Weak regularity of degenerate elliptic equations

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Abstract

Let φ: Ω → D be a conformal mapping of a bounded simply connected planar domain Ω onto the unit disc D ⊂ ℝ2. We prove existence and uniqueness in Ω of weak solutions of a degenerate Poisson equation for a hyperbolic weight h(z) = |φz |2 in a corresponding two weighted Sobolev space W2 1 (Ω, h, 1).Here φz is a complex derivative. We also study weak regularity of the solutions in conformal regular domains. The domain Ω is a conformal regular domain [4] if (φ−1)w ∈ Lα(D) for some α > 2.

Original languageEnglish
Pages (from-to)262-270
Number of pages9
JournalLobachevskii Journal of Mathematics
Volume38
Issue number2
DOIs
StatePublished - 1 Mar 2017

Keywords

  • Conformal mappings
  • Sobolev spaces
  • elliptic equations

ASJC Scopus subject areas

  • General Mathematics

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