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Extremal Values of Half-Eigenvalues for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F690342/MediaObjects/13661_2010_Article_947_IEq1_HTML.gif -Laplacian with Weights in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F690342/MediaObjects/13661_2010_Article_947_IEq2_HTML.gif Balls

Boundary Value Problems20102010:690342

DOI: 10.1155/2010/690342

Received: 24 May 2010

Accepted: 21 October 2010

Published: 2 November 2010

Abstract

For one-dimensional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F690342/MediaObjects/13661_2010_Article_947_IEq3_HTML.gif -Laplacian with weights in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F690342/MediaObjects/13661_2010_Article_947_IEq4_HTML.gif balls, we are interested in the extremal values of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F690342/MediaObjects/13661_2010_Article_947_IEq5_HTML.gif 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.

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Authors’ Affiliations

(1)
Department of Mathematical Sciences, Tsinghua University

Copyright

© Ping Yan. 2010

This article is published under license to BioMed Central Ltd. 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.