Harmonic synthesis of solutions of elliptic equation with periodic coefficients

Victor P. Palamodov

Annales de l'institut Fourier (1993)

  • Volume: 43, Issue: 3, page 751-768
  • ISSN: 0373-0956

Abstract

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An elliptic system in ℝ n , which is invariant under the action of the group ℤ n is considered. We construct a holomorphic family of finite-dimensional subrepresentations of the group in the space of solutions (Floquet solutions), such that any solution of the growth O ( exp ( a | x | ) ) at infinity can be rewritten in the form of an integral over the family.

How to cite

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Palamodov, Victor P.. "Harmonic synthesis of solutions of elliptic equation with periodic coefficients." Annales de l'institut Fourier 43.3 (1993): 751-768. <http://eudml.org/doc/75018>.

@article{Palamodov1993,
abstract = {An elliptic system in $\{\Bbb R\}^ n$, which is invariant under the action of the group $\{\Bbb Z\}^n$ is considered. We construct a holomorphic family of finite-dimensional subrepresentations of the group in the space of solutions (Floquet solutions), such that any solution of the growth $O(\{\rm exp\}( a \vert x \vert ) )$ at infinity can be rewritten in the form of an integral over the family.},
author = {Palamodov, Victor P.},
journal = {Annales de l'institut Fourier},
keywords = {representation of translation group; coherent analytic sheaf; Noether operator for a coherent sheaf; approximation; Floquet solutions; Lasker- Noether decomposition; elliptic system},
language = {eng},
number = {3},
pages = {751-768},
publisher = {Association des Annales de l'Institut Fourier},
title = {Harmonic synthesis of solutions of elliptic equation with periodic coefficients},
url = {http://eudml.org/doc/75018},
volume = {43},
year = {1993},
}

TY - JOUR
AU - Palamodov, Victor P.
TI - Harmonic synthesis of solutions of elliptic equation with periodic coefficients
JO - Annales de l'institut Fourier
PY - 1993
PB - Association des Annales de l'Institut Fourier
VL - 43
IS - 3
SP - 751
EP - 768
AB - An elliptic system in ${\Bbb R}^ n$, which is invariant under the action of the group ${\Bbb Z}^n$ is considered. We construct a holomorphic family of finite-dimensional subrepresentations of the group in the space of solutions (Floquet solutions), such that any solution of the growth $O({\rm exp}( a \vert x \vert ) )$ at infinity can be rewritten in the form of an integral over the family.
LA - eng
KW - representation of translation group; coherent analytic sheaf; Noether operator for a coherent sheaf; approximation; Floquet solutions; Lasker- Noether decomposition; elliptic system
UR - http://eudml.org/doc/75018
ER -

References

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  1. [1] V.P. PALAMODOV, Linear differential operators with constant coefficients, Moscow, Nauka, 1967, Springer-Verlag, 1970. Zbl0191.43401
  2. [2] L. EHRENPREIS, Fourier analysis in several complex variables, N.Y., 1970. Zbl0195.10401MR44 #3066
  3. [3] I.M. GEL'FAND, Eigenfunction decomposition of equation with periodic coefficients, Doklady ANSSSR, 73, n° 6 (1950), 1117-1120 (Russian). 
  4. [4] P.A. KUCHMENT, Floquet theory for partial differential equations, Russian Math. Surveys, 37, n° 4 (1982), 1-50. Zbl0519.35003MR84b:35018
  5. [5] L. BUNGART, On analytic fibre bundles I. Holomorphic fiber bundles with infinite dimensional fibres, Topology, 7 (1968), 55-68. Zbl0153.10202MR36 #5390
  6. [6] V.P. PALAMODOV, The projective limit on the category of linear topological spaces, Mathematics of the USSR Sbornik, 4 (1968), 529-559. Zbl0175.41801
  7. [7] O. ZARISKI, P. SAMUEL, Commutative algebra, Ch. IV, Van Nostrand, 1958. 
  8. [8] N. BOURBAKI, L'algèbre commutative, Paris, Hermann, 1967. 
  9. [9] V.P. PALAMODOV, Differential operators in coherent analytic sheaves, Mathematics of the USSR Sbornik, 6 (1968), 365-391. Zbl0187.07903MR38 #3471
  10. [10] Y.T. SIU, Noether-Lasker decomposition of coherent analytic subsheaves, Trans. of A.M.S., 135 (1969), 375-385. Zbl0175.37403MR38 #2340
  11. [11] R. GUNNING, H. ROSSI, Analytic functions of serveral complex variables, Englewood Cliffs, Prentice Hall, 1965. Zbl0141.08601
  12. [12] B. MALGRANGE, Existence et approximation des solutions des équations aux dérivées partielles et des équations des convolution, Ann. Inst. Fourier, Grenoble, 6 (1955-1956), 271-355. Zbl0071.09002MR19,280a
  13. [13] V.P. PALAMODOV, Deformation of complex spaces, Russian Math. Surveys, 31 (1976), 129-197. Zbl0347.32009MR58 #22671

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