4.3. PLATEAU PROBLEM 83
with L D
q
1 C u
2
x
C u
2
y
. is gives
1 C u
2
y
u
xx
2u
x
u
y
u
xy
C
1 C u
2
x
u
yy
D 0: (4.85)
It is well known that solutions of (4.85) can be generated by the Enneper–Weierstrass formulas
given by
x D Re
Z
f .1 g
2
/dz;
y D Re
Z
if .1 C g
2
/dz; (4.86)
u D Re
Z
2 f g dz:
For example, choosing
f
D
1
and
g
D
z
gives
x D Re
z
z
3
3
;
y D Re i
z C
z
3
3
; (4.87)
u D Re z
2
;
and denoting z D p C i q gives rise to
x D p
p
3
3
C pq
2
;
y D q p
2
q C
q
3
3
; (4.88)
u D p
2
q
2
;
commonly referred to as Enneper’s minimal surface. Here we present a new way of generating
a solution to (4.85). We will show that (4.85) is linearizable and furthermore, we can represent
its solution parametrically in terms of solutions of Laplace’s equation.
4.3.1 LINEARIZATION
Under the Legendre transformation
x D v
r
; y D v
s
u D rv
r
C sv
s
v; (4.89)
first-order derivatives transform as
u
x
D r; u
y
D s; (4.90)
84 4. POINT AND CONTACT TRANSFORMATIONS
while second-order derivatives transform as
u
xx
D
v
ss
v
rr
v
ss
v
2
rs
; u
xy
D
v
rs
v
rr
v
ss
v
2
rs
; u
yy
D
v
rr
v
rr
v
ss
v
2
rs
; (4.91)
thus transforming (4.85) to the linear PDE
1 C r
2
v
rr
C 2rsv
rs
C
1 C s
2
v
ss
D 0: (4.92)
If we denote A D 1 C r
2
, B D 2rs and C D 1 C s
2
then B
2
4AC < 0 so that the PDE (4.92)
is elliptic. We now show that (4.92) is transformable to Laplace’s equation. Introducing polar
coordinates
r D cos b; s D sin b (4.93)
transforms (4.92) to
4
C
2
v
C v
bb
C v
D 0: (4.94)
Further, if we let
D
1
sinh a
; (4.95)
then (4.94) becomes
v
aa
C v
bb
C
2v
a
sinh a cosh a
D 0: (4.96)
Next we transform the dependent variable v as
v D
w
tanh a
; (4.97)
leading to
w
aa
C w
bb
C
2w
cosh
2
a
D 0: (4.98)
Finally, if we introduce the first-order Darboux transformation
w D W
a
tanh a W; (4.99)
then equation (4.98) becomes
@
@a
r
2
W
tanh
a
r
2
W
D
0;
(4.100)
where r
2
W D 0 is Laplace’s equation. Integrating (4.100) gives
r
2
W D F .b/ cosh a; (4.101)
where F .b/ is arbitrary, but we can set F .b/ D 0 without loss of generality. e rationale is as
follows. If F .b/ is arbitrary, then we can set F D G
00
C G so
r
2
W D
G
00
.b/ C G.b/
cosh a: (4.102)
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