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:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis left-parenthesis upper M right-parenthesis right-parenthesis EndLabel StartFraction partial-differential u Over partial-differential x EndFraction minus StartFraction partial-differential v Over partial-differential y EndFraction equals a u plus b v plus f comma StartFraction partial-differential u Over partial-differential y EndFraction plus StartFraction partial-differential v Over partial-differential x EndFraction equals c u plus d v plus g period EndLayout">

(

(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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis left-parenthesis upper N right-parenthesis right-parenthesis EndLabel integral Subscript 0 Superscript y Baseline b left-parenthesis x comma t right-parenthesis d t plus integral Subscript 0 Superscript x Baseline d left-parenthesis t comma y right-parenthesis d t equals integral Subscript 0 Superscript y Baseline b left-parenthesis 0 comma t right-parenthesis d t plus integral Subscript 0 Superscript x Baseline d left-parenthesis t comma 0 right-parenthesis d t EndLayout">

(

(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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar f double-vertical-bar Subscript p Baseline equals sup Underscript 0 less-than-or-slanted-equals r greater-than 1 Endscripts left-brace left-parenthesis 1 slash 2 pi right-parenthesis integral Subscript negative pi Superscript pi Baseline StartAbsoluteValue f left-parenthesis r e Superscript i theta Baseline right-parenthesis EndAbsoluteValue Superscript p Baseline d theta right-brace Superscript 1 slash p Baseline">

‖
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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column double-vertical-bar v double-vertical-bar Subscript p Baseline less-than-or-slanted-equals upper A Subscript p Baseline double-vertical-bar u double-vertical-bar Subscript p Baseline plus upper B Subscript p Baseline comma 2nd Column a m p semicolon 1 greater-than p greater-than normal infinity comma 2nd Row 1st Column double-vertical-bar v double-vertical-bar Subscript p Baseline less-than-or-slanted-equals upper A Subscript p Baseline double-vertical-bar u double-vertical-bar Subscript 1 Baseline plus upper B Subscript p Baseline comma 2nd Column a m p semicolon 0 greater-than p greater-than 1 comma EndLayout">

‖
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\}

.