Fredholm theory connected with a Douglis-Nirenberg system of differential equations over $\mathbb{R}^n$

Abstract

We consider a spectral problem over $\mathbb{R}^n$ for a Douglis-Nirenberg system of differential operators under limited smoothness assumptions and under the assumption of parameter-ellipticity in a closed sector $\mathcal{L}$ in the complex plane with vertex at the origin. We pose the problem in an $L_p$ Sobolev-Bessel potential space setting, $1 < p < \infty$, and denote by $A_p$ the operator induced in this setting by the spectral problem. We then derive results pertaining to the Fredholm theory for $A_p$ for values of the spectral parameter $\lambda$ lying in $\mathcal{L}$ as well as results pertaining to the invariance of the Fredholm domain of $A_p$ with $p$.

Key words: Parameter-ellipticity, Douglis-Nirenberg system, Fredholm properties.

Full Text

Article Information

Title	Fredholm theory connected with a Douglis-Nirenberg system of differential equations over $\mathbb{R}^n$
Source	Methods Funct. Anal. Topology, Vol. 22 (2016), no. 4, 330-345
MathSciNet	MR3591084
zbMATH	06742115
Milestones	Received 27/04/2016
Copyright	The Author(s) 2016 (CC BY-SA)

Authors Information

M. Faierman
School of Mathematics and Statistics, The University of New South Wales, UNSW Sydney, NSW 2052, Australia

Citation Example

M. Faierman, Fredholm theory connected with a Douglis-Nirenberg system of differential equations over $\mathbb{R}^n$, Methods Funct. Anal. Topology 22 (2016), no. 4, 330-345.

BibTex

@article {MFAT913,
    AUTHOR = {Faierman, M.},
     TITLE = {Fredholm theory connected with a Douglis-Nirenberg system of differential equations over $\mathbb{R}^n$},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {22},
      YEAR = {2016},
    NUMBER = {4},
     PAGES = {330-345},
      ISSN = {1029-3531},
  MRNUMBER = {MR3591084},
 ZBLNUMBER = {06742115},
       URL = {http://mfat.imath.kiev.ua/article/?id=913},
}

References

1. Robert A. Adams, Sobolev spaces, Academic Press, New York-London, 1975.  MathSciNet
2. S. Agmon, The $L_p$ approach to the Dirichlet problem. I. Regularity theorems, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 405-448.  MathSciNet
3. S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35-92.  MathSciNet
4. M. Agranovich, R. Denk, and M. Faierman, Weakly smooth nonselfadjoint spectral elliptic boundary problems, Math. Top. 14 (1997), 138-199.  MathSciNet
5. M. S. Agranovich and M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type, Russ. Math. Surv. 19 (1964), no. 3, 53-157.  MathSciNet CrossRef
6. Ju. M. Berezanskii, Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968.  MathSciNet
7. Jacques Chazarain and Alain Piriou, Introduction to the theory of linear partial differential equations, North-Holland Publishing Co., Amsterdam-New York, 1982.  MathSciNet
8. R. Denk and M. Faierman, Estimates for solutions of a parameter-elliptic multi-order system of differential equations, Integr. Equ. Oper. Theory 66 (2010), no. 3, 327-365.  MathSciNet CrossRef
9. Robert Denk, Reinhard Mennicken, and Leonid Volevich, The Newton polygon and elliptic problems with parameter, Math. Nachr. 192 (1998), 125-157.  MathSciNet CrossRef
10. M. Faierman, Elliptic problems for a Douglis-Nirenberg system over $\bf R^n$ and over an exterior subregion, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), no. 3, 579-594.  MathSciNet CrossRef
11. Gerd Grubb and Niels Jorgen Kokholm, A global calculus of parameter-dependent pseudodifferential boundary problems in $L_ p$ Sobolev spaces, Acta Math. 171 (1993), no. 2, 165-229.  MathSciNet CrossRef
12. Tosio Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin-New York, 1976.  MathSciNet
13. A N Kozhevnikov, Spectral problems for pseudodifferential systems elliptic in the Douglis-Nirenberg sense, and their applications, Math. USSR Sb. 21 (1973), no. 1, 63-90.  MathSciNet CrossRef
14. J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972.  MathSciNet CrossRef
15. Robert B. Lockhart and Robert C. McOwen, On elliptic systems in $\bf R^n$, Acta Math. 150 (1983), no. 1-2, 125-135.  MathSciNet CrossRef
16. A. S. Markus, Introduction to the spectral theory of polynomial operator pencils, Translations of Mathematical Monographs, vol. 71, American Mathematical Society, Providence, RI, 1988.  MathSciNet
17. Vladimir A. Mikhailets and Aleksandr A. Murach, Hormander spaces, interpolation, and elliptic problems, De Gruyter Studies in Mathematics, vol. 60, De Gruyter, Berlin, 2014.  MathSciNet CrossRef
18. A. A. Murach, On elliptic systems in Hormander spaces, Ukrainian Math. J. 61 (2009), no. 3, 467-477.  MathSciNet CrossRef
19. Patrick J. Rabier, Fredholm and regularity theory of Douglis-Nirenberg elliptic systems on $\Bbb R^ N$, Math. Z. 270 (2012), no. 1-2, 369-393.  MathSciNet CrossRef
20. E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Vol. 2, Oxford, at the Clarendon Press, 1958.  MathSciNet
21. Hans Triebel, Interpolation theory, function spaces, differential operators, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978.  MathSciNet
22. T. N. Zinchenko and A. A. Murach, Douglis-Nirenberg elliptic systems in Hormander spaces, Ukrainian Math. J. 64 (2013), no. 11, 1672-1687.  MathSciNet CrossRef