## Functiones et Approximatio Commentarii Mathematici

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

Paweł Domański

#### Abstract

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.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 44, Number 1 (2011), 79-109.

Dates
First available in Project Euclid: 30 March 2011

https://projecteuclid.org/euclid.facm/1301497748

Digital Object Identifier
doi:10.7169/facm/1301497748

Mathematical Reviews number (MathSciNet)
MR2807900

Zentralblatt MATH identifier
1221.35050

#### Citation

Domański, Paweł. Real analytic parameter dependence of solutions of differential equations over Roumieu classes. Funct. Approx. Comment. Math. 44 (2011), no. 1, 79--109. doi:10.7169/facm/1301497748. https://projecteuclid.org/euclid.facm/1301497748