Functiones et Approximatio Commentarii Mathematici

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

Paweł Domański

Full-text: Open access

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


Export citation

References

  • J. Bonet, P. Domański, Real analytic curves in Fréchet spaces and their duals, Mh. Math. 126 (1998), 13–36.
  • J. Bonet, P. Domański, Parameter dependence of solutions of partial differential equations in spaces of real analytic functions, Proc. Amer. Math. Soc. 129 (2000), 495–503.
  • J. Bonet, P. Domański, Parameter dependence of solutions of differential equations on spaces of distributions and the splitting of short exact sequences, J. Funct. Anal., 230 (2006), 329–381.
  • J. Bonet, P. Domański, The structure of spaces of quasianalytic functions of Roumieu type, Arch. Math. 89 (2007), 430–441.
  • J. Bonet, P. Domański, The splitting of exact sequences of PLS-spaces and smooth dependence of solutions of linear partial differential equations, Advances Math. 217 (2008), 561–585.
  • J. Bonet, R. Meise, S. N. Melikhov, A comparison of two different ways to define classes of ultradifferentiable functions, Bull. Belg. Math. Simon Stevin 14 (2007), 425–444.
  • R. W. Braun, Surjektivität partieller Differentialoperatoren auf Roumieu-Klassen, Habilitationsschrift, Düsseldorf 1993.
  • R. W. Braun, R. Meise, B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Results Math. 17 (1990), 207–237.
  • R. W. Braun, R. Meise, B. A. Taylor, Characterization of the linear partial differential equations that admit solution operators on Gevrey classes, J. reine angew. Math. 588 (2005), 169–220.
  • R. W. Braun, R. Meise, D. Vogt, Existence of fundamental solutions and surjectivity of convolution operators on classes of ultradifferentiable functions, Proc. London Math. Soc. 61 (1990), 344–370
  • F. E. Browder, Analyticity and partial differential equations I, Amer. J. Math. 84 (1962), 660–710
  • P. Domański, Classical PLS-spaces: spaces of distributions, real analytic functions and their relatives, in: Orlicz Centenary Volume, Banach Center Publications, 64, Z. Ciesielski, A. Pełczyński and L. Skrzypczak (eds.), Institute of Mathematics, Warszawa 2004, pp. 51–70.
  • P. Domański, Real analytic parameter dependence of solutions of differential equations, Rev. Mat. Iberoamericana 26 (2010), 175–238.
  • P. Domański, Notes on real analytic functions and classical operators, preprint 2010.
  • P. Domański and M. Langenbruch, Vector valued hyperfunctions and boundary values of vector valued harmonic and holomorphic functions, Publ. RIMS Kyoto Univ. 44(4) (2008), 1097–1142.
  • P. Domański, D. Vogt, The space of real analytic functions has no basis, Studia Math. 142 (2000), 187–200.
  • U. Franken, On the equivalence of holomorphic and plurisubharmonic Phragmén-Lindelöf principles, Michigan Math. J. 42 (1995), no. 1, 163–173.
  • L. Frerick, A splitting theorem for nuclear Fréchet spaces, in: Functional Analysis, Proc. of the First International Workshop held at Trier University, S. Dierolf, P. Domański, S. Dineen (eds.), Walter de Gruyter, Berlin 1996, pp. 165–167.
  • L. Frerick, J. Wengenroth, A sufficient condition for the vanishing of the derived projective limit functor, Arch. Math., 67 (1996), 296–301.
  • L. Hörmander, On the existence of real analytic solutions of partial differential equations with constant coefficients, Inventiones Math. 21 (1973), 151–182.
  • L. Hörmander, The Analysis of Linear Partial Differential Operators, Springer, Berlin 1983.
  • H. Jarchow, Locally Convex Spaces, B. G. Teubner, Stuttgart 1981.
  • M. Klimek, Pluripotential Theory, London Mathematical Society Monographs, New Series, 6, Oxford University Press, New York, 1991.
  • H. Komatsu, Ultradistributions I. Structure theorems and characterization, J. Fac. Sci. Univ. Tokyo, Sec. 1 A. Math. 20 (1973), 25–105.
  • A. Kriegl, P. W. Michor, The convenient setting for real analytic mappings, Acta Math. 165 (1990), 105–159.
  • M. Langenbruch, Continuous linear right inverses for convolution operators in spaces of real analytic functions, Studia Math. 110 (1994), 65–82.
  • M. Langenbruch, Surjectivity of partial differential operators on Gevrey classs and extension of regularity, Math. Nachr. 196 (1998), 103–140.
  • M. Langenbruch, Characterization of surjective partial differential operators on spaces of real analytic functions, Studia Math. 162 (2004) 53–96.
  • M. Langenbruch, Inheritance of surjectivity for partial differential operators on spaces of real analytic functions, J. Math. Anal. Appl. 297 (2004), 696–719.
  • A. Martineau, Sur la topologie des espaces de fonctions holomorphes, Math. Ann. 163 (1966), 62–88.
  • R. Meise, Sequence space representations for zero-solutions of convolution equations on ultradifferentiable functions of Roumieu type, Studia Math. 92 (1989), 211–230.
  • R. Meise, B. A. Taylor, D. Vogt, Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse, Ann. Inst. Fourier, Grenoble 40 (1990), 619–655.
  • R. Meise, B. A. Taylor, D. Vogt, Equivalence of analytic and plurisubharmonic Phragmén-Lindelöf conditions, Proc. Symposia Pure Math. 52 (1991), part 3, 287–308.
  • R. Meise, B. A. Taylor, D. Vogt, Extremal plurisubharmonic functions of linear growth on algebraic varieties, Math. Z. 219 (1995), 515–537.
  • R. Meise, B. A. Taylor, D. Vogt, Continuous linear right inverses for partial differential operators on non-quasianalytic classes and on ultradistributions, Math. Nachr. 180 (1996), 213–242.
  • R. Meise, B. A. Taylor, D. Vogt, Continuous linear right inverses for partial differential operators of order 2 and fundamental solutions in half spaces, Manuscr. Math. 90 (1996), 449–464.
  • R. Meise, B. A. Taylor, D. Vogt, $\omega$-Hyperbolicity of linear partial differential operators with constant coefficients, Complex Analysis, Harmonic Analysis and Applications, R. Deville, J. Esterle, V. Petkov, A Sebbar, A Yger, (eds.), Pitman Research Notes, Math. Ser. 347 (1996), 157–182.
  • R. Meise, B. A. Taylor, D. Vogt, Phragmén-Lindelöf principles on algebraic varieties, J. Amer. Math. Soc. 11 (1998), 1–39.
  • R. Meise, D. Vogt, Characterization of convolution operators on spaces of $C\sp\infty$-functions admitting a continuous linear right inverse, Math. Ann. 279 (1987), 141–155.
  • R. Meise, D. Vogt, Introduction to Functional Analysis, Clarendon Press, Oxford 1997.
  • T. Meyer, Surjectivity of convolution operators on spaces of ultradifferentiable functions of Roumieu type, Studia Math. 125 (1997), 101–129.
  • V. P. Palamodov, Linear Differential Operators with Constant Coefficients, Nauka, Moscow 1967 (Russian), English transl., Springer, Berlin 1971.
  • V. P. Palamodov, Functor of projective limit in the category of topological vector spaces, Mat. Sb. 75 (1968), 567-603 (in Russian); English transl., Math. USSR Sbornik 17 (1972), 289–315.
  • V. P. Palamodov, Homological methods in the theory of locally convex spaces, Uspekhi Mat. Nauk 26 (1) (1971), 3–66 (in Russian); English transl., Russian Math. Surveys 26 (1) (1971), 1–64.
  • K. Piszczek, On a property of PLS-spaces inherited by their tensor products, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), 155–170.
  • L. Rodino, Linear Partial Differential Operators in Gevrey Classes, World Scientific, Singapore 1993.
  • T. Rösner, Surjektivität partieller Differentialoperatoren auf quasianalytischen Roumieu-Klassen, Dissertation, Düsseldorf 1997.
  • D. Vogt, Sequence space representations of spaces of test functions and distributions, in: Functional Analysis, Holomorphy and Approximation Theory, G. L. Zapata (ed.), Lecture Notes Pure Appl. Math. 83, Marcel Dekker, New York 1983, pp. 405–443.
  • D. Vogt, Lectures on projective spectra of DF-spaces, Seminar Lectures, AG Funktionalanalysis, Düsseldorf/Wuppertal 1987.
  • D. Vogt, Topics on projective spectra of LB-spaces, in: Advances in the Theory of Fréchet Spaces, T. Terzio\v glu (ed.), Kluwer, Dordrecht 1989, pp. 11–27.
  • D. Vogt, Invariants and spaces of zero solutions of linear partial differential operators, Arch. Math. 87 (2006), 163–171.
  • D. Vogt, Real analytic zero solutions of linear partial differential operators with constant coefficients, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), 577–586.
  • J. Wengenroth, Derived Functors in Functional Analysis, Lecture Notes Math. 1810, Springer, Berlin 2003.