PORTUGALIAE MATHEMATICA Vol. 53, No. 3, pp. 305324 (1996) 

Differential Operators with Generalized Constant CoefficientsS. Pilipovi\'c and D. ScarpalézosUniversity of Novi Sad, Faculty of Science, Institute for Mathematics,Trg D. Obradovi\'ca 4, 21000 Novi Sad  YUGOSLAVIA U.F.R. de Mathématiques, Université Paris 7, 2 place Jussieu, Paris 5\eme, 75005  FRANCE Abstract: The classical method of solving the equation $P(D)\,g=f$ is adapted to a method of solving the family of equations with respect to $\varepsilon$ with a prescribed growth rate. More precisely, the equation $P_{\varepsilon}(D)\,U_{\varepsilon}=H_{\varepsilon}$ where $H_{\varepsilon}$ is Colombeau's moderate function ($H\in\calc{E}_{M}(\R^{n})$) and $P_{\varepsilon}(D)$ is a differential operator with moderate coefficients in Colombeau's sense, is solved. If $P^{j}(D)\to P(D)$, $j\to\infty$, in the sense that the coefficients converge in the sharp topology, then there is a sequence $E^{j}$ of solutions of $P^{j}(D)\,U=H$ which converges in the sharp topology to a solution $E$ of $P(D)\,U=H$ in $\calc{G}(\R^{n})=\calc{E}_{M}(\R^{n})/ \calc{N}(\R^{n})$. The moderate functions $E_{\varepsilon}^{j}\in \calc{E}_{M}(\R^{n})$ which converge sharply to $E_{\varepsilon}\in \calc{E}_{M}(\R^{n})$, such that $P_{\varepsilon}^{j}(D) (E_{\varepsilon}^{j}_{\Omega})=H_{\varepsilon}_{\Omega}$ and $P_{\varepsilon}(D)(E_{\varepsilon}_{\Omega})=H_{\varepsilon}_{\Omega}$, where $\Omega$ is a bounded open set, are constructed. Full text of the article:
Electronic version published on: 29 Mar 2001. This page was last modified: 27 Nov 2007.
© 1996 Sociedade Portuguesa de Matemática
