Boundary Value Problems
Volume 2010 (2010), Article ID 690342, 21 pages
doi:10.1155/2010/690342
Research Article

Extremal Values of Half-Eigenvalues for 𝑝 -Laplacian with Weights in 𝐿 𝟏 Balls

Ping Yan

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

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.

Abstract

For one-dimensional 𝑝 -Laplacian with weights in ℒ 𝛾 ∶ = 𝐿 𝛾 ( [ 0 , 1 ] , ℝ ) ( 1 ≤ 𝛾 ≤ ∞ ) balls, we are interested in the extremal values of the 𝑚 th 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.