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November  2013, 33(11&12): 4967-4990. doi: 10.3934/dcds.2013.33.4967

## Boundary value problem for elliptic differential equations in non-commutative cases

 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, France, France

Received  November 2011 Revised  February 2012 Published  May 2013

This paper is devoted to abstract second order complete elliptic differential equations set on $\left[ 0,1\right]$ in non-commutative cases. Existence, uniqueness and maximal regularity of the strict solution are proved. The study is performed in $C^{\theta }\left( \left[ 0,1\right] ;X\right)$.
Citation: Angelo Favini, Rabah Labbas, Stéphane Maingot, Maëlis Meisner. Boundary value problem for elliptic differential equations in non-commutative cases. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4967-4990. doi: 10.3934/dcds.2013.33.4967
##### References:
 [1] G. Da Prato, Abstract differential equations, maximal regularity and linearization, in "Nonlinear Functional Analysis and its Applications, Part 1 (Berkeley, Calif., 1983)," Proc. Sympos. Pure Math., 45, Amer. Math. Soc., Providence, RI (1986), 359-370. [2] G. Da Prato and P. Grisvard, Sommes d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl. (9), 54 (1975), 305-387. [3] A. Favini, R. Labbas, S. Maingot and M. Meisner, Study of complete abstract elliptic differential equations in non-commutative cases, Appl. Anal., 91 (2012), 1495-1510. doi: 10.1080/00036811.2011.635652. [4] A. Favini, R. Labbas, S. Maingot, H. Tanabe and A. Yagi, Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces, Discrete Contin. Dyn. Syst., 22 (2008), 973-987. doi: 10.3934/dcds.2008.22.973. [5] P. Grisvard, Spazi di tracce e applicazioni, Rend. Mat. (6), 5 (1972), 657-729. [6] B. H. Haak, M. Haase and P. C. Kunstmann, Perturbation, interpolation, and maximal regularity, Adv. Differential Equations, 11 (2006), 201-240. [7] J. L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Études Sci. Publ. Math., 19 (1964), 5-68. [8] A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhaüser Verlag, Basel, 1995. [9] E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl., 107 (1985), 16-66. doi: 10.1016/0022-247X(85)90353-1. [10] H. Triebel, "Interpolation Theory, Functions Spaces, Differential Operators," North-Holland Publishing Co., Amsterdam, New York, 1978.

show all references

##### References:
 [1] G. Da Prato, Abstract differential equations, maximal regularity and linearization, in "Nonlinear Functional Analysis and its Applications, Part 1 (Berkeley, Calif., 1983)," Proc. Sympos. Pure Math., 45, Amer. Math. Soc., Providence, RI (1986), 359-370. [2] G. Da Prato and P. Grisvard, Sommes d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl. (9), 54 (1975), 305-387. [3] A. Favini, R. Labbas, S. Maingot and M. Meisner, Study of complete abstract elliptic differential equations in non-commutative cases, Appl. Anal., 91 (2012), 1495-1510. doi: 10.1080/00036811.2011.635652. [4] A. Favini, R. Labbas, S. Maingot, H. Tanabe and A. Yagi, Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces, Discrete Contin. Dyn. Syst., 22 (2008), 973-987. doi: 10.3934/dcds.2008.22.973. [5] P. Grisvard, Spazi di tracce e applicazioni, Rend. Mat. (6), 5 (1972), 657-729. [6] B. H. Haak, M. Haase and P. C. Kunstmann, Perturbation, interpolation, and maximal regularity, Adv. Differential Equations, 11 (2006), 201-240. [7] J. L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Études Sci. Publ. Math., 19 (1964), 5-68. [8] A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhaüser Verlag, Basel, 1995. [9] E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl., 107 (1985), 16-66. doi: 10.1016/0022-247X(85)90353-1. [10] H. Triebel, "Interpolation Theory, Functions Spaces, Differential Operators," North-Holland Publishing Co., Amsterdam, New York, 1978.
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