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Miklós Schweitzer
1998 Miklós Schweitzer
5
5
Part of
1998 Miklós Schweitzer
Problems
(1)
analysis
Source: miklos schweitzer 1998 q5
9/14/2021
Let
K
1
K_1
K
1
be an open disk in the complex plane whose boundary passes through the points -1 and +1, and let
K
2
K_2
K
2
be the mirror image of
K
1
K_1
K
1
across the real axis. Also, let
D
1
=
K
1
∩
K
2
D_1 = K_1 \cap K_2
D
1
=
K
1
∩
K
2
, and let
D
2
D_2
D
2
be the outside of
D
1
D_1
D
1
. Suppose that the function
u
1
(
z
)
u_1( z )
u
1
(
z
)
is harmonic on
D
1
D_1
D
1
and continuous on its closure,
u
2
(
z
)
u_2(z)
u
2
(
z
)
harmonic on
D
2
D_2
D
2
(including
∞
\infty
∞
) and continuous on its closure, and
u
1
(
z
)
=
u
2
(
z
)
u_1(z) = u_2(z)
u
1
(
z
)
=
u
2
(
z
)
at the common boundary of the domains
D
1
D_1
D
1
and
D
2
D_2
D
2
. Prove that if
u
1
(
x
)
≥
0
u_1( x )\geq 0
u
1
(
x
)
≥
0
for all
−
1
<
x
<
1
-1 < x <1
−
1
<
x
<
1
, then
u
2
(
x
)
≥
0
u_2 ( x )\geq 0
u
2
(
x
)
≥
0
for all
x
>
1
x>1
x
>
1
and
x
<
−
1
x<-1
x
<
−
1
.
complex analysis