XIAOFENG REN. equation by minimizing the corresponding functional near some approximate

Transcription

1 MULTI-LAYER LOCAL MINIMUM SOLUTIONS OF THE BISTABLE EQUATION IN AN INFINITE TUBE XIAOFENG REN Abstract. We construct local minimum solutions of the semilinear bistable equation by minimizing the corresponding functional near some approximate solutions, assuming that some global minimum solutions are isolated. The key step is to prove with the help of a characterization of Palais-Smale sequences that the functional takes higher values away from the approximate solutions than it does near the approximate solutions.. Introduction In this note we continue to study the problem posed in []. Consider the semilinear elliptic equation ( u + f(u) =0on@ where f is a bistable function, and is an unbounded tube-shaped domain in R d. () is the Euler-Lagrange equation of the functional E dened by (2) E(u) = [ 2 jruj2 + W (u)]dx where W (u) = u 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 (3) A = fu 2 L loc() : ru 2 (L 2 ()) d ; W(u)2 L ()g: Here L loc () is the space of measurable functions that belong to L (K) for every compact subset K of. We present an alternative approach to the construction of multi-layer solutions of () by minimizing E near some proper approximate solutions. The solutions being local minima of Ereects the advantage of this approach since in Theorem 4.4 [] solutions are found through an indirect deformation argument and they are only known as critical points of (2). The hypothesis here that guarantees the existence of the solutions is also weaker than that in [] (see Remark 2). The precise conditions on f, W and are given as follows. 99 Mathematics Subject Classication. Primary 35J20; Secondary 35J60.

2 2 XIAOFENG REN H-: W is a C 2 function that has exactly two global minimaat and where W ( ) = W () = 0, W 00 ( ) > 0, and W 00 () > 0. H-2: There exists > 0 such that W (u) =u 2 for all juj > 2. H-3: is a smooth innite tube periodic in x -direction, i.e., x =(x ;x 0 )2 R R d = R d is in if and only if x +(;0; :::; 0) is in and x 0 lies in a bounded subset of R d. Note that H-2 is more or less of technical nature. Indeed for each W satisfying H- and W 00 (u) > 0 for juj >, the maximum principle implies that bounded solutions of () lie between and, so one can always modify W to satisfy H-2 without aecting bounded solutions. As a consequence of H- and H-2 there exists C>0such that for all u 2 R (4) (5) f 2 (u) CW(u): Dene a segment S(x ;t), x and t 2 R,ofby S(x ;t):=f(y ;y 0 )2: jy x j<tg: For every u 2A(dened in (3)) dene a continuous function bu : R! R by (6) bu(x )= u(y)dy; x 2 R js(x ;=2)j S(x ;=2) where js(x ; =2)j denotes the Lebesgue measure of S(x ; =2) in R d. Corollary 2.2 [] states that for every u 2A, x! bu(x ) and x! bu(x ) exist and equal or. Setting (7) A = fu 2A: x! bu(x )=; x! bu(x )=g; ;2f ;g; we decompose A = A [A [A [A. If u 2A, then Lemma 2.3 [] states that every v 2 L loc () is in A if and only if v u 2 W ;2 (). Therefore each subclass A is a complete ane space, a translate of W ;2 (), where the tangent space at each pointisw ;2 (), the distance of u; v 2 A is ku vk W ;2 (). InA we use B(u; r) to denote fv 2A :ku vk W ;2 () <rg. Lemma 2.5 [] states that E : A! R belongs to C 2 (A ;R ) where E 0 (u) = [ru r+f(u)]dx; u 2A ;2W;2 (); E 00 (u)(; ) = [rr +f 0 (u) ]dx; u 2A ;; 2 W;2 (): We use ke 0 (u)k (ke 00 (u)k, respectively) to denote the norm of the liner (bilinear, respectively) form E 0 (u) (E 00 (u), respectively). It is clear from H- and H-2 that ke 00 (u)k is bounded uniformly in u. u is a critical point ofeif u 2Aand for every 2 W ;2 () [ru r+f(u)]dx =0: A critical point of Eis a classical solution of () by the standard elliptic regularity theory. The set of all critical points in A is denoted K.

3 MULTI-LAYER LOCAL MINIMUM SOLUTIONS 3 The global minima ina and in A are isolated critical points in the sense of Lemma 2.6 [] which states that there exists 0 > 0 such that for every u 2A \K,u6=,wehaveku k W ;2 () > 0. Theorem 3.2 [] asserts that in each A, there is a global minimum of E, i.e., there exists u 2A such that E(u) = inf v2a E(v): Wenow take U 2A,U 2 2A,..., U M 2A ( )M+ ( ) M to be M global minimaof E in their own subclasses. We say that U, U 2,...,U M are isolated global minimaif there exists 0 > 0 such that for every u 2B(U i ; 0 )nfu i g and every i =;2; :::; M, we havee(u)>e(u i ). The U i 's being isolated implies that the domain can not be a cylinder, i.e., 6= R 0 where 0 R d, since in a cylinder no global minimum in A or A is isolated due to the translational invariance. Two important operators are dened on A. Let k be an integer, and dene the shift operator k : A!A for ; 2f ;gby (8) k u(x)=u(x (k; 0; :::; 0)); x2: Dene the paste operator : A A!A for ;; 2f ;gby (9) (u; v) =u+v : A recursive use of (9) extend to : A 2 (0) A3 2 ::: A k!a by (u ;u 2 ; :::; u k )=(u ;(u 2 ;(:::; (u k ;u k )))): These two operators are often used together. If there is no danger of confusion, we write j u for ( j u ; j2 u 2 ; :::; jk u k ). The main result in this paper is the following existence theorem, which improves Theorem 4.4 []. Theorem. Let U 2A,U 22A,...,U M 2A ( )M+ ( ) M be isolated global minima in their own subclasses. Then for each r 2 (0; minf 0 ; 0 ;2 p js(0; =2)jg) there exists L>0such that as long as minfj 2 j ;j 3 j 2 ; :::; j M j M g >Lthere exists V 2B( j U; r=2) with E(V )= inf E(u); u2b( ju;r) i.e., there isalocal minimum of E in B( j U; r) A ( )M+. Recall that j U = ( j U ; j2 U 2 ; :::; jm U M ), 0 measures how isolated the U i 's are, 0 measures how isolated and are, and js(0; =2)j is the Lebesgue measure of S(0; =2). Remark 2. In Theorem 4.4 [] 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.

4 4 XIAOFENG REN 2. Proof of Theorem To make the proof of Theorem more readable, we assume M = 2. The general case can be handled along the same line. We use C, C, 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. Let U 2 A and U 2 2 A be two isolated global minima ofe, and 0 be the radius of the balls around U and U 2 in which there is no other global minima. Take twointegers j and j 2, j <j 2, and look for a local minimum of E in B( j U; r) A for some r 2 (0; minf 0; 0 ;2 p js(0; =2)jg): Here j U = ( j U ; j2 U 2 ) serves as an approximate solution. We rst show that E(u) is large for all u 2B( j U; r)nb( j U; r=2). Lemma 4. Fix r 2 (0; minf 0 ; 0 ;2 p js(0; =2)jg). There exist L>0and >0 such that for every pair of integers (j ;j 2 ) with j 2 j >L E(u) for all u 2B( j U; r)nb( j U; r=2). inf E(v)+ 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 2B( j U; r) such that () n! E(u n)= inf E(v): Because of Lemma 4, we can safely assume u n 2B( j U; r=2). We now show that fu n g isapalais-smale sequence. Recall that a sequence fg n g is a Palais-Smale sequence if E(g n )! c 2 R and ke 0 (g n )k!0asn!. If fu n g is not a Palais-Smale sequence, then we can assume ke 0 (u n )k!>0as n!. Find n 2 W ;2 () with k n k W ;2 () = such that E 0 (u n ) n =2. Then consider for t 2 (0;r=2) E(u n t n )=E(u n ) te 0 (u n ) n +(t 2 =2)E 00 (u n t n n )( n ; n ) E(u n ) (=2)t + Ct 2 inf E(v) +o() (=2)t + Ct 2 where t n 2 (0;t) is guaranteed by the Taylor expansion formula and the constant C comes from the fact that E 00 is bounded. o() stands for a quantity that approaches 0 as n!. Choosing n sucient large and t suciently small, we deduce E(u n t n ) < inf E(v) which is impossible since u n t n 2B( j U; r). This proves that fu n g isapalais- Smale sequence. We quote a characterization of Palais-Smale sequences from [].

5 MULTI-LAYER LOCAL MINIMUM SOLUTIONS 5 Proposition 3. []. Let fu n g be a Palais-Smale sequence ina, and 2 f ; g. If n! E(u n)=0, then A = A for some 2f ;gand If n! ku n k W ;2 () =0: n! E(u n) > 0, then there exist w ;w 2 ; :::; w k 2 Knf ; g, k, w i 2 A i+ i, = and k+ =, and k integral sequences fl ;n g, fl 2;n g,...,fl k;n g with n! (l i+;n l i;n )=for each i =;2; :::; k such that n! ku n ( l;n w ; l2;n w 2 ; :::; lk;n w k )k W ;2 () =0; along a subsequence offu n g. n! E(u n)=e(w )+E(w 2 )+::: + E(w k ) Applying Proposition 3. [] tou n we nd k integral sequences fl ;n g, fl 2;n g,..., fl k;n g, and k nontrivial critical points w, w 2,...,w k such that ku n ln wk W ;2 () = o(). Then k j U ln wk W ;2 () r=2+o(). If one of the l i;n 's approaches or, sayl k;n!,asn!, then r=2 k j U ln wk W ;2 () + o() kw k +k W ;2 () + o() 0 + o() by Lemma 2.6 [], which is inconsistent with the assumption on r. Therefore k = and l ;n is bounded in n. We can select a proper subsequence of fl ;n g and shift w to assume l ;n = 0. Then with the help of () ku n w k W ;2 ()! 0; E(w ) = n! E(u n)= inf E(v); and w 2B( j U; r=2), i.e., w is a local minimum of E in B( j U; r). The proof of Theorem is complete after we set V = w. 3. Proof of Lemma 4 Suppose the lemma is not true. Then there exist r satisfying 0 <r<minf 0 ; 0 ;2 p js(0; =2)jg; a sequence of pairs of integers (j ;n ;j 2;n ), with j 2;n fu n gb( jn U; r)nb( jn U; r=2) such that (2) E(u n ) inf E(v) =o() v2b( jn U;r) as n!. We can nd a constant C independent ofnsuch that (3) E(u) <C j ;n!, and a sequence for all u 2B( jn U; r). To see (3) we estimate for each u 2B( jn U; r) = j 2 je(u) E( jn U)j jr(u jn U)+r jn Uj 2 jr jn Uj [W (u) W ( jn U)]j

6 6 XIAOFENG REN = j 2 jr(u jn U)j 2 + r(u jn U) r jn U+ [W(u) W( jn U)]j 2 kr(u j n U)k 2 L 2 () + kr(u j n U)k L 2 ()kr jn Uk L 2 () +kf( jn U)k L 2 ()ku jn Uk L 2 () + Cku jn Uk 2 L 2 () C (kr jn Uk L 2 () + kf( jn U)k L 2 ())ku jn Uk W ;2 () + C 2 ku jn Uk 2 W ;2 () C qe( jn U)ku jn Uk W ;2 () + C 2 ku jn Uk 2 W ;2 () : The last inequality follows from (4). The last line is bounded independent ofn since E( jn U)=E(U )+E(U 2 )+o() and ku jn Uk W ;2 () r. This proves (3). If we write = [ k= S(k; =2), we can nd a sequence fm ng of integers with (4) and (5) by (3). (4) and (5) actually imply (6) n! (m n j ;n ) = n! (j 2;n m n )= n! [ n! S(m n;) 2 jru nj 2 + W (u n )] = 0 To see (6) we use Lemma 2. [] to obtain for some 2f (7) Consider From (5) we know (8) About (u n n! [jr(u n )j 2 + ju n j 2 ]=0: S(m n;=2) sup jbu n (x ) j =0 x 2(m n =2;mn+=2) ;g. In particular, ) 2 we set bu n (m n ) = o(): [jr(u n )j 2 + ju n j 2 ]: S(m n;=2) jr(u n S(m n;=2) G n = fx 2 S(m n ; =2) : ju n (x) )j 2 = o(): j <g; B n = S(m n ;=2)nG n where is so small that for all u 2 ( ;+), c (u ) 2 W (u) c 2 (u ) 2 for some positive c and c 2. The reader may think G n as a good set and B n as a bad set. On the good set G n by (5) we nd (9) ju n j 2 C W (u n )=o(): G n G n

7 MULTI-LAYER LOCAL MINIMUM SOLUTIONS 7 On the bad set B n we note that ju n (x) j 2ju n (x) bu n (m n )jif we choose n large enough because of (7). Therefore with the help of the Poincare inequality (20) ju n j 2 4 ju n bu n (m n )j 2 C jru n j 2 = o() B n S(m n;=2) S(m n;=2) by (5). Then (8), (9) and (20) imply [jr(u n )j 2 + ju n j 2 ]=o(): S(m n;=2) We need to show =. Assume =. Then it follows r 2 ku n jn Uk 2 W ;2 () [jr(u n jn U)j 2 + ju n jn Uj 2 ] S(m n;=2) = ( ) 2 + o()=4js(0; =2)j + o() S(m n;=2) with the help of (4). This is inconsistent with our assumption on r, and (6) is proved. We then truncate u n at S(m n ; =2) to dene (2) v n = un (x) if x 62 S(m n ; =2) (u n (x) )(x m n )+ if x 2 S(m n ; =2) where is a smooth function such that 0 if x 2 [ =4; =4] (x) = if x 62 [ =2; =2] : It follows from (6) and (2) (22) We set E(v n ) E(u n )=o(); and kv n u n k W ;2 () = o(): v ;n (x) = vn (x) if x m n if x m if x ; v >m 2;n (x) = n n v n (x) if x : >m n Clearly v ;n 2A,v 2;n 2A,v n=(v ;n ;v 2;n ) and (23) E(v n )=E(v ;n )+E(v 2;n ): From E(v ;n ) E(U ), E(v 2;n ) E(U 2 ), (2), (22) and (23) we deduce (24) On the other hand by (4) which implies inf E(v) E(U )+E(U 2 )+o(): v2b( jn U;r) E( jn U)=E(U )+E(U 2 )+o(); (25) inf E(v) E(U )+E(U 2 )+o(): v2b( jn U;r)

8 8 XIAOFENG REN Combing (24) and (25) we deduce and (26) inf E(v) =E(U )+E(U 2 )+o() v2b( jn U;r) E(v ;n )=E(U )+o(); E(v 2;n )=E(U 2 )+o(): We turn our attention to the distance between v ;n and j;n U, and the distance between v 2;n and j2;n U 2. Clearly kv ;n j;n U k W ;2 () + kv 2;n j2;n U 2 k W ;2 () kv n jn Uk W ;2 () r=2+o() by the triangle inequality and (22). Then either (27) or (28) kv ;n kv 2;n j;n U k W ;2 () r=4+o() j2;n U 2 k W ;2 () r=4+o(): Assume without the loss of generality that the former occurs. We look for an upper bound for kv ;n j;n U k W ;2 (). Consider, with the help of (4), jrv n r jn Uj 2 jrv n r jn Uj 2 x <m n = j(rv ;n r j;n U )+r j2;n U 2 j 2 x <m n j[ jrv ;n x <m n r j;n U j 2 ] =2 [ jr j2;n U 2 j 2 ] =2 j 2 x <m n (29) = j[ jrv ;n x <m n r j;n U j 2 ] =2 + o()j 2 : Also note jrv ;n r j;n U j 2 = jrv ;n r j;n U j 2 + jr j;n U j 2 x <m n x >m n (30) = x <m n jrv ;n r j;n U j 2 + o() again by (4). Then we deduce from (29) and (30) jrv ;n r j;n U j 2 jrv n r jn Uj 2 +o(): A similar argument shows jv ;n j;n U j 2 We then nd, with the help of (22), (3) jv n jn Uj 2 + o(): kv ;n j;n U k W ;2 () kv n jn Uk W ;2 () + o() r + o():

9 We shift v ;n back by (32) Note that (26) implies MULTI-LAYER LOCAL MINIMUM SOLUTIONS 9 j ;n to consider j;n v ;n.from (27) and (3) we deduce r=4+o() k j;n v ;n E(v ;n )! E(U )= U k W ;2 () r + o(): inf E(v) v2a as n!, i.e., fv ;n g is a global minimizing sequence of E in A. Asinthe proof of Theorem, fv ;n g (as well as f j;n v ;n g)isapalais-smale sequence. Applying Proposition 3. [] tof j;n v ;n g,we nd k integral sequences fl ;n g, fl 2;n g,...,fl k;n g, and k nontrivial critical points w, w 2,...,w k satisfying k j;n v ;n ln wk W ;2 ()! 0asn!, which implies with the help of (32) (33) r=4+o() ku If one of the l i;n 's approaches r ku ln wk W ;2 () r + o(): or as n!,sayl k;n!, then (33) implies ln wk W ;2 () + o() kw k +k W ;2 () + o() 0 + o(); which is again inconsistent with the assumption on r. We conclude that k = and l ;n is bounded in n. By passing to a subsequence of fl ;n g and shifting w we can assume l ;n = 0. Then we deduce with the help of (33) k j;n v ;n w k W ;2 () = o(); r=4ku w k W ;2 () r: Since f j;n v ;n g is a minimizing sequence in A,we nd E(w )=E(U ) and r=4 ku w k W ;2 () r< 0. This is inconsistent with the assumption that U is the only minimum in B(U ; 0 ). The proof of Lemma 4 is complete. References. X. Ren, Variational approach to multiple layers of the bistable equation in long tubes, Arch. Rational Mech. Anal., to appear. Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN address: ren@ima.umn.edu