Weak regularity of degenerate elliptic equations

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 W₂¹ (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.