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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 & Continuous Dynamical Systems - A, 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, 45 (1986), 359.   Google Scholar

[2]

G. Da Prato and P. Grisvard, Sommes d'opérateurs linéaires et équations différentielles opérationnelles,, J. Math. Pures Appl. (9), 54 (1975), 305.   Google Scholar

[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.  doi: 10.1080/00036811.2011.635652.  Google Scholar

[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.  doi: 10.3934/dcds.2008.22.973.  Google Scholar

[5]

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

[6]

B. H. Haak, M. Haase and P. C. Kunstmann, Perturbation, interpolation, and maximal regularity,, Adv. Differential Equations, 11 (2006), 201.   Google Scholar

[7]

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

[8]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Birkhaüser Verlag, (1995).   Google Scholar

[9]

E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions,, J. Math. Anal. Appl., 107 (1985), 16.  doi: 10.1016/0022-247X(85)90353-1.  Google Scholar

[10]

H. Triebel, "Interpolation Theory, Functions Spaces, Differential Operators,", North-Holland Publishing Co., (1978).   Google Scholar

show all references

References:
[1]

G. Da Prato, Abstract differential equations, maximal regularity and linearization,, in, 45 (1986), 359.   Google Scholar

[2]

G. Da Prato and P. Grisvard, Sommes d'opérateurs linéaires et équations différentielles opérationnelles,, J. Math. Pures Appl. (9), 54 (1975), 305.   Google Scholar

[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.  doi: 10.1080/00036811.2011.635652.  Google Scholar

[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.  doi: 10.3934/dcds.2008.22.973.  Google Scholar

[5]

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

[6]

B. H. Haak, M. Haase and P. C. Kunstmann, Perturbation, interpolation, and maximal regularity,, Adv. Differential Equations, 11 (2006), 201.   Google Scholar

[7]

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

[8]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Birkhaüser Verlag, (1995).   Google Scholar

[9]

E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions,, J. Math. Anal. Appl., 107 (1985), 16.  doi: 10.1016/0022-247X(85)90353-1.  Google Scholar

[10]

H. Triebel, "Interpolation Theory, Functions Spaces, Differential Operators,", North-Holland Publishing Co., (1978).   Google Scholar

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