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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A sharp regularity result of solutions of a transmission problem


Authors: Giovanna Citti and Fausto Ferrari
Journal: Proc. Amer. Math. Soc. 140 (2012), 615-620
MSC (2010): Primary 35J20; Secondary 35B65
Published electronically: June 13, 2011
MathSciNet review: 2846330
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Abstract: In this paper we study the regularity of solutions of a transmission problem arising in studying a fiber-reinforced composite media. It is described through a divergence type equation div$ (A \nabla u) = h,$ in an open set $ D$, where $ h$ is a bounded function and $ A$ is a uniformly elliptic matrix, bounded and with piecewise Hölder continuous coefficients. The subdomains $ D_i$ where $ A$ is of class $ C^{\alpha}$ have disjoint closure and are of class $ C^{1,\alpha}$. Exploiting an idea contained in a paper by Li and Vogelius, we obtain the optimal regularity result for solutions, proving that they are of class $ C^{1,\alpha}(\bar D_i)$.


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Giovanna Citti
Affiliation: Dipartimento di Matematica, Università Degli Studi di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna, Italy
Email: citti@dm.unibo.it

Fausto Ferrari
Affiliation: Dipartimento di Matematica, Università Degli Studi di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna, Italy
Email: ferrari@dm.unibo.it

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10916-X
PII: S 0002-9939(2011)10916-X
Keywords: Transmission problems, sharp regularity
Received by editor(s): August 2, 2010
Received by editor(s) in revised form: November 29, 2010
Published electronically: June 13, 2011
Additional Notes: The first and second authors were partially supported by M.U.R.S.T., Italy, and CG-DICE project, EU VII Framework Program.
The second author was partially supported by the GNAMPA project “Equazioni non lineari su varietà: proprietà qualitative e classificazione delle soluzioni”.
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2011 American Mathematical Society