Hua C., Wong R. Differential Equations and Asymptotic Theory in Mathematical Physics
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372
Here
q
is one of the
K
eigenvalues
of
the tridiagonal matrix
where
2m7ra 2m7i-b 2m7ra 2m7rb
+coth
-),
K
K
and
cQ
is
a
corresponding eigenvector of
Q.
We remark that such kind of
matrices also appeared in the study of K-peaked solutions
[8]
and
[ll].
,
d
=
m7r(coth
-
,
p
=
m7rcsch
-
K
a
=
mrcsch
-
K
4.
Existence
of
Wriggled Lamellar Solutions
We use bifurcation analysis to construct wriggled lamellar solutions. We use
y
as
a bifurcation parameter. Let
X(y)
be one of the
K
eigenvalues of order
e2
found in Part
3
of Theorem
3.1,
associated with
a
positive integer m.
Generically this eigenvalue is simple. To have multiplicity there would be
another m‘
#
m
so
that
X(y)
=
A,,
for a
A,,
associated with m’. Because
of
(3.4)
the latter case happens rarely, so we assume that
X(y)
is simple.
It
is continued smoothly to a curve of simple eigenvalues
X(y)
of
L,
as
y
varies. Let
YB
be a particular value of
y
so
that
X(~B)
=
0.
The existence
of such
YB
follows from
(3.4).
The sign of
X(y)
is determined, to the leading
order term, by
$(A
-
$)
+m27r2. This quantity is positive when
y
is small
and negative when
y
is large. See
[18,
Section
71
for more details. Denote
the eigenfunction associated with
X(~B)
by
cp~(r~,
y)
=
+B(z)
cos(m7ry). We
write
UB
:=
uyB
and
LB
:=
LyB
for simplicity. Let
x
:=
{W
E
w2l2(o)
:
aVw
=
Oon
ao,
?~i
=
0},
Y
:=
{Z
E
~~(0)
:
z
=
o}.
(4.1)
Here
X
is
a
dense subspace of
Y.
Y
is an Hilbert space with the usual
inner product
(.,
.)
inherited from
L2(D).
A
nonlinear map
F
:
(0,
CQ)
x
X
3
Y
is defined by
F(y,
w)
:=
-c2A(u,
+
w)
+
f
(u,
+
w)
-
f
(u,
+
w)
+
q(
-A)-’
(u,
+
w
-
a).
(4.2)
Obviously the “trivial branch” (y,O) is a solution branch of
F(y,w)
=
0.
It
corresponds to the K-interface, perfect lamellar solution
u,
of (1.2),
parameterized by
y.
We look for another solution branch, a bifurcating
branch,
(y(s), w(s))
of
F.
It
gives another solution
u,(~)
+
w(s)
of (1.2).
373
Using the classical Crandall and Rabinowitz’s bifurcation theorem
[5],
we obtain the following
Theorem
4.1.
At
y
=
YB
another solution branch
(y(s),
w(s))
bifurcates
from the “trivial branch”
(y,O).
Here
w(s)
=
S~B
+
sg(s)
where the param-
eter
s
is
in
a neighborhood
of
0
with
y(0)
=
YB
and
w(0)
=
0.
Moreover
g(s)
E
X
satisfies
g(s)
I
p~
and
g(0)
=
0.
Note that
u,(~)
+w(s)
is approximately
uy(S)
(X)+S+B(Z)
cos(rn.rry) since
g(s)
is
a smaller term compared to
~B(x,
y)
=
+B(~c)
cos(rn.rry). Plot
2
of
Figure
2
is made based on this observation.
5.
Stability
of
the Bifurcating Solutions
The eigenvalue
X(y)
of the “trivial” branch
uy
corresponds to an eigenvalue
X,(s)
of the bifurcating solution
ur(s)
+
w(s).
The sign of
X,(s)
may be
determined from the shape of
y(s).
Thus we proceed to compute
y’(0)
and
~”(0).
However the overall stability of
u~(~)+
w(s)
is interesting only when
X(y)
is the principal, i.e. the smallest, eigenvalue
of
L,.
Otherwise, both
uy
and
u~(~)
+
w(s)
are unstable.
Place,w(s)
=
SV)B
+
sg(s)
into
F(y,
w)
=
0
and divide by
s:
(5.1)
+
f(U-dS
+.W(.))
-E2A(?
+
p~
+
g(S))
+
cy(s)(-A)-’(-
+
VB
+
g(s))
=
Const.
where Const. refers to the term coming from the average
of
f,
which is
independent of
(x,y).
Here we do not need its exact value. On the other
hand divide the equation
(1.2)
of by
s
and subtract the result from
(5.1):
f(..,(,)
+.w(s))-f(.u,(s)
1
+
--E~A(VB
+
g(s))
+
S
(54
~y(s)(-A)-’((p~
+
g(s))
=
Const.
Differentiate
(5.2)
with respect to
s
and set
s
=
0
afterwards:
du,
1
dy
2
LBg’(0)
+
~’(0)
{
f”(uB)
-
Then we multiply
(5.3)
by
p~
and integrate over
D:
pB
+
c(
-A)
-‘pB}
+
-
f”(uB)pi
=
Const..
(5.3)
374
Clearly the right side of
(5.4)
is
0
since
C~B(Z,
y)
=
$B(z)
cos(mry) and
integration with respect to
y
yields
0.
A
simple computation shows that
y’(0)
=
0.
(5.5)
(5.5) implies that the bifurcation digram has the shape of a pitchfork.
There are two possibilities illustrated in Figure
3.
To
determine which of
Wriggled,
Unstable
Wriggled,
Stable
-->
Y
Figure
3.
The two possible diagrams of wriggled lamellar solutions bifurcating out of
perfect lamellar solutions. The bifurcating solutions are unstable in the first case where
~”(0)
<
0,
and stable in the second case where
~”(0)
>
0.
the two cases occurs, we need to find y”(0) which can be calculated
as
follows
:
~”(0)
JD
{Y(uB)
3
IY=~
‘PB
+
VB(-A)-’PB}
(5.6)
=
-
JD{2f”(UB)&g’(O)
+
~f”’(UB)&}.
We need to compute the integrals on both sides of
(5.6).
The computations
are formidable. We have to expand the quantity to the
c5
order term,
because all the lower order terms up to
c4
vanish. Our main idea is to
expand
UB,
$B,
2g’(O)
as
(...)
+
c2(
...)
near each interface
xj.
This is a very
long computation. We don’t know if there is a simpler proof.
To
state our results, we now assume that at
y
=
y~,
the principle eigen-
value
X(~B)
of
UB
is
0,
and this eigenvalue is associated with
a
particular
m.
There are
K
eigenvalues of order
6’
associated with this particular
m.
Here
the
0
eigenvalue
is
the smallest. Hence
A
now is the smallest eigenvalue
of
[G,(T$,
z;)].
According to
[18,
Section
71
1
mr(tanh(mra)
+
tanh(mrb))’
1
mr(coth(mra)
+
cot(mrb)
-
csch
(mra)
+
csch
(mxb))
’
1
d
+
Ja2
+
p2
3-
2apcosO
if
K=
1,
A=
A=
A=
if
K
=
2,
,
0=27~/K, if
K
23.
(5.7)
375
Recall that
a,
p
and
d
are defined after (3.8).
Define
”-
I
cosh(mn(1-
2x7))
-
5(mn)37
i-
4sinh(mn)
8yB
where
yBO/‘r
can be determined and m is associated with the principal
eigenvalue
0.
Note that
S(a,K)
depends.on
a
and
K
only.
It
does not
depend on
r.
Since
r
depends on the shape of
W,
S(a,
K)
is independent
of the exact shape of
W.
Then we obtain the following result:
(5.9)
As
a consequence, we have
Theorem
5.1.
When
E
is suBciently
small,
the bifurcating solution
u~(~)
+
W(S)
of
K
wriggled interfaces is stable
if
S(a,
K)
>
0
and
it
is
unstable
if
S(a,K)
<
0.
Let us use Theorem 5.1 to work out some examples.
The quantity
S(a,
K)
may be accurately calculated. Tables
1
and 2 report our numerical
calculations for the cases
a
=
1/2
and 7/8. In each table the first column is
the number of the interfaces in the perfect lamellar solution
UB.
The second
column gives the value of m associated with the principal eigenvalue
0
of
UB.
Note that m does not increase
as
fast as
K
does. The third column
has the value
of
yBO/‘r.
We will explain the fourth in a moment. The fifth
column has the value of
S(a,
K).
The last column indicates the stability
of
the bifurcating solution with
K
wriggled interfaces.
There an interesting relationship between the perfect lamellar solution
UB
whose principal eigenvalue is
0,
and the 1-D global minimizer. In [18,
Section 81 it is shown that the 1-D global minimizer (the global minimizer of
11
in Theorem
2.1,
also a perfect lamellar solution on
D
after trivial exten-
sion), which is one of the 1-D local .minimizers, has the number
of
interfaces
Ko,t
which minimizes (among positive integers
N)
TN
+
ya2b2/(6N2).
If
we take
y
=
TB
so
that the K-interface, perfect lamellar solution
UB
has
0
principal eigenvalue, we find the 1-D global minimizer corresponding to
YB.
The number of interfaces
Kept
of
this 1-D global minimizer is reported
in the fourth columns in Tables
1
and 2. For most
a
and
K
the 1-D global
minimizer has one more interface than
UB
does. In some other cases the
376
Km
YB'/T
KO,,
S(1/8,
K)
1
3 1.7317e+03
2
-1.5234e-01
2
5
1.3418et04
4 -8.9872e-03
Stability
Unstable
Unstable
3
4
1-D
global minimizer is exactly
UB.
Because
UB
is on the borderline of
2-D
stability, the 1-D global minimizer sits near the
2-D
stability borderline.
2
1.0218e+04 3 5.7102e-02 Stable
3 2.3798e+04 5 5.1520e-02 Stable
Acknowledgments.
I
thank Professor Chen Hua for inviting me to give
a talk at this conference. The work reported here
is
a
joint project with
Prof. Xiaofeng Ren. This research is supported in part by
a
Direct Grant
from CUHK and an Earmarked Grant
df
RGC of Hong Kong.
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