关于一类二维奇异积分方程的唯一解

A unique solution of a kind of the two dimensional singular integral equations

设 D是一个边界 Γ∈C1 α( 0 <α≤ 1 )的有界单连通域 ,复函数 q( z)∈ C1 α( D) ,|q( z) |≤ 1 ,等式只能在Γ上成立 ,且在Γ上等式 q( z( t) ) z′( t) /z′( t) +1 |z∈Γ=0最多在有限个点上成立 .本文给出以满足上述条件的复伸张 q1 ( z)及 q2 ( z)∈ C1 α( D)为系数的二维奇异积分方程 ω( z) +q1 ( z)π Dω( t)( t-z) 2 dσz +q2 ( z)π Dω( t)t-zdσz =f ( z)的唯一解 .同时给出了一个实例 .

Suppose that D is a simply connected region whose boundary Γ∈C 1 α(0<α≤1), the complex function q(z)∈C 1 α(),｜q(z)｜≤1, equality may be only for z∈Γ, equality q(z(t)) z′(t) /z′(t)+1｜ z∈Γ =0 may hold only at finite points.Solutions of a kind of two dimensional singular integral equations with the coefficient q 1(z) satisfies above conditions and the coefficients q 2(z)∈C 1 α(): ω(z)+q 1(z) π Dω(t)(t-z) 2 d σ z+q 2(z) π Dω(t)t-z d σ z=f(z), z∈D are obtained.Meanwhile,an example is given.