functions and \(\overline C\) denote the ring of Colombeau’s generalized complex numbers. Let \(P(D)=\sum_{| \beta| \leq m}a_\beta D^\beta = [P_{\phi,\varepsilon}(i\frac\partial{\partial x})]\), \(D^\beta=i^{| \beta| }\partial^\beta\), where \(\sum_{| \beta| \leq m}a_\beta x^\beta\), \(a_\beta\in\overline C\), is a polynomial in \(G\), be the operator for which \(| \sum_{| \beta| \leq m}a_{\beta,\phi,\varepsilon}c^\beta| \geq C\varepsilon^r\), \(\varepsilon\in(0,\eta)\), holds for some \(c=(c_1,\dots,c_n)\in R^n\), \(C>0\), \(r>0\) and \(\eta>0\). This operator is called hypoelliptic. The main result gives necessary and sufficient conditions that a given operator \(P(D)\) is hypoelliptic.

Reviewer: Bogoljub Stanković (Novi Sad)

MSC:

35H10	Hypoelliptic equations
46F30	Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)

Keywords:

Colombeau’s generalized functions; necessary and sufficient condition