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

Angelo Favini; Rabah Labbas; Stéphane Maingot; Hiroki Tanabe; Atsushi Yagi

doi:10.3934/dcds.2008.22.973

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2008, Volume 22, Issue 4: 973-987. Doi: 10.3934/dcds.2008.22.973

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

1.
Università degli Studi di Bologn, Dipartimento di Matematica, Piazza di Porta S. Donato, 5, 40126 Bologna
2.
Laboratoire de Mathématiques, U.F.R Sciences, et Techniques, Université du Havre, B.P 540, 76058 Le Havre Cedex
3.
Hirai Sanso 12-13, Takarazuka 665-0817
4.
Department of Applied Physics, Osaka University, Suita, Osaka, 565-0871

Received: May 2007

Revised: September 2007

Published: October 2008

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Abstract

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:

Second order abstract differential equation,
Boundary condition,
Analytic semigroup,
Maximal regularity,
Elliptic equation,
Hölderian regularity.

Mathematics Subject Classification: 34G10, 34K10, 35J25, 35J40.

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

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