Deficient Coerciveness Estimate for an Abstract Differential Equation with a Parameter Dependent Boundary Conditions
Aibeche, Aissa ; Favini, Angelo ; Mezoued, Chahrazed
Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007), p. 535-547 / Harvested from Biblioteca Digitale Italiana di Matematica
Access to full text
Full (PDF)
Full (DjVu)

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.

In questo lavoro si considera un problema differenziale astratto di tipo ellittico, in cui sia l'equazione che le condizioni ai limiti possono contenere un parametro spettrale. Prima si prova che questo operatore è un isomorfismo tra appropriati spazi funzionali e poi si dimostra una stima coerciva con difetto. I risultati ottenuti sono applicati allo studio di alcune classi di problemi ellittici, anche possibilmente degeneri.

Publié le : 2007-10-01
@article{BUMI_2007_8_10B_3_535_0,
     author = {Aissa Aibeche and Angelo Favini and Chahrazed Mezoued},
     title = {Deficient Coerciveness Estimate for an Abstract Differential Equation with a Parameter Dependent Boundary Conditions},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {10-A},
     year = {2007},
     pages = {535-547},
     zbl = {1129.35027},
     mrnumber = {2351540},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2007_8_10B_3_535_0}
}

Aibeche, Aissa; Favini, Angelo; Mezoued, Chahrazed. Deficient Coerciveness Estimate for an Abstract Differential Equation with a Parameter Dependent Boundary Conditions. Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007) pp. 535-547. http://gdmltest.u-ga.fr/item/BUMI_2007_8_10B_3_535_0/

[1] Agmon, S. - Nirenberg, L., Properties of Solutions of Ordinary Differential Equation in Banach Spaces, Comm. Pure Appl. Math., 16 (1963), 121-239. | MR 155203 | Zbl 0117.10001

[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. | MR 192188

[3] Aibeche, A., Coerciveness Estimates for a Class of Elliptic Problems, Diff. Equ. Dynam. Syst., 4 (1993), 341-351. | MR 1259173 | Zbl 0875.35021

[4] Aibeche, A., Completeness of Generalized Eigenvectors for a Class of Elliptic Problems, Result. Math., 31 (1998), 1-8. | MR 1610103 | Zbl 0894.35076

[5] Cobos, F. - Fernandez, D. L., On Interpolation of Compact Operators, Ark. Mat., 27 (1989), 211-217. | MR 1022277 | Zbl 0691.46047

[6] Denche, M., Abstract Differential Equation with a Spectral Parameter in the Boundary Conditions, Result. Math., 35 (1999), 217-227. | MR 1694903 | Zbl 0927.35032

[7] Dunford, N. - Schwartz, J. T., Linear Operators, Vol. II, Interscience, New York (1963). | MR 188745

[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. | MR 1440417 | Zbl 0889.35039

[9] Favini, A. - Yagi, A., Degenerate Differential Equations in Banach Spaces, Marcel Dekker, New York (1999). | MR 1654663 | Zbl 0913.34001

[10] Krein, S. G., Linear Differential Equations in Banach Spaces, American Mathematical Society, Providence (1971). | MR 342804

[11] Lions, J. L., Sur les Espaces d'Interpolation: Dualité, Math. Scand., 9 (1961), 147-177. | MR 159212 | Zbl 0103.08102

[12] Lions, J. L. - Magenes, E., Problèmes aux Limites non Homogènes et applications, Vol. I, Dunod, Paris (1968). | MR 247243

[13] Lions, J. L. - Peetre, J., Sur une classe d'espaces d'interpolation, Inst. Hautes Etudes Sci. Publ. Math., 19 (1964), 5-68. | MR 165343 | Zbl 0148.11403

[14] Triebel, H., Interpolation Theory, Functions Spaces, Differential Operators, North Holland, Amsterdam (1978). | MR 503903 | Zbl 0387.46033

[15] Yakubov, S. Y., Completeness of Root Functions of Regular Differential Operators, Longman, Scientific and Technical, New York (1994). | MR 1401350

[16] Yakubov, S. Y., Linear Differential Equations and Applications, Baku, elm (1985). (in Russian). | Zbl 0622.34001

[17] Yakubov, S. Y., Noncoercive Boundary Value Problems for the Laplace Equation with a Spectral Parameter, Semigroup Forum, 53, (1996), 298-316. | MR 1406776 | Zbl 0857.35095

[18] Yakubov, S. Y. - Yakubov, Y. Y., Differential-Operator Equations. Ordinary and Partial Differential Equations, Chapman and Hall/CRC Press, New York (2000). | MR 1739280 | Zbl 0936.35002