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Multi-layer Local Minimum Solutions of the Bistable Equation in Noncylindrical Domains
Journal of Differential Equations DE3284 journal of differential equations 141, 1324 (1997) article no. DE973284
Multi-layer Local Minimum Solutions of the Bistable Equation in Noncylindrical Domains Xiaofeng Ren* Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, Minnesota 55455 Received July 9, 1996; revised March 11, 1997
We construct local minimum solutions for the semilinear bistable equation by minimizing the corresponding functional near some approximate solutions, under the hypothesis that certain global minimum solutions are isolated. The key is a certain characterization of PalaisSmale sequences and a proof that the functional takes higher values away from the approximate solutions. 1997 Academic Press
1. INTRODUCTION In this note we continue to study the problem posed in [1]. Consider the semilinear elliptic equation
{
&2u+f (u)=0
in 0
u =0 &
on 0.
(1.1)
where f is a bistable function, and 0 is an unbounded tube-shaped domain in R d. (1.1) is the EulerLagrange equation of the functional E defined by E(u)=
|
0
[ 12 |{u| 2 +W(u)] dx
(1.2)
where W(u)= u&1 f (w) dw, and W is a double well potential function of equal depth. The domain of the functional E is taken to be the class A=[u # L 1loc(0): {u # (L 2(0)) d, W(u) # L 1(0)].
(1.3)
Here L 1loc(0) is the space of measurable functions that belong to L 1(K ) for every compact subset K of 0. * Current address: Department of Mathematics and Statistics, Utah State University, Logan, UT 84322.
13 0022-039697 25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.
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14
XIAOFENG REN
We present an alternative approach to the construction of multi-layer solutions of (1.1) by minimizing E near some proper approximate solutions. The solutions being local minima of E reflects the advantage of this approach since in Theorem 4.4 [1] solutions are found through an indirect deformation argument and they are only known as critical points of (1.2). The hypothesis here that guarantees the existence of the solutions is also weaker than that in [1] (see Remark 2). The precise conditions on f, W and 0 are given as follows. H-1. W is a C 2 function that has exactly two global minima at &1 and 1, where W(&1)=W(1)=0, W"(&1)>0, and W"(1)>0. H-2. There exists 3>0 such that W(u)=3u 2 for all |u| >2. H-3. 0 is a smooth infinite tube periodic in x 1-direction, i.e., x=(x 1, x$) # R 1_R d&1 =R d is in 0 if and only if x+(1, 0, ..., 0) is in 0 and x$ lies in a bounded subset of R d&1. Note that H-2 is more or less of technical nature. Indeed for each W satisfying H-1 and W"(u)>0 for |u| >1, the maximum principle implies that bounded solutions of (1.1) lie between &1 and 1, so one can always modify W to satisfy H-2 without affecting bounded solutions. As a consequence of H-1 and H-2 there exists C>0 such that for all u # R1 f 2(u) CW(u).
(1.4)
Define a segment S(x 1, t), x 1 and t # R 1, of 0 by S(x 1, t) :=[( y 1, y$) # 0: | y 1 &x 1 |
(1.5)
For every u # A (defined in (1.3)) define a continuous function u^: R 1 R 1 by u^(x 1 )=
1 |S(x , 12)| 1
|
u( y) dy,
x 1 # R 1,
(1.6)
S(x 1, 12)
where |S(x 1, 12)| denotes the Lebesgue measure of S(x 1, 12) in R d. Corollary 2.2 [1] states that for every u # A, lim x 1 & u^(x 1 ) and lim x 1 u^(x 1 ) exist and equal &1 or 1. Setting
{
A '` = u # A:
lim
x 1 &
=
u^(x 1 )=`, lim u^(x 1 )=' , x1
`, ' # [&1, 1],
1 &1 1 we decompose A=A &1 &1 _ A &1 _ A 1 _ A 1 .
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(1.7)
MULTI-LAYER LOCAL MINIMUM SOLUTIONS
15
If u # A '` , then Lemma 2.3 [1] states that every v # L 1loc(0) is in A '` if and only if v&u # W 1, 2(0). Therefore each subclass A '` is a complete affine space, a translate of W 1, 2(0), where the tangent space at each point is W 1, 2(0), the distance of u, v # A '` is &u&v& W 1, 2 (0) . In A '` we use B(u, r) to denote [v # A '` : &u&v& W 1, 2 (0)
|
[{u } {,+f (u) ,] dx,
u # A '` ,
, # W 1, 2(0),
|
[{, } {+f $(u) ,] dx,
u # A '` ,
,, # W 1, 2(0).
0
E"(u)(,, )=
0
We use &E$(u)& (&E"(u)&, respectively) to denote the norm of the linear (bilinear, respectively) form E$(u) (E"(u), respectively). It is clear from H-1 and H-2 that &E"(u)& is bounded uniformly in u. u is a critical point of E if u # A and for every , # W 1, 2(0)
|
[{u } {,+f (u),] dx=0.
0
A critical point of E is a classical solution of (1.1) by the standard elliptic regularity theory. The set of all critical points in A is denoted by K. 1 The global minima &1 in A &1 &1 and 1 in A 1 are isolated critical points in the sense of Lemma 2.6 [1] which states that there exists * 0 >0 such that for every u # A %% & K, u{%, we have &u&%& W 1, 2 (0) >* 0 . Theorem 3.2 [1] asserts that in each A '` , there is a global minimum of E, i.e., there exists u # A '` such that E(u)= inf E(v). '
v # A`
M+1
(&1) to be M global We now take U 1 # A 1&1 , U 2 # A &1 1 ,..., U M # A (&1) M minima of E in their own subclasses. We say that U 1 , U 2 ,..., U M are isolated global minima if there exists + 0 >0 such that for every u # B(U i , + 0 )"[U i ] and every i=1, 2, ..., M, we have E(u)>E(U i ). The U i 's being isolated implies that the domain 0 can not be a cylinder, i.e., 0{R 1_0$ where 0$/R d&1, since in a cylinder no global minimum in is isolated due to the translational invariance. A 1&1 or A &1 1 Two important operators are defined on A. Let k be an integer, and define the shift operator { k : A '` A '` for `, ' # [&1, 1] by
{ k u(x)=u(x&(k, 0, ..., 0)),
x # 0.
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(1.8)
16
XIAOFENG REN
Define the paste operator ?: A %`_A '% A '` for `, %, ' # [&1, 1] by ?(u, v)=u+v&%. %
(1.9) %
A recursive use of (1.9) extend ? to ?: A ` 2_A %32_ } } } _A '%k A '` by ?(u 1 , u 2 , ..., u k )=?(u 1 , ?(u 2 , ?(..., ?(u k&1 , u k )))).
(1.10)
These two operators are often used together. If there is no danger of confusion, we write ? j u for ?({ j1 u 1 , { j2 u 2 , ..., { jk u k ). The main result in this paper is the following existence theorem, which improves Theorem 4.4 [1]. M+1
(&1) Theorem 1. Let U 1 # A 1&1 , U 2 # A &1 1 ,..., U M # A (&1) M global minima in their own subclasses. Then for each
be isolated
r # (0, min[+ 0 , * 0 , 2 - |S(0, 12)|]) there exists L>0 such that as long as j M &j M&1 ]>L there exists V # B(? j U, r2) with E(V)=
inf
min[ j 2 &j 1 , j 3 &j 2 , ...,
E(u),
u # B(?j U, r) (&1) i.e., there is a local minimum of E in B(? j U, r)/A &1
M+1
.
Recall that ? j U=?({ j1 U 1 , { j2 U 2 , ..., { jM U M ), + 0 measures how isolated the U i 's are, * 0 measures how isolated &1 and 1 are, and |S(0, 12)| is the Lebesgue measure of S(0, 12). Remark 2. In Theorem 4.4 [1] the U i 's are assumed isolated as critical points, while here they are merely isolated as global minima. Remark 3. Each U i can be regarded as a single layer and ? j U as a function of k layers. Since the local minimum V is close to ? j U, V is a k-layer solution.
2. PROOF OF THEOREM 1 To make the proof of Theorem 1 more readable, we assume M=2. The general case can be handled along the same lines. We use C, C 1 , C 2 ,... to denote generic constants that may vary from line to line. We often do not mention passing to a subsequence when we do so. be two isolated global minima of E, and Let U 1 # A 1&1 and U 2 # A &1 1 +0 be the radius of the balls around U 1 and U 2 in which there is no
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MULTI-LAYER LOCAL MINIMUM SOLUTIONS
17
other global minima. Take two integers j 1 and j 2 , j 1 < j 2 , and look for a local minimum of E in B(? j U, r)/A &1 &1 for some r # (0, min[+ 0 , * 0 , 2 - |S(0, 12)| ]). Here ? j U=?({ j1 U 1 , { j2 U 2 ) serves as an approximate solution. We first show that E(u) is large for all u # B(? j U, r)"B(? j U, r2). Lemma 4. Fix r # (0, min[+ 0 , * 0 , 2 - |S(0, 12)| ]). There exist L>0 and =>0 such that for every pair of integers ( j 1 , j 2 ) with j 2 &j 1 >L E(u)
E(v)+=
inf v # B(?j U, r)
for all u # B(? j U, r)"B(? j U, r2). Remark 5. L is a lower bound of the distance between the layers. In general the larger r is, the smaller L can be. We postpone the proof of Lemma 4 to next section. Take u n # B(? j U, r) such that lim E(u n )=
n
inf
E(v).
(2.11)
v # B(?j U, r)
Because of Lemma 4, we can safely assume u n # B(? j U, r2). We now show that [u n ] is a PalaisSmale sequence. Recall that a sequence [ g n ] is a PalaisSmale sequence if E(g n ) c # R 1 and &E$( g n )& 0 as n . If [u n ] is not a PalaisSmale sequence, then we can assume &E$(u n )& $>0 as n . Therefore we can find , n # W 1, 2(0) with &, n & W 1, 2 (0) =1 such that E$(u n ) , n $2. Then consider for t # (0, r2) E(u n &t, n )=E(u n )&tE$(u n ) , n +(t 22) E"(u n &t n , n )(, n , , n ) E(u n )&($2) t+Ct 2
inf
E(v)+o(1)&($2) t+Ct 2,
v # B(?j U, r)
where t n # (0, t) is guaranteed by the Taylor expansion formula and the constant C comes from the fact that E" is bounded. o(1) stands for a quantity that approaches 0 as n . Choosing n sufficient large and t sufficiently small, we deduce E(u n &t, n )<
inf
E(v),
v # B(?j U, r)
which is impossible since u n &t, n # B(? j U, r). This proves that [u n ] is a PalaisSmale sequence. We quote a characterization of PalaisSmale sequences from [1].
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18
XIAOFENG REN
Proposition 3.1 [1]. Let [u n ] be a PalaisSmale sequence in A '` , ` and ' # [&1, 1]. If lim n E(u n )=0, then A '` =A %% for some % # [&1, 1] and lim &u n &%& W 1, 2 (0) =0.
n
If lim n E(u n )>0, then there exist w 1 , w 2 , ..., w k # K"[&1, 1], k 1, % , % 1 =` and % k+1 =', and k integral sequences [l 1, n ], w i # A %i+1 i [l 2, n ],..., [l k, n ] with lim n (l i+1, n &l i, n )= for each i=1, 2, ..., k&1 such that lim &u n &?({ l1, n w 1 , { l2, n w 2 , ..., { lk, n w k )& W 1, 2 (0) =0,
n
lim E(u n )=E(w 1 )+E(w 2 )+ } } } +E(w k )
n
along a subsequence of [u n ]. Applying Proposition 3.1 [1] to u n we find k integral sequences [l 1, n ], [l 2, n ],..., [l k, n ], and k nontrivial critical points w 1 , w 2 ,...,w k such that &u n &? ln w& W 1, 2 (0) =o(1). Then &? j U&? ln w& W 1, 2 (0) r2+o(1). If one of the l i, n 's approaches & or , say l k, n , as n , then r2 &? j U&? ln w& W 1, 2 (0) +o(1) &w k +1& W 1, 2 (0) +o(1) * 0 +o(1) by Lemma 2.6 [1], which is inconsistent with the assumption on r. Therefore k=1 and l 1, n is bounded in n. We can select a proper subsequence of [l 1, n ] and shift w 1 to assume l 1, n =0. Then with the help of (2.11) &u n &w 1 & W 1, 2 (0) 0,
E(w 1 )= lim E(u n )= n
inf
E(v),
v # B(?j U, r)
and w 1 # B(? j U, r2), i.e., w 1 is a local minimum of E in B(? j U, r). The proof of Theorem 1 is complete after we set V=w 1 .
3. PROOF OF LEMMA 4 Suppose the lemma is not true. Then there exist r satisfying 0
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19
MULTI-LAYER LOCAL MINIMUM SOLUTIONS
a sequence of pairs of integers ( j 1, n , j 2, n ), with j 2, n &j 1, n , and a sequence [u n ]/B(? jn U, r)"B(? jn U, r2) such that E(u n )&
E(v)=o(1)
inf
(3.12)
v # B(?j U, r) n
as n . We can find a constant C independent of n such that E(u)
(3.13)
for all u # B(? jn U, r). To see (3.13) we estimate for each u # B(? jn U, r) |E(u)&E(? jn U)| =
}
|
|{(u&? jn U)+{? jn U | 2 & 12
|
[W(u)&W(? jn U)]
1 2
0
+
0
=
}
1 2
|
0
|{(u&? jn U)| 2 +
} {?jn U+
|
0
1 2
|
0
&{(u&? jn U)&
0
|{? jn U| 2
} {(u&? jn U )
[W(u)&W(? jn U )] 2 L 2 (0)
|
}
+&{(u&? jn U)& L 2 (0) &{?jn U& L 2 (0)
+& f (? jn U)& L 2 (0) &u&? jn U& L 2 (0) +C &u&? jn U& 2L 2 (0) C 1(&{? jn U& L 2 (0) +& f(? jn U)& L 2 (0) )&u&? jn U& W 1, 2 (0) +C2 &u&? jn U& 2W 1, 2 (0) C 1 - E(? jn U) &u&? jn U& W 1, 2 (0) +C 2 &u&? jn U& 2W 1, 2 (0) . The last inequality follows from (1.4). The last line is bounded independent of n since E(? jn U)=E(U 1 )+E(U 2 )+o(1) and &u&? jn U& W 1, 2 (0) r. This proves (3.13). If we write 0= k= & S(k, 12), we can find a sequence [m n ] of integers with lim (m n &j 1, n )= lim ( j 2, n &m n )=
n
n
(3.14)
and lim
n
|
S(mn , 1)
[ 12 |{u n | 2 +W(u n )]=0
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(3.15)
20
XIAOFENG REN
by (3.13). (3.14) and (3.15) actually imply lim
n
|
[ |{(u n &1)| 2 + |u n &1| 2 ]=0.
(3.16)
S(mn , 12)
To see (3.16) we use Lemma 2.1 [1] to obtain |u^ n(x 1 )&%| =0
sup
lim
n x 1 # (mn &12, mn +12)
for some % # [&1, 1]. In particular, u^ n(m n )&%=o(1).
(3.17)
Consider
|
[ |{(u n &%)| 2 + |u n &%| 2 ].
S(mn , 12)
From (3.15) we know
|
|{(u n &%)| 2 =o(1).
(3.18)
S(mn , 12)
About (u n &%) 2 we set G n =[x # S(m n , 12): |u n(x)&%| <$],
B n =S(m n , 12)"G n ,
where $ is so small that for all u # (%&$, %+$), c 1(u&%) 2 W(u) c 2(u&%) 2 for some positive c 1 and c 2 . The reader may think of G n as a good set and B n as a bad set. On the good set G n by (3.15) we find
|
|u n &%| 2 C
Gn
|
W(u n )=o(1).
(3.19)
Gn
On the bad set B n we note that |u n(x)&%| 2 |u n(x)&u^ n(m n )| if we choose n large enough because of (3.17). Therefore with the help of the Poincare inequality
|
Bn
|u n &%| 2 4
|
|u n &u^ n(m n )| 2 C S(mn , 12)
|
|{u n | 2 =o(1)
S(mn , 12)
by (3.15). Then (3.18)(3.20) imply
|
[ |{(u n &%)| 2 + |u n &%| 2 ]=o(1).
S(mn , 12)
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(3.20)
21
MULTI-LAYER LOCAL MINIMUM SOLUTIONS
We need to show %=1. Assume %=&1. Then it follows r 2 &u n &? jn U& 2W 1, 2 (0) =
|
|
S(mn , 12)
[ |{(u n &? jn U )| 2 + |u n &? jn U | 2 ]
(&1&1) 2 +o(1)=4 |S(0, 12)| +o(1)
S(mn , 12)
with the help of (3.14). This is inconsistent with our assumption on r, and (3.16) is proved. We then truncate u n at S(m n , 12) to define vn =
u n(x)
{ (u (x)&1) !(x&m )+1 n
n
if x S(m n , 12) if x # S(m n , 12),
(3.21)
where ! is a smooth function such that !(x)=
{
if x 1 # [&14, 14] . if x 1 [&12, 12]
0 1
It follows from (3.16) and (3.21) E(v n )&E(u n )=o(1),
and
&v n &u n & W 1, 2 (0) =o(1).
(3.22)
We set
v 2, n
v n(x) 1
{ 1 (x)= {v (x)
v 1, n(x)=
n
if x 1 m n , if x 1 >m n if x 1 m n . if x 1 >m n
Clearly v 1, n # A 1&1 , v 2, n # A &1 1 , v n =?(v 1, n , v 2, n ) and E(v n )=E(v 1, n )+E(v 2, n ).
(3.23)
From E(v 1, n ) E(U 1 ), E(v 2, n ) E(U 2 ), (3.12), (3.22) and (3.23) we deduce inf v # B(?j U, r)
E(v) E(U 1 )+E(U 2 )+o(1).
n
On the other hand by (3.14) E(? jn U )=E(U 1 )+E(U 2 )+o(1),
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(3.24)
22
XIAOFENG REN
which implies inf v # B(?j U, r)
E(v) E(U 1 )+E(U 2 )+o(1).
(3.25)
n
Combining (3.24) and (3.25) we deduce E(v)=E(U 1 )+E(U 2 )+o(1)
inf v # B(?j U, r) n
and E(v 1, n )=E(U 1 )+o(1),
E(v 2, n )=E(U 2 )+o(1).
(3.26)
We turn our attention to the distance between v 1, n and { j1, n U 1 , and the distance between v 2, n and { j2, n U 2 . Clearly &v 1, n &{ j1, n U 1 & W 1, 2 (0) +&v 2, n &{ j2, n U 2 & W 1, 2 (0) &v n &? jn U& W 1, 2 (0) r2+o(1) by the triangle inequality and (3.22). Then either &v 1, n &{ j1, n U 1 & W 1, 2 (0) r4+o(1)
(3.27)
&v 2, n &{ j2, n U 2 & W 1, 2 (0) r4+o(1).
(3.28)
or
Assume without the loss of generality that the former occurs. We look for an upper bound for &v 1, n &{ j1, n U 1 & W 1, 2 (0) . Consider, with the help of (3.14),
|
0
|{v n &{? jn U | 2
|
|{v n &{? jn U| 2
=
|
|({v 1, n &{{ j1, n U 1 )+{{ j2, n U 2 | 2
x 1
x 1
}_| & | _ = | }_
x 1
|{v 1, n &{{ j1, n U 1 | 2
&
12
12 2
x 1
x 1
|{{ j2, n U 2 | 2
& }
|{v 1, n &{{ j1, n U 1 | 2
&
12
}
2
+o(1) . (3.29)
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MULTI-LAYER LOCAL MINIMUM SOLUTIONS
Also note
|
0
|{v 1, n &{{ j1, n U 1 | 2 =
|
=
|
x 1
x 1
|{v 1, n &{{ j1, n U 1 | 2 +
|
x 1 >mn
|{{ j1, n U 1 | 2
|{v 1, n &{{ j1, n U 1 | 2 +o(1)
(3.30)
again by (3.14). Then we deduce from (3.29) and (3.30)
|
0
|{v 1, n &{{ j1, n U 1 | 2
|
0
|{v n &{? jn U | 2 +o(1).
A similar argument shows
|
0
|v 1, n &{ j1, n U 1 | 2
|
0
|v n &? jn U | 2 +o(1).
We then find, with the help of (3.22), &v 1, n &{ j1, n U 1 & W 1, 2 (0) &v n &? jn U& W 1, 2 (0) +o(1) r+o(1). (3.31) We shift v 1, n back by &j 1, n to consider { &j1, n v 1, n . From (3.27) and (3.31) we deduce r4+o(1) &{ &j1, n v 1, n &U 1 & W 1, 2 (0) r+o(1).
(3.32)
Note that (3.26) implies E(v 1, n ) E(U 1 )= inf E(v) 1
v # A &1
as n , i.e., [v 1, n ] is a global minimizing sequence of E in A 1&1 . As in the proof of Theorem 1, [v 1, n ] (as well as [{ &j1, n v 1, n ]) is a PalaisSmale sequence. Applying Proposition 3.1 [1] to [{ &j1, n v 1, n ], we find k integral sequences [l 1, n ], [l 2, n ],..., [l k, n ], and k nontrivial critical points w 1 , w 2 ,..., w k satisfying &{ &j1, n v 1, n &? ln w& W 1, 2 (0) 0 as n , which implies with the help of (3.32) r4+o(1) &U 1 &? ln w& W 1, 2 (0) r+o(1).
(3.33)
If one of the l i, n 's approaches & or as n , say l k, n , then (3.33) implies r &U 1 &? ln w& W 1, 2 (0) +o(1) &w k +1& W 1, 2 (0) +o(1) * 0 +o(1),
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XIAOFENG REN
which is again inconsistent with the assumption on r. We conclude that k=1 and l 1, n is bounded in n. By passing to a subsequence of [l 1, n ] and shifting w 1 we can assume l 1, n =0. Then we deduce with the help of (3.33) &{ &j1, n v 1, n &w 1 & W 1, 2 (0) =o(1),
r4 &U 1 &w 1 & W 1, 2 (0) r.
Since [{ &j1, n v 1, n ] is a minimizing sequence in A 1&1 , we find E(w 1 )=E(U 1 ) and r4 &U 1 &w 1 & W 1, 2 (0) r<+ 0 . This is inconsistent with the assumption that U 1 is the only minimum in B(U 1 , + 0 ). The proof of Lemma 4 is complete.
REFERENCE 1. X. Ren, A variational approach to multiple layers of the bistable equation in long tubes, Arch. Rational Mech. Anal., to appear.
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Multi-layer Local Minimum Solutions of the Bistable Equation in Noncylindrical Domains Xiaofeng Ren* Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, Minnesota 55455 Received July 9, 1996; revised March 11, 1997
We construct local minimum solutions for the semilinear bistable equation by minimizing the corresponding functional near some approximate solutions, under the hypothesis that certain global minimum solutions are isolated. The key is a certain characterization of PalaisSmale sequences and a proof that the functional takes higher values away from the approximate solutions. 1997 Academic Press
1. INTRODUCTION In this note we continue to study the problem posed in [1]. Consider the semilinear elliptic equation
{
&2u+f (u)=0
in 0
u =0 &
on 0.
(1.1)
where f is a bistable function, and 0 is an unbounded tube-shaped domain in R d. (1.1) is the EulerLagrange equation of the functional E defined by E(u)=
|
0
[ 12 |{u| 2 +W(u)] dx
(1.2)
where W(u)= u&1 f (w) dw, and W is a double well potential function of equal depth. The domain of the functional E is taken to be the class A=[u # L 1loc(0): {u # (L 2(0)) d, W(u) # L 1(0)].
(1.3)
Here L 1loc(0) is the space of measurable functions that belong to L 1(K ) for every compact subset K of 0. * Current address: Department of Mathematics and Statistics, Utah State University, Logan, UT 84322.
13 0022-039697 25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.
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14
XIAOFENG REN
We present an alternative approach to the construction of multi-layer solutions of (1.1) by minimizing E near some proper approximate solutions. The solutions being local minima of E reflects the advantage of this approach since in Theorem 4.4 [1] solutions are found through an indirect deformation argument and they are only known as critical points of (1.2). The hypothesis here that guarantees the existence of the solutions is also weaker than that in [1] (see Remark 2). The precise conditions on f, W and 0 are given as follows. H-1. W is a C 2 function that has exactly two global minima at &1 and 1, where W(&1)=W(1)=0, W"(&1)>0, and W"(1)>0. H-2. There exists 3>0 such that W(u)=3u 2 for all |u| >2. H-3. 0 is a smooth infinite tube periodic in x 1-direction, i.e., x=(x 1, x$) # R 1_R d&1 =R d is in 0 if and only if x+(1, 0, ..., 0) is in 0 and x$ lies in a bounded subset of R d&1. Note that H-2 is more or less of technical nature. Indeed for each W satisfying H-1 and W"(u)>0 for |u| >1, the maximum principle implies that bounded solutions of (1.1) lie between &1 and 1, so one can always modify W to satisfy H-2 without affecting bounded solutions. As a consequence of H-1 and H-2 there exists C>0 such that for all u # R1 f 2(u) CW(u).
(1.4)
Define a segment S(x 1, t), x 1 and t # R 1, of 0 by S(x 1, t) :=[( y 1, y$) # 0: | y 1 &x 1 |
(1.5)
For every u # A (defined in (1.3)) define a continuous function u^: R 1 R 1 by u^(x 1 )=
1 |S(x , 12)| 1
|
u( y) dy,
x 1 # R 1,
(1.6)
S(x 1, 12)
where |S(x 1, 12)| denotes the Lebesgue measure of S(x 1, 12) in R d. Corollary 2.2 [1] states that for every u # A, lim x 1 & u^(x 1 ) and lim x 1 u^(x 1 ) exist and equal &1 or 1. Setting
{
A '` = u # A:
lim
x 1 &
=
u^(x 1 )=`, lim u^(x 1 )=' , x1
`, ' # [&1, 1],
1 &1 1 we decompose A=A &1 &1 _ A &1 _ A 1 _ A 1 .
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(1.7)
MULTI-LAYER LOCAL MINIMUM SOLUTIONS
15
If u # A '` , then Lemma 2.3 [1] states that every v # L 1loc(0) is in A '` if and only if v&u # W 1, 2(0). Therefore each subclass A '` is a complete affine space, a translate of W 1, 2(0), where the tangent space at each point is W 1, 2(0), the distance of u, v # A '` is &u&v& W 1, 2 (0) . In A '` we use B(u, r) to denote [v # A '` : &u&v& W 1, 2 (0)
|
[{u } {,+f (u) ,] dx,
u # A '` ,
, # W 1, 2(0),
|
[{, } {+f $(u) ,] dx,
u # A '` ,
,, # W 1, 2(0).
0
E"(u)(,, )=
0
We use &E$(u)& (&E"(u)&, respectively) to denote the norm of the linear (bilinear, respectively) form E$(u) (E"(u), respectively). It is clear from H-1 and H-2 that &E"(u)& is bounded uniformly in u. u is a critical point of E if u # A and for every , # W 1, 2(0)
|
[{u } {,+f (u),] dx=0.
0
A critical point of E is a classical solution of (1.1) by the standard elliptic regularity theory. The set of all critical points in A is denoted by K. 1 The global minima &1 in A &1 &1 and 1 in A 1 are isolated critical points in the sense of Lemma 2.6 [1] which states that there exists * 0 >0 such that for every u # A %% & K, u{%, we have &u&%& W 1, 2 (0) >* 0 . Theorem 3.2 [1] asserts that in each A '` , there is a global minimum of E, i.e., there exists u # A '` such that E(u)= inf E(v). '
v # A`
M+1
(&1) to be M global We now take U 1 # A 1&1 , U 2 # A &1 1 ,..., U M # A (&1) M minima of E in their own subclasses. We say that U 1 , U 2 ,..., U M are isolated global minima if there exists + 0 >0 such that for every u # B(U i , + 0 )"[U i ] and every i=1, 2, ..., M, we have E(u)>E(U i ). The U i 's being isolated implies that the domain 0 can not be a cylinder, i.e., 0{R 1_0$ where 0$/R d&1, since in a cylinder no global minimum in is isolated due to the translational invariance. A 1&1 or A &1 1 Two important operators are defined on A. Let k be an integer, and define the shift operator { k : A '` A '` for `, ' # [&1, 1] by
{ k u(x)=u(x&(k, 0, ..., 0)),
x # 0.
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(1.8)
16
XIAOFENG REN
Define the paste operator ?: A %`_A '% A '` for `, %, ' # [&1, 1] by ?(u, v)=u+v&%. %
(1.9) %
A recursive use of (1.9) extend ? to ?: A ` 2_A %32_ } } } _A '%k A '` by ?(u 1 , u 2 , ..., u k )=?(u 1 , ?(u 2 , ?(..., ?(u k&1 , u k )))).
(1.10)
These two operators are often used together. If there is no danger of confusion, we write ? j u for ?({ j1 u 1 , { j2 u 2 , ..., { jk u k ). The main result in this paper is the following existence theorem, which improves Theorem 4.4 [1]. M+1
(&1) Theorem 1. Let U 1 # A 1&1 , U 2 # A &1 1 ,..., U M # A (&1) M global minima in their own subclasses. Then for each
be isolated
r # (0, min[+ 0 , * 0 , 2 - |S(0, 12)|]) there exists L>0 such that as long as j M &j M&1 ]>L there exists V # B(? j U, r2) with E(V)=
inf
min[ j 2 &j 1 , j 3 &j 2 , ...,
E(u),
u # B(?j U, r) (&1) i.e., there is a local minimum of E in B(? j U, r)/A &1
M+1
.
Recall that ? j U=?({ j1 U 1 , { j2 U 2 , ..., { jM U M ), + 0 measures how isolated the U i 's are, * 0 measures how isolated &1 and 1 are, and |S(0, 12)| is the Lebesgue measure of S(0, 12). Remark 2. In Theorem 4.4 [1] the U i 's are assumed isolated as critical points, while here they are merely isolated as global minima. Remark 3. Each U i can be regarded as a single layer and ? j U as a function of k layers. Since the local minimum V is close to ? j U, V is a k-layer solution.
2. PROOF OF THEOREM 1 To make the proof of Theorem 1 more readable, we assume M=2. The general case can be handled along the same lines. We use C, C 1 , C 2 ,... to denote generic constants that may vary from line to line. We often do not mention passing to a subsequence when we do so. be two isolated global minima of E, and Let U 1 # A 1&1 and U 2 # A &1 1 +0 be the radius of the balls around U 1 and U 2 in which there is no
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MULTI-LAYER LOCAL MINIMUM SOLUTIONS
17
other global minima. Take two integers j 1 and j 2 , j 1 < j 2 , and look for a local minimum of E in B(? j U, r)/A &1 &1 for some r # (0, min[+ 0 , * 0 , 2 - |S(0, 12)| ]). Here ? j U=?({ j1 U 1 , { j2 U 2 ) serves as an approximate solution. We first show that E(u) is large for all u # B(? j U, r)"B(? j U, r2). Lemma 4. Fix r # (0, min[+ 0 , * 0 , 2 - |S(0, 12)| ]). There exist L>0 and =>0 such that for every pair of integers ( j 1 , j 2 ) with j 2 &j 1 >L E(u)
E(v)+=
inf v # B(?j U, r)
for all u # B(? j U, r)"B(? j U, r2). Remark 5. L is a lower bound of the distance between the layers. In general the larger r is, the smaller L can be. We postpone the proof of Lemma 4 to next section. Take u n # B(? j U, r) such that lim E(u n )=
n
inf
E(v).
(2.11)
v # B(?j U, r)
Because of Lemma 4, we can safely assume u n # B(? j U, r2). We now show that [u n ] is a PalaisSmale sequence. Recall that a sequence [ g n ] is a PalaisSmale sequence if E(g n ) c # R 1 and &E$( g n )& 0 as n . If [u n ] is not a PalaisSmale sequence, then we can assume &E$(u n )& $>0 as n . Therefore we can find , n # W 1, 2(0) with &, n & W 1, 2 (0) =1 such that E$(u n ) , n $2. Then consider for t # (0, r2) E(u n &t, n )=E(u n )&tE$(u n ) , n +(t 22) E"(u n &t n , n )(, n , , n ) E(u n )&($2) t+Ct 2
inf
E(v)+o(1)&($2) t+Ct 2,
v # B(?j U, r)
where t n # (0, t) is guaranteed by the Taylor expansion formula and the constant C comes from the fact that E" is bounded. o(1) stands for a quantity that approaches 0 as n . Choosing n sufficient large and t sufficiently small, we deduce E(u n &t, n )<
inf
E(v),
v # B(?j U, r)
which is impossible since u n &t, n # B(? j U, r). This proves that [u n ] is a PalaisSmale sequence. We quote a characterization of PalaisSmale sequences from [1].
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18
XIAOFENG REN
Proposition 3.1 [1]. Let [u n ] be a PalaisSmale sequence in A '` , ` and ' # [&1, 1]. If lim n E(u n )=0, then A '` =A %% for some % # [&1, 1] and lim &u n &%& W 1, 2 (0) =0.
n
If lim n E(u n )>0, then there exist w 1 , w 2 , ..., w k # K"[&1, 1], k 1, % , % 1 =` and % k+1 =', and k integral sequences [l 1, n ], w i # A %i+1 i [l 2, n ],..., [l k, n ] with lim n (l i+1, n &l i, n )= for each i=1, 2, ..., k&1 such that lim &u n &?({ l1, n w 1 , { l2, n w 2 , ..., { lk, n w k )& W 1, 2 (0) =0,
n
lim E(u n )=E(w 1 )+E(w 2 )+ } } } +E(w k )
n
along a subsequence of [u n ]. Applying Proposition 3.1 [1] to u n we find k integral sequences [l 1, n ], [l 2, n ],..., [l k, n ], and k nontrivial critical points w 1 , w 2 ,...,w k such that &u n &? ln w& W 1, 2 (0) =o(1). Then &? j U&? ln w& W 1, 2 (0) r2+o(1). If one of the l i, n 's approaches & or , say l k, n , as n , then r2 &? j U&? ln w& W 1, 2 (0) +o(1) &w k +1& W 1, 2 (0) +o(1) * 0 +o(1) by Lemma 2.6 [1], which is inconsistent with the assumption on r. Therefore k=1 and l 1, n is bounded in n. We can select a proper subsequence of [l 1, n ] and shift w 1 to assume l 1, n =0. Then with the help of (2.11) &u n &w 1 & W 1, 2 (0) 0,
E(w 1 )= lim E(u n )= n
inf
E(v),
v # B(?j U, r)
and w 1 # B(? j U, r2), i.e., w 1 is a local minimum of E in B(? j U, r). The proof of Theorem 1 is complete after we set V=w 1 .
3. PROOF OF LEMMA 4 Suppose the lemma is not true. Then there exist r satisfying 0
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19
MULTI-LAYER LOCAL MINIMUM SOLUTIONS
a sequence of pairs of integers ( j 1, n , j 2, n ), with j 2, n &j 1, n , and a sequence [u n ]/B(? jn U, r)"B(? jn U, r2) such that E(u n )&
E(v)=o(1)
inf
(3.12)
v # B(?j U, r) n
as n . We can find a constant C independent of n such that E(u)
(3.13)
for all u # B(? jn U, r). To see (3.13) we estimate for each u # B(? jn U, r) |E(u)&E(? jn U)| =
}
|
|{(u&? jn U)+{? jn U | 2 & 12
|
[W(u)&W(? jn U)]
1 2
0
+
0
=
}
1 2
|
0
|{(u&? jn U)| 2 +
} {?jn U+
|
0
1 2
|
0
&{(u&? jn U)&
0
|{? jn U| 2
} {(u&? jn U )
[W(u)&W(? jn U )] 2 L 2 (0)
|
}
+&{(u&? jn U)& L 2 (0) &{?jn U& L 2 (0)
+& f (? jn U)& L 2 (0) &u&? jn U& L 2 (0) +C &u&? jn U& 2L 2 (0) C 1(&{? jn U& L 2 (0) +& f(? jn U)& L 2 (0) )&u&? jn U& W 1, 2 (0) +C2 &u&? jn U& 2W 1, 2 (0) C 1 - E(? jn U) &u&? jn U& W 1, 2 (0) +C 2 &u&? jn U& 2W 1, 2 (0) . The last inequality follows from (1.4). The last line is bounded independent of n since E(? jn U)=E(U 1 )+E(U 2 )+o(1) and &u&? jn U& W 1, 2 (0) r. This proves (3.13). If we write 0= k= & S(k, 12), we can find a sequence [m n ] of integers with lim (m n &j 1, n )= lim ( j 2, n &m n )=
n
n
(3.14)
and lim
n
|
S(mn , 1)
[ 12 |{u n | 2 +W(u n )]=0
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(3.15)
20
XIAOFENG REN
by (3.13). (3.14) and (3.15) actually imply lim
n
|
[ |{(u n &1)| 2 + |u n &1| 2 ]=0.
(3.16)
S(mn , 12)
To see (3.16) we use Lemma 2.1 [1] to obtain |u^ n(x 1 )&%| =0
sup
lim
n x 1 # (mn &12, mn +12)
for some % # [&1, 1]. In particular, u^ n(m n )&%=o(1).
(3.17)
Consider
|
[ |{(u n &%)| 2 + |u n &%| 2 ].
S(mn , 12)
From (3.15) we know
|
|{(u n &%)| 2 =o(1).
(3.18)
S(mn , 12)
About (u n &%) 2 we set G n =[x # S(m n , 12): |u n(x)&%| <$],
B n =S(m n , 12)"G n ,
where $ is so small that for all u # (%&$, %+$), c 1(u&%) 2 W(u) c 2(u&%) 2 for some positive c 1 and c 2 . The reader may think of G n as a good set and B n as a bad set. On the good set G n by (3.15) we find
|
|u n &%| 2 C
Gn
|
W(u n )=o(1).
(3.19)
Gn
On the bad set B n we note that |u n(x)&%| 2 |u n(x)&u^ n(m n )| if we choose n large enough because of (3.17). Therefore with the help of the Poincare inequality
|
Bn
|u n &%| 2 4
|
|u n &u^ n(m n )| 2 C S(mn , 12)
|
|{u n | 2 =o(1)
S(mn , 12)
by (3.15). Then (3.18)(3.20) imply
|
[ |{(u n &%)| 2 + |u n &%| 2 ]=o(1).
S(mn , 12)
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(3.20)
21
MULTI-LAYER LOCAL MINIMUM SOLUTIONS
We need to show %=1. Assume %=&1. Then it follows r 2 &u n &? jn U& 2W 1, 2 (0) =
|
|
S(mn , 12)
[ |{(u n &? jn U )| 2 + |u n &? jn U | 2 ]
(&1&1) 2 +o(1)=4 |S(0, 12)| +o(1)
S(mn , 12)
with the help of (3.14). This is inconsistent with our assumption on r, and (3.16) is proved. We then truncate u n at S(m n , 12) to define vn =
u n(x)
{ (u (x)&1) !(x&m )+1 n
n
if x S(m n , 12) if x # S(m n , 12),
(3.21)
where ! is a smooth function such that !(x)=
{
if x 1 # [&14, 14] . if x 1 [&12, 12]
0 1
It follows from (3.16) and (3.21) E(v n )&E(u n )=o(1),
and
&v n &u n & W 1, 2 (0) =o(1).
(3.22)
We set
v 2, n
v n(x) 1
{ 1 (x)= {v (x)
v 1, n(x)=
n
if x 1 m n , if x 1 >m n if x 1 m n . if x 1 >m n
Clearly v 1, n # A 1&1 , v 2, n # A &1 1 , v n =?(v 1, n , v 2, n ) and E(v n )=E(v 1, n )+E(v 2, n ).
(3.23)
From E(v 1, n ) E(U 1 ), E(v 2, n ) E(U 2 ), (3.12), (3.22) and (3.23) we deduce inf v # B(?j U, r)
E(v) E(U 1 )+E(U 2 )+o(1).
n
On the other hand by (3.14) E(? jn U )=E(U 1 )+E(U 2 )+o(1),
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(3.24)
22
XIAOFENG REN
which implies inf v # B(?j U, r)
E(v) E(U 1 )+E(U 2 )+o(1).
(3.25)
n
Combining (3.24) and (3.25) we deduce E(v)=E(U 1 )+E(U 2 )+o(1)
inf v # B(?j U, r) n
and E(v 1, n )=E(U 1 )+o(1),
E(v 2, n )=E(U 2 )+o(1).
(3.26)
We turn our attention to the distance between v 1, n and { j1, n U 1 , and the distance between v 2, n and { j2, n U 2 . Clearly &v 1, n &{ j1, n U 1 & W 1, 2 (0) +&v 2, n &{ j2, n U 2 & W 1, 2 (0) &v n &? jn U& W 1, 2 (0) r2+o(1) by the triangle inequality and (3.22). Then either &v 1, n &{ j1, n U 1 & W 1, 2 (0) r4+o(1)
(3.27)
&v 2, n &{ j2, n U 2 & W 1, 2 (0) r4+o(1).
(3.28)
or
Assume without the loss of generality that the former occurs. We look for an upper bound for &v 1, n &{ j1, n U 1 & W 1, 2 (0) . Consider, with the help of (3.14),
|
0
|{v n &{? jn U | 2
|
|{v n &{? jn U| 2
=
|
|({v 1, n &{{ j1, n U 1 )+{{ j2, n U 2 | 2
x 1
x 1
}_| & | _ = | }_
x 1
|{v 1, n &{{ j1, n U 1 | 2
&
12
12 2
x 1
x 1
|{{ j2, n U 2 | 2
& }
|{v 1, n &{{ j1, n U 1 | 2
&
12
}
2
+o(1) . (3.29)
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23
MULTI-LAYER LOCAL MINIMUM SOLUTIONS
Also note
|
0
|{v 1, n &{{ j1, n U 1 | 2 =
|
=
|
x 1
x 1
|{v 1, n &{{ j1, n U 1 | 2 +
|
x 1 >mn
|{{ j1, n U 1 | 2
|{v 1, n &{{ j1, n U 1 | 2 +o(1)
(3.30)
again by (3.14). Then we deduce from (3.29) and (3.30)
|
0
|{v 1, n &{{ j1, n U 1 | 2
|
0
|{v n &{? jn U | 2 +o(1).
A similar argument shows
|
0
|v 1, n &{ j1, n U 1 | 2
|
0
|v n &? jn U | 2 +o(1).
We then find, with the help of (3.22), &v 1, n &{ j1, n U 1 & W 1, 2 (0) &v n &? jn U& W 1, 2 (0) +o(1) r+o(1). (3.31) We shift v 1, n back by &j 1, n to consider { &j1, n v 1, n . From (3.27) and (3.31) we deduce r4+o(1) &{ &j1, n v 1, n &U 1 & W 1, 2 (0) r+o(1).
(3.32)
Note that (3.26) implies E(v 1, n ) E(U 1 )= inf E(v) 1
v # A &1
as n , i.e., [v 1, n ] is a global minimizing sequence of E in A 1&1 . As in the proof of Theorem 1, [v 1, n ] (as well as [{ &j1, n v 1, n ]) is a PalaisSmale sequence. Applying Proposition 3.1 [1] to [{ &j1, n v 1, n ], we find k integral sequences [l 1, n ], [l 2, n ],..., [l k, n ], and k nontrivial critical points w 1 , w 2 ,..., w k satisfying &{ &j1, n v 1, n &? ln w& W 1, 2 (0) 0 as n , which implies with the help of (3.32) r4+o(1) &U 1 &? ln w& W 1, 2 (0) r+o(1).
(3.33)
If one of the l i, n 's approaches & or as n , say l k, n , then (3.33) implies r &U 1 &? ln w& W 1, 2 (0) +o(1) &w k +1& W 1, 2 (0) +o(1) * 0 +o(1),
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24
XIAOFENG REN
which is again inconsistent with the assumption on r. We conclude that k=1 and l 1, n is bounded in n. By passing to a subsequence of [l 1, n ] and shifting w 1 we can assume l 1, n =0. Then we deduce with the help of (3.33) &{ &j1, n v 1, n &w 1 & W 1, 2 (0) =o(1),
r4 &U 1 &w 1 & W 1, 2 (0) r.
Since [{ &j1, n v 1, n ] is a minimizing sequence in A 1&1 , we find E(w 1 )=E(U 1 ) and r4 &U 1 &w 1 & W 1, 2 (0) r<+ 0 . This is inconsistent with the assumption that U 1 is the only minimum in B(U 1 , + 0 ). The proof of Lemma 4 is complete.
REFERENCE 1. X. Ren, A variational approach to multiple layers of the bistable equation in long tubes, Arch. Rational Mech. Anal., to appear.
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