# American Institute of Mathematical Sciences

June  2011, 4(3): 523-538. doi: 10.3934/dcdss.2011.4.523

## Complete abstract differential equations of elliptic type with general Robin boundary conditions, in UMD spaces

 1 Département de Mathématiques et Informatique, ENSET d'Oran, B.P 1523 Oran El M'Naouer, Algeria 2 Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna 3 Laboratoire de Mathématiques, U.F.R Sciences, et Techniques, Université du Havre, B.P 540, 76058 Le Havre Cedex 4 Laboratoire de Mathématiques Pures et Appliquées, Université de Mostaganem, 27000, Algeria

Received  May 2009 Revised  January 2010 Published  November 2010

In this paper we prove some new results concerning a complete abstract second-order differential equation with general Robin boundary conditions. The study is developped in UMD spaces and uses the celebrated Dore-Venni Theorem. We prove existence, uniqueness and maximal regularity of the strict solution. This work completes previous one [3] by authors; see also [11].
Citation: Mustapha Cheggag, Angelo Favini, Rabah Labbas, Stéphane Maingot, Ahmed Medeghri. Complete abstract differential equations of elliptic type with general Robin boundary conditions, in UMD spaces. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 523-538. doi: 10.3934/dcdss.2011.4.523
##### References:
 [1] A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math., 10 (1960), 419-437. [2] D. L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab., 9 (1981), 997-1011. [3] M. Cheggag, A. Favini, R. Labbas, S. Maingot and A. Medeghri, Sturm-Liouville problems for an abstract differential equation of elliptic type in UMD spaces, Differential and Integral Equations, 21 (2008), 981-1000. [4] G. Dore and A. Venni, On the closedness of the sum of two closed operators, Mathematische Zeitschrift, 196 (1987), 270-286. [5] H. O. Fattorini, "The Cauchy Problem," Encyclopedia of Mathematics and its Applications, Vol. 18, Addison-Wesley, Reading, MA, 1983. [6] A. Favini, R. Labbas, S. Maingot, H. Tanabe and A. Yagi, Complete abstract differential equations of elliptic type in UMD spaces, Funkcialaj Ekvacioj, 49 (2006), 193-214. [7] A. Favini, R. Labbas, S. Maingot, H. Tanabe and A. Yagi, A simplified approach in the study of elliptic differential equations in UMD spaces and new applications, Funkcial. Ekvac., 51 (2008), 165-187. [8] P. Grisvard, Spazi di tracce e applicazioni (Italian), Rend. Mat. (6), 5 (1972), 657-729. [9] M. Haase, "The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, Vol. 169," Birkhäuser Verlag, Basel-Boston-Berlin, 2006. [10] S. G. Krein, "Linear Differential Equations in Banach Spaces," Moscou, 1967. [11] R. Labbas and M. Mechdene, Problème à dérivée oblique pour une equation differentielle opérationnelle du second ordre, Maghreb Math. Rev., 2 (1992), 177-200. [12] A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser, Basel, 1995. [13] R. E. Showalter, "Hilbert Space Methods for Partial Differential Equations," Pitman, London-San Francisco-Melbourne, 1977. [14] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North Holland, Amsterdam, 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. [2] D. L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab., 9 (1981), 997-1011. [3] M. Cheggag, A. Favini, R. Labbas, S. Maingot and A. Medeghri, Sturm-Liouville problems for an abstract differential equation of elliptic type in UMD spaces, Differential and Integral Equations, 21 (2008), 981-1000. [4] G. Dore and A. Venni, On the closedness of the sum of two closed operators, Mathematische Zeitschrift, 196 (1987), 270-286. [5] H. O. Fattorini, "The Cauchy Problem," Encyclopedia of Mathematics and its Applications, Vol. 18, Addison-Wesley, Reading, MA, 1983. [6] A. Favini, R. Labbas, S. Maingot, H. Tanabe and A. Yagi, Complete abstract differential equations of elliptic type in UMD spaces, Funkcialaj Ekvacioj, 49 (2006), 193-214. [7] A. Favini, R. Labbas, S. Maingot, H. Tanabe and A. Yagi, A simplified approach in the study of elliptic differential equations in UMD spaces and new applications, Funkcial. Ekvac., 51 (2008), 165-187. [8] P. Grisvard, Spazi di tracce e applicazioni (Italian), Rend. Mat. (6), 5 (1972), 657-729. [9] M. Haase, "The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, Vol. 169," Birkhäuser Verlag, Basel-Boston-Berlin, 2006. [10] S. G. Krein, "Linear Differential Equations in Banach Spaces," Moscou, 1967. [11] R. Labbas and M. Mechdene, Problème à dérivée oblique pour une equation differentielle opérationnelle du second ordre, Maghreb Math. Rev., 2 (1992), 177-200. [12] A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser, Basel, 1995. [13] R. E. Showalter, "Hilbert Space Methods for Partial Differential Equations," Pitman, London-San Francisco-Melbourne, 1977. [14] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North Holland, Amsterdam, 1978.
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