Nonlinear quasiconformal glue theorems

Abstract

Let Γ₀, Γ₁, …, Γ_N, L₁, …, L_M be the boundary contours of an (N + M + 1)-connected plane domain D, where Γ₁, …, Γ_N, L₁, …, L_M are situated inside Γ₀. In D there are some mutually exclusive contours γ₁,…,γ_n,1₁,…,1_m. We assume that

(0.1)

and denote

(0.2)

where and are the domains surrounded by γ_j and 1_j, respectively. We deal with the nonlinear uniformly elliptic complex equation of the first order , F = Q (z, w, w_z)w_z, zεD. We suppose that it satisfies the condition C: 1) Q(z,w,s) is continuous in wεIE (the whole plane) for almost every point zεD and sεIE, and is measurable in zεD for all continuous functions w(z) and all measurable functions s(z) in D⁺\{z_o}, z_o εD⁺; 2) the equation satisfies the uniformly elliptic condition. We prove then that the equation has a homeomorphic solution w(z) which maps quasiconformally D⁺ and D⁻ onto G⁺ and G⁻, respectively, with w(z_o) = ∞, z_oεD⁺, and satisfies the gluing conditions

(0.3)

where α(t) maps each of γ_j, l_j, L_j, and Γ_j topologically onto itself; they give positive shifts on γ ∪ Γ and reverse shifts on l ∪ L, etc.