Differential Operators with Generalized Constant Coefficients

S. Pilipovi\'c and D. Scarpalézos

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.
The main problems in presented investigations are the estimates with respect to $\varepsilon$ which makes this theory a non-trivial generalization of the classical one.

Full text of the article:

• Compressed PostScript file (120 kilobytes)
• PDF file (224 kilobytes)