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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 & Continuous Dynamical Systems, 2013, 33 (11&12) : 4991-5014. doi: 10.3934/dcds.2013.33.4991
##### References:
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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] H. Cartan, "Théorie Elémentaire des Fonctions Analytiques d'une ou Plusieurs Variables Complexes," Hermann, Paris, 1961.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] H. Komatsu, Fractional powers of operators, Pacific J. Math., 19 (1966), 285-346. doi: 10.2140/pjm.1966.19.285.  Google Scholar [15] J. L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Études Sci. Publ. Math., 19 (1964), 5-68.  Google Scholar [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. Google Scholar [17] H. Triebel, "Interpolation Theory, Functions Spaces, Differential Operators," North-Holland Publishing Co., Amsterdam, New York, 1978.  Google Scholar
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