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

 

Continuity in weak topology and extremal problems of eigenvalues of the $ p$-Laplacian


Authors: Ping Yan and Meirong Zhang
Journal: Trans. Amer. Math. Soc. 363 (2011), 2003-2028
MSC (2010): Primary 47J10, 34L15; Secondary 58C07, 58C40
Published electronically: November 5, 2010
MathSciNet review: 2746673
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Abstract: We will study the dependence of eigenvalues of the one-dimensional $ p$-Laplacian on potentials or weights. Two results are obtained. One is the continuity of eigenvalues in potentials with respect to the weak topologies of $ L^\gamma$ spaces, $ 1\le \gamma\le \infty$, and the other is the continuous differentiability of eigenvalues in potentials with respect to $ L^\gamma$ norms. As applications, we will study some extremal problems of eigenvalues by developing some analytical methods.


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Additional Information

Ping Yan
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
Email: pyan@math.tsinghua.edu.cn

Meirong Zhang
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China – and – Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, People’s Republic of China
Email: mzhang@math.tsinghua.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05051-2
PII: S 0002-9947(2010)05051-2
Keywords: Eigenvalue, $p$-Laplacian, potential, weight, continuity, weak topology, differentiability, extremal value
Received by editor(s): September 12, 2008
Received by editor(s) in revised form: March 17, 2009
Published electronically: November 5, 2010
Additional Notes: This research was supported by the National Basic Research Program of China (Grant no. 2006CB805903), the National Natural Science Foundation of China (Grant no. 10531010) and the 111 Project of China (2007).
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.