American Institute of Mathematical Sciences

Advanced
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 & Continuous Dynamical Systems - A, 2008, 22 (4) : 973-987. doi: 10.3934/dcds.2008.22.973
[1]

Jeremy LeCrone, Gieri Simonett. Continuous maximal regularity and analytic semigroups. Conference Publications, 2011, 2011 (Special) : 963-970. doi: 10.3934/proc.2011.2011.963
[2]

Annamaria Canino, Elisa De Giorgio, Berardino Sciunzi. Second order regularity for degenerate nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4231-4242. doi: 10.3934/dcds.2018184
[3]

Alassane Niang. Boundary regularity for a degenerate elliptic equation with mixed boundary conditions. Communications on Pure & Applied Analysis, 2019, 18 (1) : 107-128. doi: 10.3934/cpaa.2019007
[4]

Yannick Privat, Emmanuel Trélat, Enrique Zuazua. Complexity and regularity of maximal energy domains for the wave equation with fixed initial data. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6133-6153. doi: 10.3934/dcds.2015.35.6133
[5]

Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635
[6]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194
[7]

Luciano Abadías, Carlos Lizama, Marina Murillo-Arcila. Hölder regularity for the Moore-Gibson-Thompson equation with infinite delay. Communications on Pure & Applied Analysis, 2018, 17 (1) : 243-265. doi: 10.3934/cpaa.2018015
[8]

Kangsheng Liu, Xu Liu, Bopeng Rao. Eventual regularity of a wave equation with boundary dissipation. Mathematical Control & Related Fields, 2012, 2 (1) : 17-28. doi: 10.3934/mcrf.2012.2.17
[9]

Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4353-4390. doi: 10.3934/dcds.2018190
[10]

Alain Haraux, Mitsuharu Ôtani. Analyticity and regularity for a class of second order evolution equations. Evolution Equations & Control Theory, 2013, 2 (1) : 101-117. doi: 10.3934/eect.2013.2.101
[11]

H. Beirão da Veiga. Vorticity and regularity for flows under the Navier boundary condition. Communications on Pure & Applied Analysis, 2006, 5 (4) : 907-918. doi: 10.3934/cpaa.2006.5.907
[12]

Yemin Chen. Analytic regularity for solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials. Kinetic & Related Models, 2010, 3 (4) : 645-667. doi: 10.3934/krm.2010.3.645
[13]

F. D. Araruna, F. O. Matias, M. P. Matos, S. M. S. Souza. Hidden regularity for the Kirchhoff equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1049-1056. doi: 10.3934/cpaa.2008.7.1049
[14]

Roberto Triggiani. Sharp regularity theory of second order hyperbolic equations with Neumann boundary control non-smooth in space. Evolution Equations & Control Theory, 2016, 5 (4) : 489-514. doi: 10.3934/eect.2016016
[15]

Marie-Françoise Bidaut-Véron, Marta García-Huidobro, Cecilia Yarur. Large solutions of elliptic systems of second order and applications to the biharmonic equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 411-432. doi: 10.3934/dcds.2012.32.411
[16]

Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1335-1343. doi: 10.3934/cpaa.2010.9.1335
[17]

Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1691-1706. doi: 10.3934/dcds.2017070
[18]

Yuri V. Rogovchenko, Fatoş Tuncay. Interval oscillation of a second order nonlinear differential equation with a damping term. Conference Publications, 2007, 2007 (Special) : 883-891. doi: 10.3934/proc.2007.2007.883
[19]

Nicolas Fourrier, Irena Lasiecka. Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions. Evolution Equations & Control Theory, 2013, 2 (4) : 631-667. doi: 10.3934/eect.2013.2.631
[20]

Angelo Favini, Rabah Labbas, Keddour Lemrabet, Stéphane Maingot, Hassan D. Sidibé. Resolution and optimal regularity for a biharmonic equation with impedance boundary conditions and some generalizations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4991-5014. doi: 10.3934/dcds.2013.33.4991

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences