The spectral problem for the Neumann-Laplace operator and $p$-quasiconformal mappings Academic Article uri icon

abstract

  • In this paper we prove that the spectrum of the Neumann-Lap\-la\-ci\-an (the free membrane problem) in a large class of rough space domains is discrete. We also obtain lower estimates for the first non-trivial eigenvalue, using the $p$-quasiconformal mappings theory. In our opinion the mainly applicable class for the spectral problem is a class of $2$-quasiconformal mappings. For planar domains this class coincides with the class of quasiconformal mappings. Proofs are based on the composition operators theory for Sobolev spaces with first weak derivatives.

publication date

  • January 1, 2016