Generalized conjugate function theorems for solutions of first-order elliptic systems on the plane
Authors: Yu Chung Ling 
Published: July 2010

Our essential aim is to generalize Privoloff’s theorem, Schwarz reflection principle, Kolmogorov’s theorem and the theorem of M. Riesz for conjugate functions to the solutions of differential equations in the

z
=
x
+
i
y

z = x + iy

plane of the following elliptic type:

(

(M)

)

∂
u

∂
x

−

∂
v

∂
y

=
a
u
+
b
v
+
f
,

∂
u

∂
y

+

∂
v

∂
x

=
c
u
+
d
v
+
g
.

\begin{equation} \tag {$\text {(M)}$} \frac {{\partial u}} {{\partial x}} - \frac {{\partial v}} {{\partial y}} = au + bv + f,\frac {{\partial u}} {{\partial y}} + \frac {{\partial v}} {{\partial x}} = cu + dv + g. \end{equation}

Theorem 1. Let the coefficients of (M) be Hölder continuous on

|
z
|

⩽
1

\left | z \right | \leqslant 1

. Let

(
u
,
v
)

(u, v)

be a solution of (M) in

|
z
|

>
1

\left | z \right | > 1

. If u is continuous on

|
z
|

⩽
1

\left | z \right | \leqslant 1

and Hölder continuous with index

α
\alpha

on

|
z
|

=
1

\left | z \right | = 1

, then

(
u
,
v
)

(u, v)

is Höolder continuous with index

α
\alpha

on

|
z
|

⩽
1

\left | z \right | \leqslant 1

. Theorem 2. Let the coefficients of (M) be continuous on

|
z
|

⩽
1

\left | z \right | \leqslant 1

and satisfy the condition

(

(N)

)

∫
0
y

b
(
x
,
t
)
d
t
+

∫
0
x

d
(
t
,
y
)
d
t
=

∫
0
y

b
(
0
,
t
)
d
t
+

∫
0
x

d
(
t
,
0
)
d
t

\begin{equation} \tag {$\text {(N)}$} \int _0^y {b(x,t)dt + \int _0^x {d(t,y)dt = \int _0^y {b(0,t)dt + \int _0^x {d(t,0)dt}}}} \end{equation}

for

|
z
|

⩽
1

\left | z \right | \leqslant 1

. And let

‖
f
‖

p

=

sup

0
⩽
r
>
1

{
(
1

/

2
π
)

∫

−
π

π

|

f
(
r

e

i
θ

)

|

p

d
θ

}

1

/

p

{\left \| f \right \|_p} = {\sup _{0 \leqslant r > 1}}\{ (1/2\pi )\int _{ - \pi }^\pi {{{\left | {f(r{e^{i\theta }})} \right |}^p}d\theta {\} ^{1/p}}}

. Then to each p,

0
>
p
>
∞

0 > p > \infty

, there correspond two constants

A
p

{A_p}

and

B
p

{B_p}

such that
\[

‖
v
‖

p

⩽

A
p

‖
u
‖

p

+

B
p

,

a
m
p
;

1
>
p
>
∞
,

‖
v
‖

p

⩽

A
p

‖
u
‖

1

+

B
p

,

a
m
p
;

0
>
p
>
1
,

\begin {array}{*{20}{c}} {{{\left \| v \right \|}_p} \leqslant {A_p}{{\left \| u \right \|}_p} + {B_p},} & {1 > p > \infty ,} \\ {{{\left \| v \right \|}_p} \leqslant {A_p}{{\left \| u \right \|}_1} + {B_p},} & {0 > p > 1,} \\ \end {array}

\]
hold for every solution

(
u
,
v
)

(u, v)

of (M) in

|
z
|

>
1

\left | z \right | > 1

with

v
(
0
)
=
0

v (0) = 0

. If

f
≡
g
≡
0

f \equiv g \equiv 0

, the theorem holds for

B
p

=
0

{B_p} = 0

. Furthermore, if b and d do not satisfy the condition (N) in

|
z
|

⩽
1

\left | z \right | \leqslant 1

, then we can relax the condition

v
(
0
)
=
0

v (0) = 0

, and still have the above inequalities. Theorem 3. Let the coefficients of (M) be analytic for x, y in

|
z
|

>
1

\left | z \right | > 1

. Let

(
u
,
v
)

(u, v)

be a solution of (M) in

{

|
z
|

>
1
}
∩
{
y
>
0
}

\{ \left | z \right | > 1\} \cap \{ y > 0\}

. If u is continuous in

{

|
z
|

>
1
}
∩
{
y
⩾
0
}

\{ \left | z \right | > 1\} \cap \{ y \geqslant 0\}

and analytic on

{
−
1
>
x
>
1
}

\{ - 1 > x > 1\}

, then

(
u
,
v
)

(u,v)

can be continued analytically across the boundary

{
−
1
>
x
>
1
}

\{ - 1 > x > 1\}

. Furthermore, if the coefficients and u satisfy some further boundary conditions, then

(
u
,
v
)

(u, v)

can be continued analytically into the whole of

{

|
z
|

>
1
}

\{ \left | z \right | > 1\}

.