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December  2008, 22(4): 973-987. doi: 10.3934/dcds.2008.22.973

Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces

Angelo Favini 1, , Rabah Labbas 2, , Stéphane Maingot 2, , Hiroki Tanabe 3, and  Atsushi Yagi 4,

1. 

Università degli Studi di Bologn, Dipartimento di Matematica, Piazza di Porta S. Donato, 5, 40126 Bologna, Italy
2. 

Laboratoire de Mathématiques, U.F.R Sciences, et Techniques, Université du Havre, B.P 540, 76058 Le Havre Cedex, France, France
3. 

Hirai Sanso 12-13, Takarazuka 665-0817, Japan
4. 

Department of Applied Physics, Osaka University, Suita, Osaka, 565-0871, Japan

Received  May 2007 Revised  September 2007 Published  September 2008

In this paper we give new results on complete abstract second order differential equations of elliptic type in the framework of Hölder spaces, extending those given in [4] and [5]. More precisely we study $u^{\prime \prime }+2Bu^{\prime }+Au=f$ in the case when $f$ is Hölder continuous and under some natural assumptions on the operators $A$ and $B$. We give necessary and sufficient conditions of compatibility to obtain a strict solution $u$ and also to ensure that the strict solution has the maximal regularity property.
Keywords: Elliptic equation, Maximal regularity, Second order abstract differential equation, Analytic semigroup, Boundary condition, Hölderian regularity..
Mathematics Subject Classification: 34G10, 34K10, 35J25, 35J4.
Citation: Angelo Favini, Rabah Labbas, Stéphane Maingot, Hiroki Tanabe, Atsushi Yagi. Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 973-987. doi: 10.3934/dcds.2008.22.973
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