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analysis

Source: miklos schweitzer 1998 q5

September 14, 2021
complex analysis

Problem Statement

Let K1K_1 be an open disk in the complex plane whose boundary passes through the points -1 and +1, and let K2K_2 be the mirror image of K1K_1 across the real axis. Also, let D1=K1∩K2D_1 = K_1 \cap K_2 , and let D2D_2 be the outside of D1D_1 . Suppose that the function u1(z)u_1( z ) is harmonic on D1D_1 and continuous on its closure, u2(z)u_2(z) harmonic on D2D_2 (including ∞\infty) and continuous on its closure, and u1(z)=u2(z)u_1(z) = u_2(z) at the common boundary of the domains D1D_1 and D2D_2 . Prove that if u1(x)≥0u_1( x )\geq 0 for all −1<x<1-1 < x <1, then u2(x)≥0u_2 ( x )\geq 0 for all x>1x>1 and x<−1x<-1.
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