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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Neumann functions for second order elliptic systems with measurable coefficients


Authors: Jongkeun Choi and Seick Kim
Journal: Trans. Amer. Math. Soc. 365 (2013), 6283-6307
MSC (2010): Primary 35J08, 35J47, 35J57
Published electronically: June 3, 2013
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Abstract: We study Neumann functions for divergence form, second-order elliptic systems with bounded measurable coefficients in a bounded Lipschitz domain or a Lipschitz graph domain. We establish existence, uniqueness, and various estimates for the Neumann functions under the assumption that weak solutions of the system enjoy interior Hölder continuity. Also, we establish global pointwise bounds for the Neumann functions under the assumption that weak solutions of the system satisfy a certain natural local boundedness estimate. Moreover, we prove that such a local boundedness estimate for weak solutions of the system is in fact equivalent to the global pointwise bound for the Neumann function. We present a unified approach valid for both the scalar and the vectorial cases.


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Additional Information

Jongkeun Choi
Affiliation: Department of Mathematics, Yonsei University, Seoul 120-749, Republic of Korea
Email: cjg@yonsei.ac.kr

Seick Kim
Affiliation: Department of Mathematics, Yonsei University, Seoul 120-749, Republic of Korea
Address at time of publication: Department of Computational Science and Engineering, Yonsei University, Seoul 120-749, Republic of Korea
Email: kimseick@yonsei.ac.kr

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05886-2
PII: S 0002-9947(2013)05886-2
Keywords: Neumann function, Green's function, Neumann boundary problem, second-order elliptic system, measurable coefficients
Received by editor(s): July 2, 2011
Received by editor(s) in revised form: March 11, 2012
Published electronically: June 3, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.