Functiones et Approximatio Commentarii Mathematici
- Funct. Approx. Comment. Math.
- Volume 44, Number 1 (2011), 79-109.
Real analytic parameter dependence of solutions of differential equations over Roumieu classes
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
Permanent link to this document
https://projecteuclid.org/euclid.facm/1301497748
Digital Object Identifier
doi:10.7169/facm/1301497748
Mathematical Reviews number (MathSciNet)
MR2807900
Zentralblatt MATH identifier
1221.35050
Subjects
Primary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 46E10: Topological linear spaces of continuous, differentiable or analytic functions
Secondary: 35E20: General theory 32U05: Plurisubharmonic functions and generalizations [See also 31C10] 46A63: Topological invariants ((DN), ($\Omega$), etc.) 46A13: Spaces defined by inductive or projective limits (LB, LF, etc.) [See also 46M40] 46F05: Topological linear spaces of test functions, distributions and ultradistributions [See also 46E10, 46E35] 46M18: Homological methods (exact sequences, right inverses, lifting, etc.)
Keywords
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
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
