tailieunhanh - báo cáo hóa học:" Research Article Extremal Values of Half-Eigenvalues for p-Laplacian with Weights in L1 Balls"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Extremal Values of Half-Eigenvalues for p-Laplacian with Weights in L1 Balls | Hindawi Publishing Corporation Boundary Value Problems Volume 2010 Article ID 690342 21 pages doi 2010 690342 Research Article Extremal Values of Half-Eigenvalues for p-Laplacian with Weights in L1 Balls Ping Yan Department of Mathematical Sciences Tsinghua University Beijing 100084 China Correspondence should be addressed to Ping Yan pyan@ Received 24 May 2010 Accepted 21 October 2010 Academic Editor V. Shakhmurov Copyright 2010 Ping Yan. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. For one-dimensional p-Laplacian with weights in LY LY 0 1 R 1 Y x balls we are interested in the extremal values of the mth positive half-eigenvalues associated with Dirichlet Neumann and generalized periodic boundary conditions respectively. It will be shown that the extremal value problems for half-eigenvalues are equivalent to those for eigenvalues and all these extremal values are given by some best Sobolev constants. 1. Introduction Occasionally we need to solve extremal value problems for eigenvalues. A classical example studied by Krein 1 is the infimum and the supremum of the mth Dirichlet eigenvalues of Hill s operator with positive weight inf f-iDfw w e Er hj sup m w w e Er hj where 0 r h TO and iz-1 w e LY 0 w h 0 In this paper we always use superscripts D N P and G to indicate Dirichlet Neumann periodic and generalized periodic boundary value conditions respectively. Similar extremal value problems for p-Laplacian were studied by Yan and Zhang 2 . For Hill s operator with weight Lou and Yanagida 3 studied the minimization problem of the positive principal 2 Boundary Value Problems Neumann eigenvalues which plays a crucial role in population dynamics. Given constants K e 0 to and a e 0 1 denote SKa t m e Lto -1 m K m 0 m t dt -a . 0 The positive principal eigenvalue pN m .