Deficient Coerciveness Estimate for an Abstract Differential Equation with a Parameter Dependent Boundary Conditions

Aissa Aibeche; Angelo Favini; Chahrazed Mezoued

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 3, page 535-547
  • ISSN: 0392-4041

Abstract

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In this paper we consider an abstract elliptic differential problem where the equation and the boundary conditions may contain a spectral parameter. We first prove that this problem generates an isomorphism between appropriate spaces and we establish a more precise estimate called coerciveness estimate with defect. The results obtained are applied to study some classes of elliptic, and also possibly degenerate, problems.

How to cite

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Aibeche, Aissa, Favini, Angelo, and Mezoued, Chahrazed. "Deficient Coerciveness Estimate for an Abstract Differential Equation with a Parameter Dependent Boundary Conditions." Bollettino dell'Unione Matematica Italiana 10-B.3 (2007): 535-547. <http://eudml.org/doc/290424>.

@article{Aibeche2007,
abstract = {In this paper we consider an abstract elliptic differential problem where the equation and the boundary conditions may contain a spectral parameter. We first prove that this problem generates an isomorphism between appropriate spaces and we establish a more precise estimate called coerciveness estimate with defect. The results obtained are applied to study some classes of elliptic, and also possibly degenerate, problems.},
author = {Aibeche, Aissa, Favini, Angelo, Mezoued, Chahrazed},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {535-547},
publisher = {Unione Matematica Italiana},
title = {Deficient Coerciveness Estimate for an Abstract Differential Equation with a Parameter Dependent Boundary Conditions},
url = {http://eudml.org/doc/290424},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Aibeche, Aissa
AU - Favini, Angelo
AU - Mezoued, Chahrazed
TI - Deficient Coerciveness Estimate for an Abstract Differential Equation with a Parameter Dependent Boundary Conditions
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/10//
PB - Unione Matematica Italiana
VL - 10-B
IS - 3
SP - 535
EP - 547
AB - In this paper we consider an abstract elliptic differential problem where the equation and the boundary conditions may contain a spectral parameter. We first prove that this problem generates an isomorphism between appropriate spaces and we establish a more precise estimate called coerciveness estimate with defect. The results obtained are applied to study some classes of elliptic, and also possibly degenerate, problems.
LA - eng
UR - http://eudml.org/doc/290424
ER -

References

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  1. AGMON, S. - NIRENBERG, L., Properties of Solutions of Ordinary Differential Equation in Banach Spaces, Comm. Pure Appl. Math., 16 (1963), 121-239. Zbl0117.10001MR155203DOI10.1002/cpa.3160160204
  2. AGRANOVICH, M. S. - VISIK, M. L., Elliptic Problems with a Parameter and Parabolic Problems of General Type, Russian Math. Surveys, 19 (1964), 53-161. MR192188
  3. AIBECHE, A., Coerciveness Estimates for a Class of Elliptic Problems, Diff. Equ. Dynam. Syst., 4 (1993), 341-351. Zbl0875.35021MR1259173
  4. AIBECHE, A., Completeness of Generalized Eigenvectors for a Class of Elliptic Problems, Result. Math., 31 (1998), 1-8. Zbl0894.35076MR1610103DOI10.1007/BF03322064
  5. COBOS, F. - FERNANDEZ, D. L., On Interpolation of Compact Operators, Ark. Mat., 27 (1989), 211-217. Zbl0691.46047MR1022277DOI10.1007/BF02386372
  6. DENCHE, M., Abstract Differential Equation with a Spectral Parameter in the Boundary Conditions, Result. Math., 35 (1999), 217-227. Zbl0927.35032MR1694903DOI10.1007/BF03322814
  7. DUNFORD, N. - SCHWARTZ, J. T., Linear Operators, Vol. II, Interscience, New York (1963). MR188745
  8. FAVINI, A. - GOLDSTEIN, J. A. - ROMANELLI, S., An analytic semigroup associated to a degenerate evolution equation, in Stochastic Processes and Functional Analysis, J. A. Goldstein, N. E. Gretsky and J. J. Uhl Jr. eds, Marcel Dekker, New York, (1997), 88-100. Zbl0889.35039MR1440417
  9. FAVINI, A. - YAGI, A., Degenerate Differential Equations in Banach Spaces, Marcel Dekker, New York (1999). Zbl0913.34001MR1654663
  10. KREIN, S. G., Linear Differential Equations in Banach Spaces, American Mathematical Society, Providence (1971). MR342804
  11. LIONS, J. L., Sur les Espaces d'Interpolation: Dualité, Math. Scand., 9 (1961), 147-177. Zbl0103.08102MR159212DOI10.7146/math.scand.a-10632
  12. LIONS, J. L. - MAGENES, E., Problèmes aux Limites non Homogènes et applications, Vol. I, Dunod, Paris (1968). MR247243
  13. LIONS, J. L. - PEETRE, J., Sur une classe d'espaces d'interpolation, Inst. Hautes Etudes Sci. Publ. Math., 19 (1964), 5-68. Zbl0148.11403MR165343
  14. TRIEBEL, H., Interpolation Theory, Functions Spaces, Differential Operators, North Holland, Amsterdam (1978). Zbl0387.46033MR503903
  15. YAKUBOV, S. Y., Completeness of Root Functions of Regular Differential Operators, Longman, Scientific and Technical, New York (1994). MR1401350
  16. YAKUBOV, S. Y., Linear Differential Equations and Applications, Baku, elm (1985). (in Russian). Zbl0622.34001
  17. YAKUBOV, S. Y., Noncoercive Boundary Value Problems for the Laplace Equation with a Spectral Parameter, Semigroup Forum, 53, (1996), 298-316. Zbl0857.35095MR1406776DOI10.1007/BF02574145
  18. YAKUBOV, S. Y. - YAKUBOV, Y. Y., Differential-Operator Equations. Ordinary and Partial Differential Equations, Chapman and Hall/CRC Press, New York (2000). Zbl0936.35002MR1739280

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