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	Extremal values of eigenvalues of Sturm-Liouville operators with potentials in L-1 balls
	Wei, Qiaoling ; Meng, Gang ; Zhang, Meirong
	2010-10-12 ; 2010-10-12
关键词	Eigenvalue Sturm-Liouville operator Extremal value Potential L-p ball Variational method Period function Singular integral HILLS EQUATIONS P-LAPLACIAN STABILITY Mathematics
中文摘要	This paper is a continuation of Zhang [M. Zhang, Continuity in weak topology: Higher order linear systems of ODE, Sci. China Set. A 51 (2008) 1036-1058; M. Zhang, Extremal values of smallest eigenvalues of Hill's operators with potentials in L-1 balls, J. Differential Equations 246 (2009) 4188-4220]. Given a potential q is an element of L-p ([0, 1], R), p is an element of [1, infinity]. We use lambda(m)(q) to denote the mth Dirichlet eigenvalue of the Sturm-Liouville operator with potential q(t), where m is an element of N. The minimal value L-m,L- p(r) and the maximal value M-m,M- p(r) of lambda(m)(q) with potentials q in the L-p ball of radius r are well defined. In this paper, we will exploit the continuity of lambda(m)(q) in q with weak topologies and the variational method to give characterizations of L-m,L- p (r) and M-m,M- p (r) when p is an element of (1, infinity). By using the limiting approach as p down arrow 1, we find that the most important extremal values L-m,L- (1)(r) and M-m,M- 1(r) can be evaluated explicitly using some elementary functions of r. The corresponding extremal problems for Neumann eigenvalues and some periodic eigenvalues will be reduced to L-m,L- p(r) and M-m,M- p(r). (C) 2009 Elsevier Inc. All rights reserved.
语种	英语 ; 英语
出版者	ACADEMIC PRESS INC ELSEVIER SCIENCE ; SAN DIEGO ; 525 B ST, STE 1900, SAN DIEGO, CA 92101-4495 USA
内容类型	期刊论文
源URL	[http://hdl.handle.net/123456789/81265]
专题	清华大学
推荐引用方式 GB/T 7714	Wei, Qiaoling,Meng, Gang,Zhang, Meirong. Extremal values of eigenvalues of Sturm-Liouville operators with potentials in L-1 balls[J],2010, 2010.
APA	Wei, Qiaoling,Meng, Gang,&Zhang, Meirong.(2010).Extremal values of eigenvalues of Sturm-Liouville operators with potentials in L-1 balls..
MLA	Wei, Qiaoling,et al."Extremal values of eigenvalues of Sturm-Liouville operators with potentials in L-1 balls".(2010).

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