Parameter dependence of solutions of partial differential equations in spaces of real analytic functions

Abstract

Abstract. Let Ω ⊆ R n be an open set and let A(Ω) denote the class of real analytic functions on Ω. It is proved that for every surjective linear partial differential operator P (D, x) : A(Ω) → A(Ω) and every family (f λ ) ⊆ A(Ω) depending holomorphically on λ ∈ C m there is a solution family (u λ ) ⊆ A(Ω) depending on λ in the same way such thatThe result is a consequence of a characterization of Fréchet spaces E such that the class of "weakly" real analytic E-valued functions coincides with the analogous class defined via Taylor series. An example shows that the analogous assertions need not be valid if C m is replaced by another set.

Real analytic parameter dependence of solutions of differential equations

“…As shown in [40], A (U, X) contains functions for which vector valued Taylor series is not convergent. In [4,5] a characterization is given for which Fréchet spaces X every function F ∈ A (U, X) has a Taylor series local representation.…”

“…and therefore the real analytic dependence differs essentially from the holomorphic one. A characterization of Fréchet spaces X such that all X-valued real analytic functions are locally representable via convergent vector valued Taylor series is given in [4] and [5] but neither the space of smooth functions on a manifold nor the space of distributions satisfies this condition.…”

Parameter dependence of solutions of the Cauchy–Riemann equation on weighted spaces of smooth functions

“…real analytic [13] on the parameter λ were considered. The case that the coefficients of the partial differential differential operator P(x, ∂) are non-constant functions in x ∈ Ω was treated for F (Ω) = A (R n ), the space of real analytic functions on R n , as well [3].…”

“…An error in part b) of this lemma, p. 229, is corrected here such that the term cos(1/2) = min |y|≤ t=1/(2C 1 ) cos(C 1 y) appears 3. The proof of[60, Satz 2.2.3, p. 44] relies on [60, Satz 2.2.1, p. 43] which is an announced version (without a proof) of our result Corollary 13 on weak reducibility.…”

Parameter dependence of solutions of the Cauchy-Riemann equation on spaces of weighted smooth functions

“…real analytic [20] on the parameter λ were considered. The case that the coefficients of the partial differential differential operator P (x, ∂) are nonconstant functions in x ∈ Ω was treated for F (Ω) = A (R n ), the space of real analytic functions on R n , as well [3,4].…”