Symmetry Theorems and Uniform Rectifiability

  • John L Lewis1Email author and

    Affiliated with

    • Andrew L Vogel2

      Affiliated with

      Boundary Value Problems20062007:030190

      DOI: 10.1155/2007/30190

      Received: 3 June 2006

      Accepted: 21 September 2006

      Published: 30 November 2006

      Abstract

      We study overdetermined boundary conditions for positive solutions to some elliptic partial differential equations of http://static-content.springer.com/image/art%3A10.1155%2F2007%2F30190/MediaObjects/13661_2006_Article_641_IEq1_HTML.gif -Laplacian type in a bounded domain http://static-content.springer.com/image/art%3A10.1155%2F2007%2F30190/MediaObjects/13661_2006_Article_641_IEq2_HTML.gif . We show that these conditions imply uniform rectifiability of http://static-content.springer.com/image/art%3A10.1155%2F2007%2F30190/MediaObjects/13661_2006_Article_641_IEq3_HTML.gif and also that they yield the solution to certain symmetry problems.

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      Authors’ Affiliations

      (1)
      Department of Mathematics, University of Kentucky
      (2)
      Department of Mathematics, Syracuse University

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      Copyright

      © J. L. Lewis and A. L. Vogel 2007

      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.