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November  2013, 33(11&12): 4991-5014. doi: 10.3934/dcds.2013.33.4991

Resolution and optimal regularity for a biharmonic equation with impedance boundary conditions and some generalizations

1. 

Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna

2. 

Laboratoire de Mathématiques Appliquées du Havre, Université du Havre, 25 rue Philippe Lebon, CS 80540, 76058 Le Havre Cedex, France

3. 

Laboratoire AMNEDP, Faculté de Maths USTHB, BP 32, El Alia Bab Ezzouar, 16111 Alger, Algeria

Received  October 2011 Revised  October 2011 Published  May 2013

In this work, a biharmonic equation with an impedance (non standard) boundary condition and more general equations are considered. The study is performed in the space $L^{p}(-1,0$ $;$ $X)$, $1 < p < \infty $, where $X$ is a UMD Banach space. The problem is obtained through a formal limiting process on a family of boundary and transmission problems $(P^{\delta})_{\delta > 0}$ set in a domain having a thin layer. The limiting problem models, for instance, the bending of a thin plate with a stiffness on a part of its boundary (see Favini et al. [13]).
    We build an explicit representation of the solution, then we study its regularity and give a meaning to the non standard boundary condition.
Citation: Angelo Favini, Rabah Labbas, Keddour Lemrabet, Stéphane Maingot, Hassan D. Sidibé. Resolution and optimal regularity for a biharmonic equation with impedance boundary conditions and some generalizations. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4991-5014. doi: 10.3934/dcds.2013.33.4991
References:
[1]

A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math., 10 (1960), 419-437. doi: 10.2140/pjm.1960.10.419.

[2]

O. Belhamiti, R. Labbas, K. Lemrabet and A. Medeghri, Transmission problems in a thin layer set in the framework of the Hölder spaces: resolution and impedance concept, J. Math. Anal. Appl., 358 (2009), 457-484. doi: 10.1016/j.jmaa.2009.05.010.

[3]

J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat., 21 (1983), 163-168. doi: 10.1007/BF02384306.

[4]

M. Bourlard, A. Maghnouji, S. Nicaise and L. Paquet, Asymptotic expansion of the solution of a mixed Dirichlet-Ventcel problem with a small parameter, Asymptot. Anal., 28 (2001), 241-278.

[5]

D. L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab., 9 (1981), 997-1011. doi: 10.1214/aop/1176994270.

[6]

G. Caloz, M. Costabel, M. Dauge and G. Vial, Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer, Asymptot. Anal., 50 (2006), 121-173.

[7]

H. Cartan, "Théorie Elémentaire des Fonctions Analytiques d'une ou Plusieurs Variables Complexes," Hermann, Paris, 1961.

[8]

M. Cowling, I. Doust, A. McIntosh and A. Yagi, Banach space operators with a bounded $H^{\infty} $ functional calculus, J. Austral. Math. Soc. Ser. A, 60 (1996), 51-89. doi: 10.1017/S1446788700037393.

[9]

G. Dore, A. Favini, R. Labbas and K. Lemrabet, An abstract transmission problem in a thin layer, I: Sharp estimates, J. Funct. Anal., 261 (2011), 1865-1922. doi: 10.1016/j.jfa.2011.05.021.

[10]

G. Dore and A. Venni, $H^{\infty} $ functional calculus for sectorial and bisectorial operators, Studia Math., 166 (2005), 221-241. doi: 10.4064/sm166-3-2.

[11]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201. doi: 10.1007/BF01163654.

[12]

M. Haase, "The Functional Calculus for Sectorial Operators," Operator Theory: Advances and Applications, 169, Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8.

[13]

A. Favini, R. Labbas, K. Lemrabet, S. Maingot and H. D. Sidibé, Transmission problem for an abstract fourth-order differential equation of elliptic type in UMD spaces, Adv. Differential Equations, 15 (2010), 43-72.

[14]

H. Komatsu, Fractional powers of operators, Pacific J. Math., 19 (1966), 285-346. doi: 10.2140/pjm.1966.19.285.

[15]

J. L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Études Sci. Publ. Math., 19 (1964), 5-68.

[16]

H. D. Sidibé, "Étude d'un Problème aux Limites et de Transmission dans une Couche Mince pour une Équation Différentielle Abstraite Elliptique d'Ordre Quatre," Ph.D thesis, Université du Havre in France, 2009.

[17]

H. Triebel, "Interpolation Theory, Functions Spaces, Differential Operators," North-Holland Publishing Co., Amsterdam, New York, 1978.

show all references

References:
[1]

A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math., 10 (1960), 419-437. doi: 10.2140/pjm.1960.10.419.

[2]

O. Belhamiti, R. Labbas, K. Lemrabet and A. Medeghri, Transmission problems in a thin layer set in the framework of the Hölder spaces: resolution and impedance concept, J. Math. Anal. Appl., 358 (2009), 457-484. doi: 10.1016/j.jmaa.2009.05.010.

[3]

J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat., 21 (1983), 163-168. doi: 10.1007/BF02384306.

[4]

M. Bourlard, A. Maghnouji, S. Nicaise and L. Paquet, Asymptotic expansion of the solution of a mixed Dirichlet-Ventcel problem with a small parameter, Asymptot. Anal., 28 (2001), 241-278.

[5]

D. L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab., 9 (1981), 997-1011. doi: 10.1214/aop/1176994270.

[6]

G. Caloz, M. Costabel, M. Dauge and G. Vial, Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer, Asymptot. Anal., 50 (2006), 121-173.

[7]

H. Cartan, "Théorie Elémentaire des Fonctions Analytiques d'une ou Plusieurs Variables Complexes," Hermann, Paris, 1961.

[8]

M. Cowling, I. Doust, A. McIntosh and A. Yagi, Banach space operators with a bounded $H^{\infty} $ functional calculus, J. Austral. Math. Soc. Ser. A, 60 (1996), 51-89. doi: 10.1017/S1446788700037393.

[9]

G. Dore, A. Favini, R. Labbas and K. Lemrabet, An abstract transmission problem in a thin layer, I: Sharp estimates, J. Funct. Anal., 261 (2011), 1865-1922. doi: 10.1016/j.jfa.2011.05.021.

[10]

G. Dore and A. Venni, $H^{\infty} $ functional calculus for sectorial and bisectorial operators, Studia Math., 166 (2005), 221-241. doi: 10.4064/sm166-3-2.

[11]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201. doi: 10.1007/BF01163654.

[12]

M. Haase, "The Functional Calculus for Sectorial Operators," Operator Theory: Advances and Applications, 169, Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8.

[13]

A. Favini, R. Labbas, K. Lemrabet, S. Maingot and H. D. Sidibé, Transmission problem for an abstract fourth-order differential equation of elliptic type in UMD spaces, Adv. Differential Equations, 15 (2010), 43-72.

[14]

H. Komatsu, Fractional powers of operators, Pacific J. Math., 19 (1966), 285-346. doi: 10.2140/pjm.1966.19.285.

[15]

J. L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Études Sci. Publ. Math., 19 (1964), 5-68.

[16]

H. D. Sidibé, "Étude d'un Problème aux Limites et de Transmission dans une Couche Mince pour une Équation Différentielle Abstraite Elliptique d'Ordre Quatre," Ph.D thesis, Université du Havre in France, 2009.

[17]

H. Triebel, "Interpolation Theory, Functions Spaces, Differential Operators," North-Holland Publishing Co., Amsterdam, New York, 1978.

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