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

Angelo Favini; Rabah Labbas; Stéphane Maingot; Maëlis Meisner

doi:10.3934/dcds.2013.33.4967

Article Contents

2013, Volume 33, Issue 11&12: 4967-4990. Doi: 10.3934/dcds.2013.33.4967

This issue Previous Article Ultraparabolic equations with nonlocal delayed boundary conditions Next Article Resolution and optimal regularity for a biharmonic equation with impedance boundary conditions and some generalizations

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

Received: October 31, 2011

Revised: January 31, 2012

Published: October 2013

Abstract Related Papers Cited by

Abstract

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) $.

Keywords:

Second order abstract elliptic differential equation,
non-commutative case,
boundary condition,
analytic semigroup,
maximal regularity.

Mathematics Subject Classification: 35A09, 35B65, 35C15, 35J25, 35R20, 47D06.

Citation:

Related Papers

Cited by

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.