Weak regularity of degenerate elliptic equations Academic Article uri icon

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) = |φ z ′ |2 in a corresponding two weighted Sobolev space W 2 1 (Ω, h, 1).Here φ z ′ 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 ′ ∈ L α (D) for some α > 2.

publication date

  • January 1, 2017