Real analytic parameter dependence of solutions of differential equations over Roumieu classes

Domański, Paweł

Funct. Approx. Comment. Math., Tome 44 (2011) no. 1, p. 79-109 / Harvested from Project Euclid

Résumé

We consider the problem of real analytic parameter dependence of solutions of the linear partial differential equation $P(D)u=f$, i.e., the question if for every family $(f_\lambda)\subseteq \mathscr_{\{\omega\}}(\Omega)$ of ultradifferentiable functions of Roumieu type (in particular, of real analytic functions or of functions from Gevrey classes) depending in a real analytic way on $\lambda\in U$, $U$ a real analytic manifold, there is a family of solutions $(u_\lambda)\subseteq \mathscr_{\{\omega\}}(\Omega)$ also depending analytically on $\lambda$ such that $$ P(D)u_\lambda=f_\lambda \text{for every $\lambda\in U$}, $$ where $\Om\subseteq \mathbb{R}^d$ an open set. We solve the problem for many types of differential operators following a similar method as in the earlier paper of the same author for operators acting on spaces of distributions. We show for an operator $P(D)$ on the space of real analytic functions $\mathscr{A}(\Omega)$, $\Omega \subseteq \mathbb{R}^d$ open convex, that it has real analytic parameter dependence if and only if its principal part $P_p(D)$ has a continuous linear right inverse on the space $C^\infty(\Omega)$ (or, equivalently, on $\mathscr{D}'(\Omega)$). In particular, the property does not depend on the set of parameters $U$. Surprisingly, in all solved non-quasianalytic cases, it follows that the solution is positive if and only if $P(D)$ has a linear continuous right inverse.

Détail

Publié le : 2011-03-15
Classification:  analytic dependence on parameters,  linear continuous right inverse,  linear partial differential operator,  convolution operator,  linear partial differential equation with constant coefficients,  space of real analytic functions,  ultradifferentiable functions of Roumieu type,  Gevrey classes,  functor $Proj^1$,  PLS-space,  locally convex space,  vector valued equation,  solvability,  35B30,  46E10,  35E20,  32U05,  46A63,  46A13,  46F05,  46M18