Conformal geometry & the composite membrane problem

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Abstract

We show that a certain eigenvalue minimization problem in two dimensions for the Laplace operator in conformal classes is equivalent to the composite membrane problem. We again establish such a link in higher dimensions for eigenvalue problems stemming from the critical GJMS operators. New free boundary problems of unstable type arise in higher dimensions linked to the critical GJMS operator. In dimension four, the critical GJMS operator is exactly the Paneitz operator.

Original languageEnglish (US)
Pages (from-to)31-35
Number of pages5
JournalAnalysis and Geometry in Metric Spaces
Volume1
Issue number1
DOIs
StatePublished - 2013

All Science Journal Classification (ASJC) codes

  • Analysis
  • Geometry and Topology
  • Applied Mathematics

Keywords

  • Composite membrane problem
  • Conformal geometry
  • Eigenvalue minimization in conformal classes
  • Free boundary problems
  • GJMS operators
  • Paneitz operator

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