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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

Mustapha Cheggag 1, , Angelo Favini 2, , Rabah Labbas 3, , Stéphane Maingot 3, and  Ahmed Medeghri 4,

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].
Keywords: UMD spaces, Boundary value problem of Robin type, Bounded imaginary powers, Dore-Venni Theorem., Abstract differential equations of second order.
Mathematics Subject Classification: Primary: 35J20; Secondary: 35J40, 47D0.
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 & 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.  Google Scholar

[2]

D. L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab., 9 (1981), 997-1011.  Google Scholar

[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.  Google Scholar

[4]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Mathematische Zeitschrift, 196 (1987), 270-286.  Google Scholar

[5]

H. O. Fattorini, "The Cauchy Problem," Encyclopedia of Mathematics and its Applications, Vol. 18, Addison-Wesley, Reading, MA, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

P. Grisvard, Spazi di tracce e applicazioni (Italian), Rend. Mat. (6), 5 (1972), 657-729.  Google Scholar

[9]

M. Haase, "The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, Vol. 169," Birkhäuser Verlag, Basel-Boston-Berlin, 2006.  Google Scholar

[10]

S. G. Krein, "Linear Differential Equations in Banach Spaces," Moscou, 1967. Google Scholar

[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.  Google Scholar

[12]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser, Basel, 1995.  Google Scholar

[13]

R. E. Showalter, "Hilbert Space Methods for Partial Differential Equations," Pitman, London-San Francisco-Melbourne, 1977.  Google Scholar

[14]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North Holland, Amsterdam, 1978.  Google Scholar

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.  Google Scholar

[2]

D. L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab., 9 (1981), 997-1011.  Google Scholar

[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.  Google Scholar

[4]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Mathematische Zeitschrift, 196 (1987), 270-286.  Google Scholar

[5]

H. O. Fattorini, "The Cauchy Problem," Encyclopedia of Mathematics and its Applications, Vol. 18, Addison-Wesley, Reading, MA, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

P. Grisvard, Spazi di tracce e applicazioni (Italian), Rend. Mat. (6), 5 (1972), 657-729.  Google Scholar

[9]

M. Haase, "The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, Vol. 169," Birkhäuser Verlag, Basel-Boston-Berlin, 2006.  Google Scholar

[10]

S. G. Krein, "Linear Differential Equations in Banach Spaces," Moscou, 1967. Google Scholar

[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.  Google Scholar

[12]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser, Basel, 1995.  Google Scholar

[13]

R. E. Showalter, "Hilbert Space Methods for Partial Differential Equations," Pitman, London-San Francisco-Melbourne, 1977.  Google Scholar

[14]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North Holland, Amsterdam, 1978.  Google Scholar

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