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- Title: 1520
- Type: Notes
- School: Minnesota
- Course: IMA 97
- Term: Fall
for Asymptotics the biharmonic equation near the tip of a crack Avner Friedman Bei Huy Juan J. L. Velazquezz 1 Introduction A mathematical model of a crack in a 2-dimensional uniform elastic medium occupying a bounded domain consists of the following system (see [9]): ' 2 H 2( ); (1.1) 2' = 0 in n (t); (1.2) ' = @' = 0 from both sides of (t); (1.3) @n ' = g; @' = h on @ ; ; (1.4) @n where ' = '(x) = '(x1; x2) is the stress function. Here is the crack which, for simplicity, we shall take to be a curve of the form x2 = f(x1); 1 6 x1 6 0 (1.5) contained in except for its initial point ( 1; f( 1)), which lies on @ ; we shall also assume, for simplicity, that f(0) = 0; f 0(0) = 0: (1.6) We are interested in the behavior of ' near the origin. From basic work by Kondrat v and Oleinik [14] [15] it follows that if f is in C 1[ 0; 0] e for some 0 > 0, then ' 2 C 3=2 near the origin; (1.7) and j'(x)j 6 Cjxj3=2; (1.8) 1=2: jr'(x)j 6 Cjxj (1.9) IMA, University of Minnesota, Minneapolis, MN 55455 Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 z Departamento de Matematica Aplicada, Falcultad de Matematicas, Universidad Complutense, 28040 Madrid, Spain y 1 On the other hand if f 2 C 1[ 0; 0] then, by [5] (and some of the papers referenced therein), ' has an asymptotic expansion near the origin. In the special case where f(x1) 0 for 1 X h 0 6 x1 6 0 the expansion is given explicitly [22] (see also [6; x16] [8; Part II, Chap. 7]) by k + 1 + b cos k 1 '(r; ) = ak cos 2 k 2 k=1 i +ck sin k 1 + dk sin k + 1 ; 2 2 rk=2+1 where 2 Z (1.10) k + 1 d for k > 2; ( 0; ) cos 2 0bk+2 + ak = k=2+1 0 k + 2 b + k + 1 a = 2 Z ( ; ) cos k + 1 d for k > 2 0 2 k+2 k r0 k=2 2 2 0 (the formulae for b1; b2; b3; a1 are little di erent) and similar relations hold for ck , dk . Hence we get C Z fj ( ; )j + j ( ; )jgd ; jak j + jbk j 6 k 0 r0 0 and the same inequality holds for jck j + jdk j. It follows that the series (1.10) is uniformly convergent for 0 6 r 6 0, for any < 1. From (1.10) we get '(r; ) = A1r3=2B1( ) + A2r3=2B2( ) + A3r2B3( ) +A4r5=2B4( ) + A5r5=2B5( ) + O(r3 ); where (1.11) B1( ) = cos 3 + 3 cos 1 ; B2( ) = sin 3 + sin 1 ; B3( ) = sin2 ; 2 2 2 2 (1.12) 5 5 cos 1 ; B ( ) = sin 5 sin 1 ; B4( ) = cos 2 5 2 2 2 note that r2B3( ) = x2. 2 The main purpose of this paper is to establish asymptotic expansions of up to order r3 , under very weak assumptions on the regularity of f(x1). Our results are: (i) If f 2 C 1+ [ 0; 0], then '(x) = A1r3=2B1( ) + A2r3=2B2( ) + O(r2 ) for any > 0 such that + > 1=2. 2 (1.13) (ii) If f 2 C 2+ [ 0; 0], then '(r; ) = A1r3=2B1( ) + A2r3=2B2( ) + A3r2B3( ) + A4r5=2B4( ) (1.14) 1 +A5r5=2B5( ) 2A2r5=2f 00(0) cos 2 + O(r3 ) for any > 0 such that + > 1=2. Note that if f 00(0) = 0, then the expansion (1.14) agrees with that of (1.11). The rst two terms in the above expansion are called, in fractural mechanics, the stress intensity factors; or mode I and mode II of fracture [16; p. 24]. The proof of the estimates (i), (ii) require maximum principles for biharmonic solutions in a domain whose boundary has a singular point. Such estimates are established in x2; for related results see Remark 2.3. In x3 we give a proof of (i) and, in x5, a proof of (ii). In x4 we derive an additional regularity result for ', near the tip O = (0; 0), for f in C 1+ , namely: '(x) 6 C jxj2 + d2(x) near the origin; (1.15) jxj3=2 jxj2 where d(x) = dist(x; ). This improves the inequality that can be obtained by sub-Schauder (2 )-estimates (stated in x9, Example 2), if jxj + 0 < d(x)=jxj 1 for some 0 > 0. The remaining part of the paper is concerned with an application of some of the above results to the crack propagation problem, where the tip of the crack is moving in time according to (see [9]) _ X(t) = h(jJ(X(t))j)jJ(X(t))j; (1.16) where X(t) is the tip of the crack at time t, h(s) is a given function, and J(X(t)) is the limit of J-integrals taken along circles that shrink to X(t). The model is described in x6. In x7 we prove that the crack propagation problem with C 1+ crack is equivalent to the following geometric problem: Find an extension x2 = f(x1); 1 6 x1 6 ( > 0) of which is C 1+ such that, at each intermediate value x1 = s, the coe cient A2 = A2(s) in the asymptotic expansion about the tip X(s) = (s; f(s)) satis es: A2(s) 0: (1.17) In x8 we make a few comments on this problem, which we hope to pursue in a future work. The paper concludes with an appendix in which we have assembled several sub-Schauder estimates used in this paper. 2 Maximum principles In this section we establish estimates for solutions of 2' = f in a bounded domain in terms of the supremum of the boundary values of ' and @'=@ on @ . When @ is C 4 such an estimate is well known: k'kW 1;1 ( ) 6 C k'kL1 (@ ) + kr'kL1(@ ) + kfkL1( ) : (2.1) 3 This estimate was derived by Miranda [18] when is 2-dimensional, and by Agmon [2] for which is n-dimensional; Agmon has actually extended the results to elliptic operators of any order with variable coe cients. For our purposes we need to deal with domains whose boundary has a singularity as, for example, in Figure 1 below. However we begin with a local version of the type (2.1). Let be a bounded n-dimensional domain, S an open subset of @ , S in C 4, and D a subdomain of such that D [ S0 where S0 @ ; S 0 intS: Lemma 2.1 Under the above assumptions there is a constant C such that for any function ' in Lp( ) \ C 1( [ S), p > 1, if 2 ' = f then in ; (2.2) (2.3) k'kW 1;1 (D) 6 C k'kL1(S) + kr'kL1(S) + k'kLp( ) + kfkL1( ) : = k'kL1(S) + kr'kL1(S) + k'kLp( ) + kfkL1( ): Without loss of generality we may assume that f = 0; otherwise we subtract from ' the special solution of (2.2) 1 Z jx yj2 log 1 f(y)dy: (2.4) 8 jx yj Introduce domains D1 D2 with Proof. Set D D1 ; D1 D2; D2 [ S0; @D \ S int(@D1 \ S); @D1 \ S int(@D2 \ S): Take a C 1 function such that = 1 in D1; = 0 in n D2 e and introduce a domain D with C 4 boundary such that e e D1 D D2; and 0 in n D: Consider the function v = '. It satis es 2v = X j j63 D ' 4 and Write v = v1 + v2 where @v e v = @ = 0 on @ D n S: e 2v1 = 0 in D; @v v1 = v; @v1 = @ @ e on @ D: By the Miranda-Agmon maximum principle kv1kW 1;1 (D) 6 C k 'kL1 (@D) + kr( ')kL1(@D) 6 C k'kL1(S) + kr'kL1(S) : (2.5) e e e e To estimate v2 we apply Theorem 8.1 of [1] which says: If for any w 2 C 4(D) with e w = @w = 0 on @ D there holds @ Z for some integer k, 0 6 k 6 4, then e D v2 2wdx 6 jwjW 4 k;p0 (D) e 1 ( 1 + p0 = 1) p (2.6) (2.7) jv2jW k;p (D) 6 C ; e e here v2 is an arbitrary function (say in Lp(D)), is a constant depending on v2, and C is a constant independent of v2, . We shall apply this result to the function v2 de ned above as v v1. By integration by parts, Z e D v2 wdx = 2 = Z Z e D e D 2 v2 wdx = Z X X e D j j63 D ' wdx j j63 '( 1) D ( w)dx so that (2.6) holds with k = 1, = Ck'kLp. Consequently, by (2.7), kv2kW 1;p (D) 6 Ck'kLp( ): e Combining this with (2.5) we get k'kW 1;p(D) 6 kvkW 1;p(D) 6 C : e e e We can now repeat the above argument with the smaller domains D1 , D, D2 (still containing D) and k = 2 in (2.6). We get kv1kW 1;1 (D) + kv2kW 2;p (D) 6 C : e e 5 If p > n, then by Sobolev embedding we deduce that and (2.3) follows. If however p 6 n, we repeat the above process with a larger value of p; in fact, if p < n, then 1 1 e ' 2 Lq (D); and k'kLq (D) 6 C ; where q = 1 n ; e p whereas if p = n, then e ' 2 Lq (D); and k'kLq (D) 6 C ; 8q < +1: e After a nite number of steps the proof of (2.3) is completed. We shall now specialize to 2-dimensional domains whose boundary has a singular point. The method of Miranda-Agmon does not extend to such domains, and, in fact, our estimates will also be quite di erent. Let ! be a domain shown in Figure 1, consisting of two line segments j j = !=2 ( < ! 6 2 ), connected by an arc on the circle jxj = 1, and regularized around j j = !=2, jxj = 1 so that @ ! n f0g 2 C 1. kv2kW 1;1 (D) 6 C e 2 ! ! U ! e ! O ! Fig. 1 Then @ ! = ! [ e! , where e! = @ ! n ! = @ ! \ fjxj = 1g and ! consists of the two line segments = !=2 initiating at the origin with two small smooth arcs attached at each endpoint. Theorem 2.2 Suppose that ' 2 H 2( ! ), 2' = f in ! ; ' = @' = 0 on @ ! n ! : @n 6 (2.8) (2.9) Then, for any p > 2, > 0, < 2=p, there exists a constant C = Cp; ; > 0, depending only on the regularity of @ ! \ fjxj > 1=2g (but not on the angle size !), such that Z ! jxj p j'jpdx 1=p 6C Z hn Z + ! jxj ! 1=2 @'(x) dS jxj (1=2+2=p+ ) @n + Z j'(x)j p=2 dS o2=p i ! jxj jf(x)jdx : 3=2 (2.10) Proof. De ne u to be the solution of the following problem: u 2 H 2( ! ); 2u = g in ! ; @u u = @n = 0 on @ ! ; (2.11) (2.12) where g is any function in L2( ! ). By integration by parts (cf. [14; equation (5)]) Z ! E[u]dx = Z ! ugdx 6 CkukL1( ! )kgkL1( ! ) where E[u] = 2 X i;j=1 @ 2u 2, 1=q + 1=e = 1, 1 < q < 2. By embedding, q @xi@xj kukL1( ! ) 6 C and hence Z Z ! E[u]dx 1=2 ; (2.13) We shall now use the inequalities (40) and (47) of [14] (see also x9): ! E[u]dx 6 Ckgk2 1( ! ): L 2+2 ju(x)j 6 Cjxj 2 kgkL1( ! ) + 2 Z jru(x)j2 6 Cjxj2 kgk2 1( ! ) + L where = (e) is the solution of ! sin2(e ) = 2 sin2 !; ! e Z ! E[u]dx ; E[u]dx ! 0 < ! (e) 6 ; e! and ! is any constant such that ! 6 ! and < ! 6 2 ; it is easy to verify that there exists e e e a unique such = (e). We take ! = 2 so that (e) = 1=2. If we substitute (2.13) into ! e ! these inequalities, we obtain (e) > 1 ; !2 ju(x)j2 6 Cjxj3kgk2 1( ! ); L 2 6 Cjxj kgk2 jru(x)j L1 ( ! ) : 7 (2.14) (2.15) To obtain the estimates for the second and third order derivatives for u, we introduce the scaling u"(x) = u("x) for 1 6 jxj 6 4: Then 2u" = "4g("x) in D = f1 < jxj < 4; ! < < ! g 2 2 @u" = 0 on @D: u" = @n e By interior-boundary Lp estimates, h ku"kW 4;e(D0) 6 Cp max ju"(x)j + " p e D 4 Z 6 Cp " kgkL1 ( ! ) + " e 3=2 4 2=e p D e jgjp("x)dx 1=ei p kgkLpe( ! \f"<jxj<4"g) ; where D0 = D \ f2 < jxj < 3g, 1=p + 1=e = 1. By embedding p p ku"kW 3;r (D0\ ) 6 Cp"3=2 kgkL1( ! ) + k jxj5=2 2=egkLpe( ! \f"<jxj<4"g) ; e 3=2 kgk 1 5=2 2=egk p pe ku"kW 2;1 (D0\ ) 6 Cp" e L ( ! ) + k jxj L ( ! \f"<jxj<4"g) ; where = f = !=2g, r = p=(2 p) = p=(p 2), and so 1=r + 1=r0 = 1 for r0 = p=2. e e Rewriting this in terms of the original variables, we have Z max jD2u(x)j 6 C" f2"<jxj<3"g\ ! f2"<jxj<3"g\ ! jD3 u(x)jrdS 1=r 6 C" 3=2+1=r kgkL1( ! ) + k jxj5=2 2=e p 2=e p gkLpe( ! ) ; 1=2 kgkL1( ! ) + k jxj5=2 r gkLpe( ! ) : Setting K = kgkL1 ( ! ) + k jxj Z 5=2 2=e p gkLpe( ! ) , it then follows that 3=2 1=r+ jD2u(x)j 6 Cjxj Z f2"<jxj<3"g\ ! g jxj jD u(x)j dS 6 3 K r; C" (2.16) (2.17) 1=2 K: 1=r+ Letting "j = (3=2) j and summing over j, we conclude that fjxj<1=2g\ ! jxj3=2 jD3u(x)j dS r 1=r 6 CK: (2.18) Inequality (2.18) is clearly valid if we integrate also over ! n fjxj < 1=2g, and similarly (2.17) is valid over all of ! . We now multiply equation (2.11) by ' and integrate by parts to obtain Z Z @ u @' u dS + Z u fdx ' gdx = ' @n @n ! ! ! 8 6 nZ ! jxj 3=2 1=r+ jD u(x)j dS 3 2 Z r o1=r n Z + sup jxj jD u(x)j 1=2 + sup jxj 3=2 ju(x)j Z ! ! jxj 1=2 @'(x) @n dx o2=p ! jxj (3=2 1=r+ ) j'(x)j dx r0 o1=r0 jxj3=2jf(x)jdx p=2 6 CK hn Z Z ! ! jxj (1=2+2=p+ ) j'(x)j dS + jxj 1=2 @'(x) + Z jxj3=2jf(x)jdxi @n dS ! Since, by H lder's inequality, o K 6 C k jxj gkLq ( ! ) for any < 2=p, the assertion (2.10) follows immediately by duality. Remark 2.1. Let = right-hand side of (2.10). Then, by Theorem 2.2, Z ! jxj pj'(x)jpdx 6 C p: 1Z p p p 2 "2 ! \f"<jyj<4"g j'(y)j dy 6 C " : h io io Take '"(x) = '("x) as before. Then Z f1<jxj<4g j'" (x)jpdx 6 and, by Lemma 2.1, for jxj = 2, j'" (x)j + jr'"(x)j 6 C " 6 C " n n 2=p + 2=p + ! ! max j' (x)j + jr'"(x)j \f1<jxj<4g " h max j'(y)j + jyjjr'(y)j : \f"<jyj<4"g h i Rewriting this in terms of the original variables, we get j'(x)j 6 C jxj 2=p + C jr'(x)j 6 C jxj ! \fjxj=2<jyj<2jxjg max j'(y)j + jyjjr'(y)j ; h (2.19) i 1 2=p + C ! \fjxj=2<jyj<2jxjg max jyj 1j'(y)j + jr'(y)j : (2.20) Later on we shall also need a maximum principle in a domain , shown in Figure 2; it is bounded by a circle jxj = 1, a circle : jx ( ; 0)j = , and a curve: x2 = f(x1); 1 < x1 < . We assume that f(x1) is in C 1+ for 1 < x1 < , and that f(0) = 0, f 0(0) = 0. 9 O Fig. 2 Theorem 2.3 Let the foregoing assumptions on , hold, and suppose that ' 2 H 2 ( ), 2' = f in ; ' = 'n = 0 on @ n : (2.21) (2.22) Then, for any p > 2, > 0, < 2=p, there exists a constant C = Cp; ; > 0 independent of such that Z jxj p j'jpdx 1=p 6C Z hn Z + jxj 1=2 @'(x) dS + Z jxj3=2jf(x)jdxi @n jxj (1=2+2=p+ ) j'(x)j p=2 dS o2=p (2.23) following problem Proof. The proof is similar to that for Theorem 2.2. Denote by u the solution of the u 2 H 2( ); 2u = g in ; (p > 2) u = @u = 0 on @ : @n (2.24) (2.25) As before, we obtain From (40) of [11], kukL1( ) + Z E[u]dx 1=2 6 CkgkL1( ): (2.26) where the constant C is independent of . ju(x)j 6 Cjxj3=2kgkL1( ) in ; 10 Now we proceed as in the derivation of (2.16), (2.17) to derive, for " < , the inequalities Z f2"<jxj<3"g\ g jxj 3=2 1=r+ jD u(x)j dS 6 3 r K r ; C" jD2 u(x)j 6 C" 1=2 K for x 2 f2" < jxj < 3"g \ ; (2.27) where K is de ned as before as above but with ! replaced by . (The assumption " < is necessary to ensure that the rescaled function satis es the equation in a domain with uniformly smooth boundary.) The rest of the proof is the same as in Theorem 2.2. Finally, we prove a maximum principle for the domain ;m, bounded by a circle jxj = 1, and ;m, where m > 2 , and o n ;m = (x ; x ) 1 (x ; x ) 2 1;m= ; 12 12 and 1;m= is a curve consisting of two semi-circles centered at (1; 0) and (m= ; 0) of radius 1, connected by two line segments, regularized near four points (1; 1), (1; 1), (m= ; 1), (m= ; 1) so that 1;m= 2 C 5. ;m O ;m Fig. 3 We follow the proof of Theorem 2.2. First, it is clear that the estimates (2.16), (2.17) are still valid in the region f2" < jxj < 3"g for " < . Next, for jxj > 3 , the argument of Theorem 2.2 leads to Z jD2u(x)j 6 Cd(x) f2"<d(x)<3"g\ ;m d(x)3=2 1=2 e 1=r+ e jD3 u(x)j dS 6 C" K ; r r (2.28) (2.29) K for jxj > 3 : 11 where d(x) = min(jxj; jx (m; 0)j); 6 " < m; e and K is de ned as K but with jxj5=2 2=q replaced by d(x)5=2 2=q and with ! replaced by ;m. Now we can take " = "j as before to conclude: Theorem 2.4 Suppose that ' 2 H 2( ;m), 2' = f in ;m; ' = 'n = 0 on @ ;m n ;m : (2.30) (2.31) Then, for any p > 2, > 0 > 0, there exists a constant C = Cp; ; > 0 independent of and m such that Z d(x) ;m p j'jpdx 1=p d(x) (1=2+2=p+ )j'(x)j dS ;m Z Z 1=2 @'(x) 3=2jf(x)jdx + ;m d(x) @n dS + ;m d(x) 6C n Z p=2 o2=p (2.32) principle for a conical region: They rst establish such a result in an in nite cylinder, and then, by a local Miranda-Agmon maximum principle (Lemma 10.1 in [17]), for a polycylinder. Finally, they map a conical region into a polycylinder. Our approach to derive a maximum principle with integral norms (e. g. Theorem 2.2) is much simpler, and our local MirandaAgmon lemma (which is sharper than Lemma 10.1 in [17]) then yields pointwise estimates for u; ru as in (2.19), (2.20). We note that the Maz'ya-Plameneveskii estimates are in weighted sup norms, similar to (2.19), (2.20); however, for our purposes, the integral estimates will be more convenient. It is also important to note that whereas the conical region in [17] is assumed to have C 4 boundary, our method allows weaker regularity on the parts of the boundary with zero Dirichlet data. Thus in the case of Theorems 2.3, 2.4, our method requires only that the curve x2 = f(x1) is in C 1+ . Remark 2.2. Remark 2.1 extends to both Theorems 2.3 and 2.4. Remark 2.3. Maz'ya and Plameneveskii [17] use a di erent method to derive a maximum 3 The stress intensity factors Throughout sections 3{6 we assume that ' 2 H 2(B1 n ); 2' = 0 in B1 n ; ' = @' = 0 from both sides of ; @n 12 (3.1) (3.2) (3.3) where B1 is the unit disc fx2 + x2 < 1g and 1 2 = fx2 = f(x1); x 6 x1 6 0g (3.4) (3.5) is a curve contained in B1 except for its end point ( x ; f( x )). We also assume that f(0) = 0; f 0(0) = 0: In this section and in section 4, we also assume that f 2 C 1+ , whereas in section 5 we shall require that f 2 C 2+ . Theorem 3.1 If f 2 C 1+ [ x ; 0], then (1.13) holds for any > 0 such that + > 1=2. We shall rst prove a weaker result: '(x) = A1r3=2B1( ) + A2r3=2B2( ) + O(r3=2+ ) for some > 0: (3.6) (3.7) (3.8) (3.9) Proof of (3.6). De ne then " (x) = '("x) ; "3=2 jxj < 1 ; " Under this change of variables, is changed to " j "(x)j 6 Cjxj3=2 for jxj < 1 : " : x2 = f" (x1) 1 f("x1); " 16x1 60) where f" 2 C 1+ , and f"(0) = f"0 (0) = 0; kf"kC1+ ( 6 C" : (3.10) Let G = G" be a function de ned on 2 ( ! is de ned in the previous section; here we take ! = 2 ) as follows: G 2 H 2( 2 ); 2G = 0 in 2 ; G = @G = 0 on f = g; (3.11) @n G = "; @G = @ on rest of @ 2 : (3.12) @n @n Setting ! = 2 2C" , we want to apply the Theorem 2.2 to " G in the domain ! . For jxj 6 1=2, G has an expansion (see x1) G= 1 X k=1 e rk=2+1Bk ( ) (r > 0; < < ); (3.13) 13 e where both Bk and their derivatives are bounded. It follows that jGj j j= jGn j j j= C" C" 6 C"2 jxj3=2; 6 C" jxj1=2: Since 2 C 3=2 by [14], similar estimates are also valid for : j j j j= C" 6 C"3 =2jxj3=2; 6 C" =2jxj1=2: " It is also clear that a similar estimate is valid for arcs near fj j = C" ; jxj = 1g. By Theorem 2.2, for any p > 2, jr j j j= C" G on two connecting small smooth k Since also " G" kLp(B1\f +C" < < C" g) 6 Cp" =2: G"j 6 j "j + jG" j 6 Cjf"j3=2 + C"2 jxj3=2 6 C" =2jxj3=2 in B1 \ fj j 6 C" g, we get " j k " G"kLp(B1) 6 Cp" =2: (3.14) Notice that j " G"j is uniformly in C 3=2. By interpolation, for any < =2, we can take p large enough such that j "(x) G" (x)j 6 C" for jxj 6 1: (3.15) Rewriting this in terms of the original variables, we have '(x) G" x 6 C" for jxj 6 ": (3.16) 3=2 " " By (3.13), q 3=2B ( ) + O(r2 ) for r = x2 + x2 6 1 ; G" (x) = r " (3.17) 1 2 2 where O(r2) means a term which is bounded by Cr2 with the constant C independent of ". Thus 2 r3=2B"( ) 6 C"3=2+ + C"3=2 r for r 6 " : '(x) " 2 Hence 3=2 1=2 " r '(x) B" ( ) 6 C" r + C " for r 6 " : (3.18) 3=2 r 2 Setting = =2, we get '(x) B"( ) 6 C" (" )3=2 + C" =2 6 C" =4 for " +1 6 r 6 2 +1" +1: (3.19) 3=2 r 14 Now take "j = 2 j , r = " +1. Then j+1 B"j ( ) B"j+1 ( ) 6 '(x) B"j ( ) + B"j+1 ( ) '(x) r3=2 r3=2 6 C2 j =4 = C" =4: j X B"j It follows that the series is convergent, and Setting we then have so that, by (3.19), where X j ( ) B"j+1 ( ) j>k B"j ( ) B"j+1 ( ) 6 C" =4: k B( ) = "lim B"j ( ); j !0 (3.20) B( ) B"j ( ) 6 C" =4; j (3.21) (3.22) '(x) A B( ) 6 Cr =[4( +1)]; 3=2 r A B( ) = A1B1( ) + A2B2( ); A = (A1; A2); B( ) = (B1( ); B2( )); and B1( ), B2( ) are de ned in (1.12). This completes the proof of (3.6). Let C1, C2 be positive constants such that 1 k'kH 2(B1n ) 6 C1; kfkC1+ 6 C2; 0 < < 2 : In order to complete the proof of Theorem 3.1, it su ces to prove: (3.23) 1 Lemma 3.2 For any 2 (0; min( ; 2 )) there exists a constant C = C depending only on C1, C2 such that j'(x) jxj3=2(A1B1( ) + A2B2( ))j 6 Cr3=2+ in B1: (3.24) Proof. It is su cient to show that (3.24) holds for all r < 1 . Setting 2 w(x) = '(x) A1r3=2B1( ) A2r3=2B2( ); we have, by (3.6), jw(x)j 6 Cr3=2+ ; for some 0 < < : 15 (3.25) The proof of (3.25) shows that C depends only on C1, C2. If 6 , then there is nothing to prove. So we may assume that > ; consequently, for any 0 < < 1, (3.26) r + r ) < C for all jxj < 1. If C can be chosen to be independent of , as well as of C1, C2, then (3.24) follows by taking ! 0 in (3.26). So it su ces to show that if such a C does not exist then we get a contradiction. Assuming that such a C does not exists, there exist sequences fn, wn, n and xn such that )j Cn = sup 3=2 jwn(x)j = 3=2 jwn(xn ! 1; + n r ) Rn (R + n R ) jxj<1 r (r n n if n ! 1, where Rn = jxnj < 1. In view of (3.25), we must then have 3=2(r jw(x)j n ! 0; Introduce a function Gn ( ) by Rn ! 0: wn(x) = CnR3=2(R + n R )Gn ( ); x = Rn : n n n Then and x Gn ( n) = 1 where n = Rn ; n 3=2 jGn ( )j 6 jxj 3=2(jxj + n jxj ) Rn (Rn + n R ) n j j + j j n 6 j j3=2 Rn (3.27) R + n n 6 j j3=2(j j + j j ): As n ! 1, the curves and, for a subsequence, n , de ned by 2 = fn( 1), converge in the -plane to the ray S0 = f( 1; 0); 1 < 1 < 0g (3.28) (3.29) (3.30) (3.31) Gn ( ) ! G( ) uniformly in compact subsets of R2 S 0, and 2G = 0 in R2 n S 0; G G = @@ = 0 from both sides of S0; jG( )j 6 j j3=2(j j + j j ) in R2 16 (0 < < < min( ; 1 )): 2 To prove (3.30), we actually need to use sub-Schauder boundary estimates (cf. x9). We rst apply Lemma 9.2 to the function u(x) = " 3=2'n("x) in a ring 1=2 < jxj < 2 to obtain, for any 2 (0; 1), j'n (x)j 6 C d(x) 1+ ; jr'n(x)j 6 C d(x) ; (3.32) jxj3=2 jxj jxj1=2 jxj where d(x) is the distance to the n . Similar estimates are valid for jxj3=2[A1B1( )+A2B2( )] with d(x) replaced by jx2j. It follows that jwn(x)j 6 C d(x) + jx2j 1+ ; jrwn(x)j 6 C d(x) + jx2j : (3.33) jxj3=2 jxj jxj1=2 jxj Since < , we can choose su ciently close to 1 so that > . Rewriting (3.33) in terms of , we then have, 3=2 (1+ ) 3=2 jGn ( )j 6 CK Rn Rn j j 6 CK j j3=2 for 2 CK ; Cn R3=2(Rn + n Rn) Cn n R1=2R j j1=2 n jr Gn ( )j 6 CK 1=2 n 6 CK j j1=2 for 2 CK ; C (3.34) (3.35) n CnRn (Rn + n Rn ) where CK = f K 6 1 6 0; j 2j 6 CR 1g (the constant C is from (3.10)). Similarly n R1=2 R ( ) CK n n (3.36) [r Gn ]C (CK ) 6 1=2 R ) 6 Cn CnRn (Rn + n n provided is taken to be small enough. With these estimates, we can now apply Theorem 9.3 (ii) to Gn ( ) with the boundary S given by j 2j = CR 1 to obtain (3.30). We also n have uniform convergence up to the boundary in (3.28), so that if n ! e = (e1; e2), then jG(e)j = 1: (3.37) We now invoke a Liouville theorem (Lemma 3.3 below) to deduce from (3.29){(3.31) that G( ) 0, which is a contradiction to (3.37). Lemma 3.3 If G is a function satisfying (3.29){(3.31), then G 0. Proof. From the expansion G( ) = 1 X k=1 rk=2+1Bk ( ) (r = j j) (3.38) near = 0 and (3.31) (with > 0) it follows that B1 = 0. Recalling that 2 r2B2( ) = c 2 ; c constant, 17 we introduce the function Then The function 2 H( ) = G( ) c 2 : jH( )j 6 Cj j5=2 near = 0: @G ( ) = @H = @ @ 1 1 2 n S and satis es the same boundary conditions as in (3.30). Since is a biharmonic in R 0 has an expansion similar to (3.38), we deduce that jrj ( )j 6 Cj j5=2 Next, for j j large, we have 1j near = 0: (3.39) By scaling and applying elliptic boundary and interior estimates we deduce that jG( )j 6 2j j3=2+ : 1j jrj ( )j 6 Cj j3=2+ for j j large (3.40) and 0 6 j 6 3. o n 1 Set DR = R < j j < R n S0. By integration by parts, 0= Z where IR is a linear combination of integrals DR 2 = Z DR j j2 IR; AR;j = BR;j = Z Z fj j=1=RgnS0 fj j=RgnS0 Dj D3 j Dj D3 j : In view of (3.39), (3.40), we clearly have 2 jAR;j j 6 C ; jBR;j j 6 CR ; R R and, since < 1=2, IR ! 0 if R ! 1. It follows that Z so that 0. Since = @ =@ = 0 on S0, we get 0. This means that @ (G c 2 ) = 0 2 @ 1 and so G = g( 2 ), g(4) = 2G = 0. Hence G is a polynomial of degree 6 3 in 2, and by invoking (3.30), (3.31) we nally conclude that G 0. 18 R2nS0 j j2 = 0; 4 A atness lemma Let the assumptions of Theorem 3.1 be satis ed and denote by d(x) the distance from x to . In this section we investigate the behavior of '(x) as x approaches the tip O while, at the same time, d(x)=jxj tends to zero. Theorem 4.1 Under the assumptions of Theorem 3.1 j'(x)j 6 C jxj2 + d2(x) jxj3=2 jxj2 where C is a constant depending only on the C1, C2 in (3.23). (4.1) Remark 4.1. If we apply the sub-Schauder estimates (x9, Lemma 9.2) to " in 1=2 < jxj < 2, where ' is as in Theorem 3.1, we get j'(x)j 6 C d(x) 1+ 8 > 0: jxj3=2 jxj The estimate (4.1) is an improvement of (4.2) when 3=2 '("x) (4.2) jxj + 0 < d(x) 1 for some 0 > 0: jxj To prove Theorem 4.1, we shall establish a lemma which is of intrinsic interest. Let = fx2 = f(x1); 1 < x1 < 1g; D = f(x1; x2); 1 < x1 < 1; 1 < x2 < f(x1)g; and let be a subarc of : = fx2 = f(x1); 1=2 < x1 < 1=2g: We assume a bound kfkC1+ 6 A (0 < < 1; A > 0); (4.3) (4.4) and the \"- atness" condition: jf(x1)j < " (0 < " < 1=2): Denote by d(x) the distance function d(x) = dist(x; ); x 2 D: 19 Lemma 4.2 (Flatness Lemma) If 2 H 2(D n ), k kH 2(Dn ) 6 C1, 2 = 0 in D; = @ = 0 on ; @ j j < 1 in D; then (4.5) (4.6) (4.7) (4.8) j (x)j 6 C(d2(x) + "2) in D where C is a constant depending only on A, and C1. Remark 4.2. By sub-Schauder estimates (see Example 2 following Theorem 9.1) j (x)j 6 Cd2 (x) 8 > 0; (4.8) is an improvement when d(x) "1+ 0 for some 0 > 0. Proof. We assume that (4.8) is not valid and derive a contradiction. If (4.8) is not true then there exist sequences = n ; = n ; D = Dn ; " = "n; = n; such that, with dn (x) = dist(x; n ), n n Cn = sup d2j(x)(x)j"2 = d2j(xn(x+)j"2 ! 1 if n ! 1; + n n n) n Dn n where xn 2 Dn ; we necessarily have dn dn (xn) ! 0; "n ! 0 if n ! 1 for otherwise the sequence Cn will remain bounded (by (4.7)). Denote by xn the point on e n such that dn = jxn xnj. e Introduce functions Gn ( ) by n (x) = Cn (dn + "n )Gn ( ) 2 2 where x xn = dn : e (4.9) (4.10) Then e dn (x) = dn dn ( ); e e Gn ( n ) = 1; n = xn d xn ; dn ( n) = 1; n e where dn ( ) is the distance from to the image en of n under the mapping x xn = dn . e Clearly, 2 2 2e 2 +2 jGn( )j 6 dnd(x) +2"n 6 dndn2( ) "2 "n (4.11) 2 +" d+ n n n n 20 It is su cient to consider the following two cases: Case (1): dn ="n ! 0 if n ! 1. Case (2): dn > c"n for some c > 0 for all large n. In case (1), (4.11) clearly implies that dn 2 + 1; 2 e ( ) jGn ( )j 6 dn " (4.12) n and in case (2), (4.11) implies that 2 (4.13) jGn ( )j 6 dnd(x) + 1 6 j j2 + 1: 2 n As n ! 1, the curves en in the -plane converges to the line f 2 = 0g and, by (4.10), Gn ( ) ! G( ) uniformly in compact subsets of f 2 < 0g, 2G( ) = 0 in f 2 < 0g; jG( )j 6 1 in f 2 < 0g in case (1); jG( )j 6 j j2 + 1 in f 2 < 0g in case (2): (4.14) (4.15) (4.16) (4.17) By sub-Schauder estimates (Lemma 9.2) applied to Gn ( ) we deduce that (4.18) G = @G = 0 on f 2 = 0g: @ We also note that the convergence in (4.10) is uniform near the boundary so that, in particular, if e n = xn d xn ! e (e = (e1; e2); e2 6 0; jej = 1); n then jG(e)j = limjGn ( n )j = 1: (4.19) By a Liouville theorem (Lemma 4.3 below) we conclude from (4.15){(4.17), (4.18) and (4.19) that 2 G( ) = K 2 ; K 6= 0: (4.20) This contradicts (4.16) (case (1)). We shall next derive a contradiction to (4.20) (case (2)). The proof depends on sharp estimates on the n (x). Since, j n(x)j 6 Cn(d2 (x) + "2 ) in Dn n n the atness condition implies that j n(x)j 6 CCn"2 in fjx1j < 1 ; 4"n < x2 < 2"ng: n 2 21 (4.21) By interior elliptic estimates we then also have We need to construct an auxiliary function. For this purpose we rst consider the problem 2' = 0 in fx2 > 0g; '(x1; 0) = f0(x1); @ '(x ; 0) = g (x ); 01 @x 1 2 1 jr n(x)j 6 CCn"n on fjx1j < 4 ; x2 = 3"ng: (4.22) we nd that 1+ where f0 2 Cloc , g0 2 Cloc. If f0, g0 are uniformly bounded, then we can write a solution in the form h @ P(f )i ' = P(f0) + x2 P(g0) @x 0 2 where P is the Poisson kernel: 1Z1 x2 P(h)(x) = 2 h( 1 )d 1 : 1 (x1 1 )2 + x2 Noting that @ P(f ) = 1 Z 1 (x1 1)2 x2 f ( )d ; 2 @x 0 [(x )2 + x2]2 0 1 1 2 (4.23) 1 1 1 2 Introduce the functions j'(x)j 6 Cjf0jL1 + Cjx2j(jg0jL1 + jf0jL1 ): (4.24) Let (x1) be a C 2 function such that (x1) = 1 if jx1j < 1=8, (x1) = 0 if jx1j > 1=4. @ f0 = n; g0 = @x ( n) on x2 = 3"n 2 and denote by n the biharmonic function in fx2 < 3"ng with Dirichlet data f0, g0 on x2 = 3"n. Using a representation similar to (4.23) (with (x1; x2) replaced by (x1; x2 3"n )) we get, from (4.24) and (4.21), (4.22), the estimate j n (x)j 6 CCn("2 + jx2j"n) if x2 6 3"n: n Consider the function (4.25) n = It satis es n n in n = fjx1j < 1 ; 1 < x2 < 3"ng: 8 2 n = 0 in n; @ n = @xn = 0 on x2 = 3"n; jx1j < 1 ; 8 2 22 and by (4.25) and (4.7). By standard elliptic regularity 1 j (x ; 3" Md )j 6 C(Md )2 if jx j < 1 n n n 1 Kn n 1 16 for any M > 0, and we shall later choose M 1 (but Mdn + 3"n < 1). Combining this with (4.25), we get j n(0; 3"n Mdn )j 6 CCn("2 + dn M"n ) + (1 + CCn"n)C(Mdn )2: (4.26) n On the other hand, by (4.10), (4.14), (4.20), 2 2b 2b 2 b n (0; 3"n Mdn ) > Cn dn Gn ( n ) Cn dn G( n ) = KCn dn j n;2 j where b b b (0; "n Mdn ) xn = dn n = dn( n;1 ; n;2) e b and j n;2j M if M 1. Comparing this with (4.26) we get KCnd2 M 2 6 CCnM"2 + C(1 + CCn"n)M 2d2 : n n n Since dn > c"n (c > 0), choosing M to be large enough (say KM > 2C), we get a contradiction as we let n ! 1. j n (x)j 6 1 + CCn"n Kn in n; Lemma 4.3 If G( ) is a function satisfying (4.15),(4.17), (4.18), and if jG( )j 6 C(1+j j2) in f 2 < 0g, then 2 G( ) K 2 (4.27) where K is a constant. Proof. The biharmonic function satis es the same boundary condition as G, and jrj H( )j 6 Cj j j if j j is large: By integration by parts, Z @2 H( ) = @ 2 G( ) 1 where IR ! 0 if R ! 1 (cf. the proof of Lemma 3.3). If follows that H 0 and, by unique continuation, H 0. Hence G = g1( 2) + g2( 2) 1 and this implies (4.27). Proof of Theorem 4.1. Applying the atness lemma to " 3=2'("x) for 1=2 < jxj < 2 and arbitrarily small ", the assertion (4.1) easily follows. 23 fj j<R; 2<0g H H = 2 Z fj j<R; 2<0g ( H)2 + IR 5 Higher order expansion In this section we assume that f 2 C 2+ and obtain higher order expansion of '. Theorem 5.1 Let ' be a solution of (3.1){(3.5) and assume that f 2 C 2+ [ x ; 0] for some > 0. Then for any 0 < < 1=2 such that + > 1=2, the expansion (1.14) holds. The assumptions of Theorem 5.1 imply that k'kH 2(B1n ) 6 C1; kfkC2+ 6 C2: For clarity we shall rst prove a special case: Lemma 5.2 Under the assumptions of Theorem 5.1, for any 2 (0; 1=2) there exists a constant C depending only on C1 , C2 and such that j'(x) r3=2[A1B1( ) + A2B2( )] A3r2B3( )j 6 Cr2+ : In the sequel we shall need the following interpolation inequality: (5.1) (5.2) krukL1 6 C kuk =(1+ )kruk1=(1+ ) + kukL1 L1 C jy xj = "; jz xj = ": Then and Hence and choosing (0 < < 1) where the norms are taken in a bounded domain . It su ces to prove (5.2) in dimension 1. For any x 2 , let y; z belong to such that u(y) u(x) = u (e) xx yx ux(z) = ux(z) ux(e) + u(y) u(x) : x yx kuxkL1 6 kuxkC " + 2 kukL1 " " +1 = kukL1=(kuxkC + kukL1) yields the assertion. In the sequel we shall also use the interpolation inequality k kC 6 C k k( 1 )= [ ] = + k kL1 L C which follows from (0 < < < 1); (5.3) j (x) (y)j = j (x) (y)j jx yj 6 [ ] + 2k kL1 C jx yj jx yj 24 by taking = (k kL1 =k kC )1= . Proof of Lemma 5.2. The function is biharmonic in B4(0) n " , where " " (x) = 3=2 '("x) " 1 is uniformly C 2; . By sub-Schauder estimates (cf. x9, Theorem 9.3), : x2 = 1 f("x1) f0(x1) " jD2 "j + [D2 "] 6 C for 1 < jxj < 2: Rewriting this inequality in terms of ', we get jD2'(x)j 6 Cjxj 1=2 ; [D2']C [jxj<r<1] 6 Cjxj 1=2 (5.4) where the constant C is independent of ". Multiplying by a constant if necessary, we may assume without loss of generality that j "j 6 1; jr "j 6 1 for jxj < 1; 0 < " < 1: Let '0 = " where " will later on be chosen very small but xed, and denote by G0 the solution of It is clear that and Hence, using (5.4), G 2 H 2(B1(0) n f = g); 2G = 0 in B1(0) n f = g; G = @G = 0 on f = g; @n G = '0; @G = @'0 on rest of @(B1(0) n f = g): @n @n (5.5) (5.6) jf000(x1)j = "jf 00("x1)j 6 "; jf0(x1)j 6 1 "jx1j2: 2 2 j'0j + jG0j 6 Cjxj 1=2 "jx1j 6 C"2jxj3=2 on j j 6 2"; 1=2 "jx j 6 C"jxj1=2 on j j 6 2"; jr'0j + jrG0j 6 Cjxj 1 Applying Theorem 2.2, we get (5.7) (5.8) (5.9) k'0 G0kLp( 2 25 2" ) 6 C": Notice that both '0 and G0 are uniformly C 3=2. It is clear that if < 1=4 =2 (recall that 0 < < 1=2), then [r('0 G0 )]C 6 C for some universal constant C (Actually, '0 G0 is bounded also in the C 1=2 norm in this rst step of the iteration; but we use the C norm for the later iterations). Thus by interpolation, for any < 1, if we choose p large enough we obtain j'0 G0j 6 C" for jxj < 1: (5.10) By (5.2) with u = '0 G0, we also have jr('0 G0 )j 6 Cj'0 G0 j =(1+ )[r('0 G0)]1=(1+ )+C" 6 C" =(1+ ) for jxj < 1: (5.11) C e Clearly, there is a constant C and P0 (x) = [r3=2B"( ) + r2B" ( )] such that jG0 P0j 6 Cr5=2 for r < 1=2; jr(G0 P0)j 6 Cr3=2 for r < 1=2: It follows that (5.12) (5.13) j'0 P0 j 6 C" + Cr5=2 for r < 1=2; jr('0 P0)j 6 C" =(1+ ) + Cr3=2 for r < 1=2: We now x small constants ", such that C" + C(2 )5=2 6 2+ ; C" =(1+ ) + C(2 )3=2 6 1+ ; we can actually take " = for large enough and any su ciently small. Since " is now xed, we shall simply write P0 as e P0 = r3=2B 0( ) + r2B 0( ); and we then have j'0 P0j 6 2+ for jxj < 2 ; jr('0 P0)j 6 1+ for jxj < 2 : Next, we de ne Then and, by (5.7), (5.8) 1 ' ( x) P ( x) for r 6 2: '1 = 2+ 0 0 (5.14) (5.15) (5.16) (5.17) j'1j 6 1; jr'1j 6 1 for r 6 2; j'1j 6 C 2 j xj 1=2(" j x1j)2 6 C"2 3=2 jxj3=2 for j j < 2(" ); (5.18) jr'1j 6 C 1 j xj 1=2(" j x1j) 6 C" 1=2 jxj1=2 for j j < 2(" ): (5.19) 26 Finally, since jD2'0j + jD2G0 j 6 Cjxj [r'1]C (fj j<2(" )g\fr<jxjg) 6 C since < 1=4 =2. We claim that 1=2, 1 j xj 1=2 (" j x 1 1 j) 6 C"1 jxj1=2 (5.20) (5.21) j'1(x)j 6 jxj1+ if jxj < 2: For clarity of exposition we shall postpone the proof until the end of this section, and in fact, establish a general result (namely Lemma 5.4) for which (5.21) follows as a special case (upon taking fj (x1) to be parabolic curves x2 = j x2, with 1 < 1 < 2 < 1; notice that '1 1 is biharmonic outside the thin region enclosed by these two parabolic curves). In view of (5.21), we can apply Theorem 9.3 (ii) to the function '1( x) 1+ which (by (5.21)) is bounded in 1=2 < jxj < 2 and whose C 1+ -norm is uniformly bounded in the sector j j 6 2(" ) (by (5.19), (5.20)). We then have [r'1]C (B1) 6 C (5.22) (5.23) (5.24) for some universal constant C. We can now proceed as above (with the same "; ) to derive j'1 P1j 6 2+ for jxj < 2 ; jr('1 P1)j 6 1+ for jxj < 2 e for some P1(x) = r3=2B 1( ) + r2B 1( ). Proceeding by induction, we de ne 1 'k+1(x) = 2+ 'k ( x) Pk ( x) = ( Clearly, (2+ ) )k+1 '0 ( k+1 x) k X j=0 ( (2+ ) )j+1 Pk j ( j+1 x): j( and k X j=0 (2+ ) )k+1 '0 ( k+1 x)j 6 C 6 C"2 (3=2 6C k X j=0 (2+ )(k+1) k+1 j xj )k jxj3=2 for j j < 2" k+1 ; 1=2 (2+ ) )j+1 k+1 j k+1 xj 2 " ( (2+ ) )j+1 jPk j ( j+1 x)j ( j j+1xj 1=2 k+1 j j+1 xj 2 " 6 C"2 (3=2 27 )k jxj3=2 for j j < 2" k+1 : It follows that j'k+1(x)j 6 C"2 (3=2 Similarly, h r )(k+1)jxj3=2 for j j < 2" k+1 : (5.25) ( (2+ ) )k+1 '0( k+1x) i 6 C (1+ )(k+1)j k+1xj 1=2 " k+1 j k+1xj 6 C" (1=2 )(k+1)jxj1=2 for j j < 2" k+1 ; and k X j=0 ( (2+ ) )j+1 r Pk j ( j+1 x) 6C k X j=0 ( (1+ ) )j+1 j j+1 xj 1=2 " k+1 j j+1 xj 6 C" (1=2 jr'k+1(x)j 6 C" (1=2 )(k+1)jxj1=2 )kjxj1=2 for j j < 2" k+1 : (5.26) (5.27) It follows that for j j < 2" k+1 : We can apply the same procedure to deduce that [r'k+1(x)]C (fj j<2" k+1g\fr<jxjg) 6 C"1 jxj1=2 : With the estimates (5.25) and (5.26) and (5.27) at hand (where the constants C are independent of "; ; k), the same procedure can be used to deduce that j'k+1 Pk+1 (x)j 6 2+ for jxj 6 2 ; jr('k+1 Pk+1 (x))j 6 1+ for jxj 6 2 : We can rewrite the inequality (5.28) in terms of the original variables: (5.28) (5.29) j'0(x) Qk (x)j 6 ( k )2+ for jxj < 2 k ; x Qk (x) = jj j=0 Since Pj (x= j ) are bounded by Cjx= j j2 + Cjx= j j3=2, the series converges, and we let ( j )2+ P 1 X where k X Q(x) = k!1 Qk (x) = x lim ( j )2+ Pj j : j=0 e Since Qk (x) are of the form r3=2B ( ) + r2B ( ), the limit function is also of the same form, and we denote it by e Q(x) r3=2B( ) + r2B( ): 28 We then have jQ(x) Qk (x)j 6 Therefore, that 1 X j=k+1 ( j )2+ h x 3=2 + x 2i 6 C n( k )1=2+ jxj3=2 + ( k ) jxj2o: j j j'0(x) Q(x)j 6 C( k )2+ for jxj < k : (5.30) For each jxj < , we choose k such that k+1 < jxj 6 k . Then the above inequality implies and the proof of Lemma 5.2 is complete. Remark 5.1. The above proof also shows that j'0(x) Q(x)j 6 Cjxj2+ ; jr('0(x) Q(x))j 6 Cjxj1+ : iterative argument used in the proof of Lemma 2.4 of [12]. Remark 5.2. The iterative argument used in the proof of Lemma 5.2 is similar to the Gj = @Gj = 0 on both sides of j ; @n Gj = 'j ; @Gj = @'j on rest of @(B1(0) n j ); @n @n where j : x2 = 1 f 00(0) j "x2; 2 < x1 < 0, and 1 2 Lemma 5.3 Suppose Gj is the solution of the following system: Gj 2 H 2(B1(0) n j ); 2Gj = 0 in B1(0) n j ; (5.31) (5.32) (5.33) jf 00(0)j 6 1; k'j kC1(B1) 6 1: Then, for any small > 0 there exist su ciently small " and such that the corresponding Gj satis es: 1 jGj (x) Pj (x)j 6 2 3 for jxj < 2 ; (5.34) jrGj (x) rPj (x)j 6 1 2 for jxj < 2 ; (5.35) 2 where _ Pj (x) = r3=2 Aj1B1( ) + Aj2B2( ) + Aj2" j rB2( )) + r2Aj3B3( ) +r5=2 Aj B4( ) + Aj B5( ) r5=2 3 Aj " j B1( ); 4 5 22 for all j > 0, 0 6 j < 1. 29 Proof. Under the above assumptions it is clear that jGj j 6 Cjxj3=2; jrGj j 6 Cjxj1=2; [rGj ]C1=2 (B1=2) 6 C: e Let Gj be the solution of the following problem e Gj 2 H 2(B1=2(0) n f = g); e 2Gj = 0 in B1=2(0) n f = g; e G e Gj = @@nj = 0 on f = g; e G e Gj = Gj ; @@nj = @Gj on rest of @(B1=2(0) n f = g): @n (5.36) (5.37) By scaling and using C 2+ estimates as before, we get jxj 3=2 e " 1jGj Gj j + jxj 1=2 e jr(Gj Gj )j 6 C" if j j 6 2": Then by maximum principle (Theorem 2.2), e kGj Gj kLp(B1=2) 6 C": For any < 1, we use C 3=2 regularity and interpolation, and take p to be large enough to obtain e e jGj Gj j 6 C" ; jr(Gj Gj )j 6 C" =3: By (1.11), we have e jGj P(x)j 6 Cr3; e jr(Gj P(x))j 6 Cr2; e e e P(x) = r3=2B( ) + r2B( ) + r5=2B( ): Notice that the extra terms in Pj (x) are of order ". Choosing " and such that 1 C" + C(2 )3 < 4 3 ; C" =3 + C(2 )2 < 1 2 ; (5.38) 4 the proof is now complete. Proof of Theorem 5.1. We shall modify the proof of Lemma 5.2 for C 2 expansion to obtain C 5=2 expansion. The Pj will be of the form n o _ Pj (x) = r3=2 Aj1B1( ) + Aj2B2( ) + Aj2" j rB2( )) n o 3 + r2Aj B3( ) + r5=2 Aj B4( ) + Aj B5( ) r5=2 2 Aj " j B1( ) ; 3 4 5 2 1 (x) + P 2 (x) Pj j where obtained from Lemma 5.3. 30 1 Notice that the terms r5=2 cos 3 in Pj1 and Pj2 cancel out, and the term r5=2 cos 2 (in 2 Pj1 and Pj2) is biharmonic. Thus Pj still satis es the biharmonic equation: 2Pj 0. We de ne 'k inductively as 'k+1 = 31 'k ( x) Pk ( x) = ( (3 ) )k+1 '0 ( k+1x) k X j=0 ( (3 ) )j+1 Pk j ( j+1 x); where k j are still to be determined. Instead of two lines = ( 2" k ), we now use the two C 1 curves k : x = 1 " k f 00(0)x2 "1+ (1+ )k x2; 2 < x1 < 0; 2 1 1 2 which are "1+ (1+ )k x2 close to the original curve (instead of just " k x1 close to the original 1 curve). Then, using (5.4), we nd that on k+1, as well as on any arc jxj =const. in the thin k+1 and k+1 , region connecting + provided 0 < < 1=2, + > 1=2, 0 < 6 0, where 0 = +2 + 1=2 ; cf. the proof of (5.18){(5.20). Next, we estimate the sum X j( (3 ))k+1'0( k+1 x)j 6 2(1+ ) (k+1)(1=2+2 + )jxj3=2; C" (3 ) )k+1 ' ( k+1 x) 6 C"1+ (k+1)( + 1=2)jxj1=2; r ( 0 h i r ( (3 ) )k+1'0( k+1 x) C ( k+1\fjxj<Rg) 6 C"(1+ )(1 )R1=2 ; (5.39) (5.40) (5.41) ( (3 ) )j+1 Pk j ( j+1 x) = X ( (3 ) )j+1 1 Pk j ( j+1x) + X ( (3 ) )j+1 2 Pk j ( j+1 x) on the curves k+1 and in the thin region lying between them. The second derivatives of those terms involving r2 and r5=2 are bounded. Therefore, we can follow the calculations as in the case for C 2 expansion to conclude that on k+1 as well as any arc jxj =const. which k+1 and k+1 , lies in the thin region lying between + X (3 X r ( ) )j+1 P 2 ( j+1 x) 6 C"2 k(1+ ) jxj3=2; kj ( (3 ))j+1 Pk2 j ( j+1 x) 6 C" k jxj1=2; i (5.42) (5.43) furthermore, h r X ( (3 ) )j+1 2 Pk j ( j+1x) 6 C" C ( k+1 \fjxj<Rg) 31 (1 ) k( 2 ) 1=2 R 6 C"(1 )R1=2 ; (5.44) provided 6 =2. Next, we split Pk1 into two parts. Pk Since we have 1 (x) = r AB Pk Pk 3=2 k ( ) + 3=2 k 11 2 11 (x) + 12 (x): _ r A B2( ) + " k rB2 ( ) @ B ( ) @ 2 B ( ) B1( ) = = @ 1 = = @ 2 1 = = 0; @ B1( ) = O(j j3); @ B1( ) = O(j j2): Using the computation for C 2 expansion, we nd that the estimates (5.42){(5.44) are valid P for ( (3 ))j+1 Pk11 j ( j+1 x). To estimate Pk12, notice that x1 = r + O(r2) on the curve x2 = 1 f 00(0)x2; x1 < 0. This 1 2 curve can be rewritten in polar coordinates: = 1 f 00(0)r + O(r2 ): 2 We take j = j f 00(0)=2 in order to make the curves k "1+ (1+ )k r2 close to the curves _ = " k r. Since B2( ) = (@=@ )B2( ) = 3 cos 3 + 1 cos 1 , we have 2 2 2 2 @ @3 @2 B2( ) = = @ B2( ) = = @ 3 B2( ) = = 0; @ 2 B2( ) = = 2; and _ B2( ) + " k rB2( ) = = 0; @ B ( ) + " rB ( ) = 2" k r; k _2 @ 2 = @ 2 B ( ) + " rB ( ) = 2; 2 k _2 2 @ = 3 @ _ B2( ) + " k rB2( ) j j<2" 6 C" k: @ 3 k Therefore _ B2( ) + " k rB2( ) = " k r " k r " k r @ hB ( ) + " rB ( )i 2 k _2 @ = h i @ B ( ) + " rB ( ) k _2 @r 2 = = (2" k r)( " k r) + 1 ( 2) ( " k r)2 + O(j" k rj3) 2 2 ); = O(j" k rj = 2" k r 2( " k r) + O(j" k rj2) = O(j" k rj2); _ = " k B2( ) = 32 " k r = " k O(j" k rj): Rewriting these estimates in terms of Pk12, we have jPk12(x)j = " r 6 C"2 2 r7=2; k k jrPk12(x)j = " r 6 C"2 2 r5=2: k k (5.45) (5.46) (5.47) From the de nition of Pk12 it is clear that jD2 Pk12(x)j 6 Cjxj X (3 X h X 1=2 for jxj < 2: Recall that k is "1+ (1+ )k r2 close to the curve = " k r. Using (5.45){(5.47), we can then derive the estimates (similarly to (5.39){(5.41)) ( r r ) )j+1 P 12 ( j+1 x) 6 C"2 (k+1)(1=2+ )jxj3=2; kj ( (3 ) )j+1Pk12 j ( j+1 x) 6 C"1+ (k+1)( + 1=2)jxj1=2; i ( (3 ) )j+1Pk12 j ( j+1 x) C ( k+1 \fjxj<Rg) 6 C"(1+ )(1 )R1=2 ; in the thin region bounded by k+1 . Combining all these estimates, we nd that j'k+1(x)j k+1 6 C"2; jr'k+1(x)j k+1 6 C"; [r'k+1]C ( k+1) 6 C"1 : (5.48) (5.49) By Theorem 2.2, we then have k'k+1 Gk+1 kLp(B1) 6 C"; where Gk+1 is given in Lemma 5.3. From this, we can argue in the same way as before (using Lemma 5.4 below in deriving H lder estimates for r('k+1 Gk+1 )) to conclude that, for o any < 1, j'k+1 Gk+1 j 6 C" ; jr('k+1 Gk+1)j 6 C" =(1+ ): Combining this with Lemma 5.3, we conclude that j'k+1 Pk+1 j 6 3 ; jr('k+1 Pk+1 )j 6 2 : We now proceed as in the proof of (5.28){(5.30) (with 2+ replaced by 3 ) to establish the estimate '0(x) = r3=2 A1B1( ) + A2B2( ) + r2A3B3( ) 33 h 1i +r5=2 A4B4( ) + A5B5( ) 2A2"f 00(0) cos 2 + O(r3 ); which, when written in terms of the original variables, becomes '(x) = r 3=2 A1B1( ) + A2B2( ) + r2A3B3( ) h + O(r3 ); 2 with di erent coe cients A3, A4, A5. This completes the proof of (1.14) (with = ). Remark 5.3. Theorem 5.1 is new even if is a parabola x2 = x2. 1 0 (0) 6= 0 then, in Theorem 5.1, the second derivative f 00 (0) should be Remark 5.4. If f replaced by the curvature at 0 and Bi( ) should be replaced by Bi( 0), where 0 = f 0(0). Remark 5.5. As in the case of Remark 5.1, the proof of Theorem 5.1 shows that (1.14) can be di erentiated, with r(O(r3 )) = O(r2 ): The same remark applies to (1.13), with r(O(r2 )) = O(r1 ) and the proof can be given by the method of x3 using (2.20). The next lemma establishes (as a special case) the estimate (5.21) which was needed in the proof of Lemma 5.2. +r5=2 A4B4( ) + A5B5( ) 2A2 f 00(0) cos 1 i Lemma 5.4 Let 0 < < 1, and let fj = fj (x1) (j = 1; 2) be curves satisfying: fj (0) = fj0 (0) = 0; [fj0 ]C [ 2;0] 6 1; x1 6 f1(x1) 6 f2(x2) 6 x1 for 2 < x1 < 0; where 0 < < 1. Let ' 2 H 2 (B2 n ff1(x1) 6 x2 6 f2(x2); x1 < 0g) satisfy (5.50) 2' = 0 in B2 n ff1(x1) 6 x2 6 f2(x2); x1 < 0g; jr'j 6 1 in B2; '(0) = 0; r'(0) = 0; [r']C (fjx2j6 x1 g\fjxj<;x1<0g) 6 1: Then, for any < 1=2, (5.51) (5.52) (5.53) (5.54) (5.55) j'(x)j 6 Cjxj1+ ; where the constant C depends only on and , but not on the fj and . Proof. From (5.52) and (5.53) we deduce that j'(x)j 6 jxj in B2: Next, (5.53) and (5.54) imply that (5.56) (5.57) jr'(x)j 6 jxj for jx2j 6 x1; x1 < 0 and jxj < 2; 34 and therefore j'(x)j 6 jxj1+ for jx2j 6 x1; x1 < 0 and jxj < 2: Consider the function jxj<1 (5.58) (5.59) (5.60) By (5.56), sup jZ (x)j < +1 for any > 0. We claim that jxj<1 '(x) Z (x) = jxj1+ + jxj 8 > 0: sup jZ (x)j 6 C for some constant C independent of , and fj ; once this is proved, we can then nish the proof of the lemma by letting ! 0. Suppose (5.60) is not true. Then there exist sequences ' = 'n , fj = fnj (j = 1; 2), n ! 0, xn ! 0, n ( n may go to 0) such that Cn = sup jZ n (x)j = jZ n (xn)j ! 1: jxj<1 De ne Gn ( ) by: 'n (x) = Cn(R1+ + nRn )Gn ( ) where x = Rn ; Rn = jxnj: n Then 1+ 1+ n jGn( )j 6 C (Rj'n(x)j R ) 6 j j (RRn ++ Rj jRn 6 j j1+ + j j; 1+ + 1+ nn nn n n) n and, from (5.58), (5.57) and (5.54) we also have 1 2 R1+ jGn ( )j 6 C (R1+ n + R ) j j1+ 6 C j j1+ for j 2j < n 1; j j 6 R ; nn nn n n R 1 j j1=2 for j j < ; j j 6 2 ; jr Gn ( )j 6 C (R n+ ) j j1=2 6 C 2 n1 R n R 1 n + n ) 6 Cn : Cn(Rn The curve x2 = fj (x1) (j = 1; 2) under the change of variables x ! becomes 1 2 2 = R fj (Rn 1 ); R < 1 < 0: n n Under the assumptions of the lemma, we have, for any K > 1, n 1 f (R ) 6 6 1 f (R )o \ fj j < Kg fj j 6 R K g; 2 2 1 n Rn 1 n 1 Rn 2 n 1 o n 1 f (R ) 6 6 1 f (R ) fj j 6 g: 2 2 n1 R 1 n1 R 2 n1 [r Gn ]C (fj 2j< n 1 g\fj j<Kg) 6 n n and, for any K > 0, n n n n 35 Therefore, for n = min(R K ; n), we have n n 1 f (R ) 6 6 1 f (R )o \ fj j < Kg fj j 6 g 2 2 n1 Rn 1 n 1 Rn 2 n 1 Just as in the proof of Lemma 3.2, we can now apply Theorem 9.3 (ii) with boundary given by the rays j 2j = n 1 (j j < K) to conclude that (for a subsequence and any K > 1) Gn ( ) ! G( ), where G( ) satis es 2G = 0 in R2 n S 0; G = @G = 0 from both sides of S0; @ jG( )j 6 j j1+ + j j in R2; x jG(e)j = 1 where e = lim Rn ; n (5.61) (5.62) (5.63) (5.64) and where S0 = f( 1; 0); 1 < 0g. This is a contradiction to the Liouville theorem stated in the following lemma. Lemma 5.5 If G satis es (5.61){(5.63), then G( ) 0. e Proof. The proof is similar to that of Lemma 3.3. Let G 2 H 2(B1 n S0) be a solution of e 2G = 0 in B1 n S0; e e G = @ G = 0 from both sides of S0; @ e e G = G; @ G = @G on @B1: @ @ e e Then jG Gj 6 Cj j and jr(G G)j 6 C near = 0. Applying the maximum principle e (Theorem 2.3) to G G in the domain B1 n (S0 [ B" ) and then letting " ! 0 we conclude e e that G G 0. Since G has an expansion at the origin, this expansion is valid also for G( ): 1 X G( ) = rk=2+1Bk ( ) (r = j j) (5.65) near = 0. Introduce the function Then k=1 H( ) = G( ) r3=2B1( ) r2B2( ): jH( )j 6 Cj j5=2 near = 0; jG( ) r3=2B1( )j 6 Cj j3=2 near = 1: 2 Since r2B2( ) = c 2 , @=@ 1[r2B2( )] = 0. The function @G @ ( ) = @H = @ @ [r3=2B1( )] @ 1 1 1 36 is biharmonic in R2 n S0 and satis es the same zero boundary conditions. It follows that jrj ( )j 6 Cj j5=2 and 1j 1j near = 0; j = 0; 1; 2; 3; near = 1 j = 0; 1; 2; 3: Now we can follow the proof of Lemma 3.3 to conclude that 0, which immediately implies that G 0. Remark 5.7. The preceding Liouville theorem does not follow from a general theorem of Kondrat v [13; Theorem 11] since one of the assumptions he makes, e Z jrj ( )j 6 Cj j3=2 1Z 1 0 is not satis es for any , in our case. Remark 5.8. The proof of Lemma 5.2 can be extended to the case where f 2 C 1+ to yield a di erent (although more complicated) proof of Theorem 3.1. In (5.16) we need to replace 2+ by 1+ , and (5.7), (5.8) need to be modi ed by using the fact that '0 is C 3=2 and applying (5.2) to = r'1 with = < 1=2. Finally, (5.9) follows from Lemma 5.4 with fj (x1) = j jx1j1+ ; 1 < 2 < 1 < 1. 0 r jGj2rdrd < 1; 6 The crack propagation model In this section we introduce a model of crack propagation. Let be a domain in R2, representing a homogeneous elastic body. Let u = (ui), " = ("ij ) and s = ( ij ) denote the displacement vector, the strain tensor and the stress tensor, respectively. The linear elasticity equations for homogeneous isotropic material consist of the constitutive law E ij = 1 + "ij + 1 2 "kk ij (6.1) and the equilibrium conditions @ = 0; (6.2) @x ij provided there are no body forces. Here E is the Young modulus, is the Poisson ratio, and the strain-displacement relations are given by 1 "ij = 2 (ui;j + uj;i); ui;j = @j ui: (6.3) Suppose there is initially a crack in , given by a non-intersecting curve 0 with initial point on @ and terminal point (the \crack tip") X0 = (x0; y0) inside . Under external forces the crack tip will generally propagate, and we shall denote it by X(t). The crack propagation problem consists of nding the displacement u and path X(t) such that j ij;j = 0 in n (t) 37 (6.4) where (t) = 0 [ fX = X(s); 0 6 s 6 tg; ij nj = 0 on (t) (no traction on (t) means both (t)) (6.5) (6.6) (6.7) sides of (t), (nj ) is the normal to the curve, ui = i on @1 ; ij nj = gi on @2 where @ is a disjoint union of @1 ; @2 , and an appropriate dynamical equation for X(t). Based on [7] [10] [19] [20] [21], Friedman and Liu [9] introduced the following dynamics: _ X(t) = v(t) v1 + v(t) J(X(t)) (6.8) 0 v1 v(t) where _ v(t) = jX(t)j; X(0) = X0: (6.9) Here 0, v1 are positive constants and J(X(t)) is described in terms of the J-integral J = (W ~ ~ Du)dl ns Z where is the strain energy density and 1 W = 2 ij "ij ~ = (si) = ( ij nj ) s is the traction vector; is a curve in n (t), initiating at Y and terminating at X, ~ is a n normal to , and dl is the arc element. It is well known [11] that J is independent of the path connecting Y to X. Denote by S"(X) the circle with center X and radius ", and set "(X(t)) = S" (X(t)) \ . By the path-independence property of the J-integral it follows that Z J(X(t)) lim (W ~ ~ Du)dl ns (6.10) "!0 is well de ned, and this is the function we use in (6.8); here ~ is the outward normal to the n circles S"(X(t)). Taking the absolute value in (6.8) we get _ 1 1 = v1 jX(t)j jJ(X(t))j; _ 0 v1 + jX(t)j or _ jX(t))j = v1 (jJ(X(t))j + 0)+ : jJ(X(t))j Hence (6.8) can also be written in the form _ X(t) = h(jJ(X(t))j)J(X(t)) 38 0 " (X(t)) (6.11) where 1 h(s) = vs (ss + 0)+ : (6.12) 0 Note that the crack cannot propagate unless jJ(X(t))j is larger than 0. In particular, if jJ(X(0))j 6 0, then the crack does not propagate, and (t) 0. Hence in the sequel we shall always assume that jJ(X(0))j > 0. As in [9] we can express the stress components in terms of the stress function ' (which is determined up to an additive linear function): 2 2 @ 2' 11 = @ ' = '22; 12 = @x@y = '12; 22 = @ ' = '11: @y2 @x2 Then the system (6.1){(6.6) becomes: 2' = 0 in n (t); ' = 0; @' = 0 on (t); from both sides, @n (t) = 0 [ fX(s); 0 6 s 6 tg: For de niteness we take boundary conditions (cf. [9]) ' = g; @' = h on @ : @n We nally recall that J(X(t)) = (J1(X(t)); J2(X(t))) can be computed in the form h i 1 2 lim Z Ji(X(t)) = 2E "!0 ( ')2ni 2~ i dl; s~ " (X(t)) (6.13) (6.14) (6.15) (6.16) (6.17) where s1 = '22n1 '12n2; s2 = '12n1 + '11n2; (6.18) ~ ~ 1 = ( '; ( ')c); 2 = ( ( ')c; ') and ( ')c is the harmonic conjugate of ' determined up to an additive constant (the constant disappears in the limit in (6.17). De nition. The crack problem, Problem (C), is the problem of solving the system (6.13){(6.18). 7 Reformulation of the crack propagation problem In this section we reformulate the crack problem by rst replacing the dynamic formulation (6.11) by a geometric condition, and then replacing the latter by the condition (1.17). 39 We assume that ('; ) form a solution to problem (C) with in C 1+ , and write J(t) = (J1(t); J2(t)) = (J1(X(t)); J2(X(t)): For simplicity we shall always assume that f(0) = f 0(0) = 0: (7.1) Lemma 7.1 1 3 '(x) = A1r3=2 cos 2 + 3 cos 2 + A2r3=2 sin 3 + sin 1 + G r3=2A B( ) + G; (7.3) 2 2 where G = O(r3=2+ ) (for any 0 < < 1 ): 2 2G = 0, by interior elliptic estimates Since jD2Gj = O(r 1=2+ ): Proof. Consider rst the case of the tip X(0). By Theorem 3.1, J(0) = (12E ) 36A2 + 4A2; 24A1A2 ; 1 2 2 (7.2) Since j 'j = O(r 1=2 ), we also have j( ')cj = O(r 1=2 ); as can be seen by writing ( ')c as a line integral of (( ')y ; ( ')x): Similarly j( G)cj = O(r 1=2+ ): From the above estimates we easily conclude that we can take G 0 in the calculation of the J-integral. Using complex variables z = x1 + ix2 = rei , we have 3 ' = A1r3=2 cos 2 + 3 cos 1 + A2r3=2 sin 3 + sin 1 ; 2 2 2 3=2 + Cz 3=2 + Dzz 1=2 + Dz 1=2z; = Cz where 1 C = 1 (A1 iA2); D = 2 (3A1 iA2): 2 Then (cf. [8; pp. 275{276]) ' = 6A1r 1=2 cos 1 + 2A2r 1=2 sin 1 = 2Dz 1=2 + 2Dz 1=2; 2 2 1 + 2A r 1=2 cos 1 = 2iDz 1=2 + 2iDz 1=2; ( ')c = 6A1r 1=2 sin 2 2 2 @ ( ') = 3A r 1=2 sin 1 + A r 1=2 cos 1 = iDz 1=2 + iDz 1=2; 1 2 @ 2 2 @ ( ')c = 3A r 1=2 cos 1 A r 1=2 sin 1 = Dz 1=2 Dz 1=2: 1 2 @ 2 2 40 It follows that Z ( ')2 n1dl = 1 (ei + e i )dl 2 Z h i2 = 2Dei =2 + 2De i =2 1 (ei + e i )d 2 Zh 2Dz 1=2 + 2Dz 1=2 i2 = 2 (2D2 + 2D 2) = 18 A2 2 A2: 1 2 (7.4) where is the circle r = ", < < , traced counterclockwise. Similarly Z Zh i2 2 n dl = 1=2 + 2Dz 1=2 1 (ei + e i )dl ( ') 2 2Dz 2i Z h i2 = 2Dei =2 + 2De i =2 1 (ei + e i )d 2i = 2 i(2D2 2D2) = 12 A1A2: (7.5) To evaluate ~, we compute s By integration by parts (both @'=@x1 and @'=@x2 vanishes on = ) we then get Z Z @ @' ; @ @' ( '; ( ')c)d ~ 1dl = ~ s @ @x1 @ @x2 Z @' ; @' @ '; @ ( ')c d = (7.6) @x2 @x1 @ @ Z @' @ ' + @' @ ( ')c d K + K ; = 1 2 @x2 @ @x1 @ Substituting the formulas obtained above into the various expression which appears in the integrand of K1, we get Z n 3 i Cei =2 Ce i =2 + i De i =2 Dei =2 K1 = 2 41 ~ = ('22n1 '12n2; '12n1 + '11n2); s 2 2 @ 2' @ 2' = @ ' cos @x @x sin ; @x @x cos + @ ' sin @x2 @x2 12 12 2 @ @' @ @' = 1 @ @x ; 1 @ @x r r 2 1 Next we compute @'=@x2 and @'=@x1. @' = @' + @' @x1 @z @z 3 Cz1=2 + Cz1=2 + Dz 1=2 + Dz1=2 + 1 Dz 1=2z + Dz 1=2z ; =2 2 @' = i @' @' @x2 @z @z 3 i Cz1=2 Cz1=2 + i Dz 1=2 Dz1=2 + 1 i Dz 1=2z Dz 1=2z : =2 2 Similarly + 1 i De i 3 =2 Dei 3 =2 iDe 2 h i = 2 3 CD 3 CD + D2 + D2 : 2 2 o i =2 + iDei =2 d K2 = K1 + K2 = 6 (CD + CD) = 3 (3A2 + A2); 1 2 and, together with (7.4), i 2h 2 J1 = (12E ) (18 A2 2 A2) 2(K1 + K2) = (12E ) (36A2 + 4A2): 1 2 1 2 In a similar way we compute Z Z @ @' ; @ @' ( ( ')c; ')d ~ 2dl = ~ s @ @x1 @ @x2 Z @' ; @' @ ( ')c; @ ' d = @x2 @x1 @ @ Z @' @ ( ')c + @' @ ' d K + K ; e e = 1 2 @x2 @ @x1 @ where Z n 3 i Cei =2 Ce i =2 + i De i =2 Dei =2 e K1 = 2 o + 1 i De i 3 =2 Dei 3 =2 Dei =2 De i =2 d 2 h 3 iCD 3 iCD iD2 + iD2i; = 2 2 2 and Z n 3 Cei =2 + Ce i =2 + De i =2 + Dei =2 e K2 = 2 o + 1 De i 3 =2 + Dei 3 =2 iDe i =2 + iDei =2 d 2 h i = 2 3 iCD + 3 iCD iD2 + iD2 : 2 2 It follows that e e K1 + K2 = 6 (iCD iCD) = 6 A1A2; 42 It follows that 3 Cei =2 + Ce i =2 + De i =2 + Dei =2 2 o + 1 De i 3 =2 + Dei 3 =2 Dei =2 De i =2 d 2 h i 3 = 2 2 CD 3 CD D2 D2 : 2 Z n (7.7) (7.8) and, together with (7.5), i 2h 2 e e J2 = (12E ) 12A1A2 2(K1 + K2) = (12E ) (24A1A2): (7.9) _ Remark 7.1. We denote by P the angle from the positive x-axis to X(t). If we rotate the coordinate system by an angle P , then the formula in Lemma 7.1 is valid in the new coordinate system. Therefore we expect the formula for J = (J1; J2) in the original coordinate system to be: i 2h J1 = (12E ) (36A2 + 4A2) cos P 24A1A2 sin P ; 1 2 (7.10) 2) h (1 (36A2 + 4A2) sin + 24A A cos i: J2 = 2E P 12 P 1 2 We shall now verify (7.10) directly. From the computations in Lemma 7.1, we get, J= Z ~ ( ')2 2 n Z Under the new coordinate system x01 = have @ @ c @ ( ') ; @ ' @ @ c: @ '; @ ( ') cos P x1 + sin P x2; x0 = sin P x1 + cos P x2, @' ; @' @x1 @x2 ! 2 we ~ = (cos ; sin ) n = cos( P ) cos P sin( P ) sin P ; cos( P ) sin P + sin( P ) cos P ; ! cos P ; sin P ; = cos( P ); sin( P ) sin P ; cos P @' ; @' = @x1 @x2 @' cos @' sin ; @' sin + @' cos P P @x01 @x02 P @x01 ! P @x02 @' @' cos P ; sin P = @x0 ; @x0 sin P ; cos P 1 2 ! ! Clearly @ @ c cos P ; sin P @ ( ') ; @ ' @ @ c sin P ; cos P @ '; @ ( ') ! @ @ c cos P ; sin P @ ( ') ; @ ' = @ '; @ ( ')c sin P ; cos P @ @ ! ; and the 2 matrix with the ' is invariant under the above change of coordinates. Substituting these relations into J, we get (7.10). Set (7.11) s = 0 [ fx1 = f(x2); 0 6 x2 6 sg 43 where for any 0 6 s 6 s0 and consider (6.13){(6.16) with n (t) replaced by n replaced by J (s) f 0(s) = J2(s) 1 s and with (6.8) (7.12) J(s) = J((s; f(s)): We shall refer to this problem as Problem (C0). Lemma 7.2 Problems (C) and (C0) are equivalent. Proof. For a solution (', X(t)) to problem (C), we have _ X2(t) = J2(X(t)) : _ X1(t) J1(X(t)) If we write X1(t) = s; X2(t) = f(s), then _ J ((s; f(s)) f 0(s) = X2(t) = J2((s; f(s)) _ 1(t) 1 X (7.13) with x2 = f(s) de ned by f(s) = X2 (X1 1(s)), which shows that ('; f(s)) forms a solution to problem (C0). Conversely, let ('; s ) be a solution to problem (C0) and de ne X(t) = (X1 (t); X2(t)) by _ X(t) = h(jJ(t)j)J(t) (7.14) b where J(t) = J(X(t)). Writing X1(t) = s or t = X1 1 (s), we introduce a function X2 = f(s) by b f(s) = X2(X1 1 (s)): Thus _ b f 0(s) = X2(t) = J2(s; f(s)) _ X1(t) J1(s; f(s)) b b which implies that f 0 at X1(t) agrees with f 0 at s, i.e., f and f de ne the same curve with di erent parameterizations. It follows that the J(t) in (7.14) is the J-integral for the tip X(t) of the curve de ned by f(s), 0 6 s 6 X1(t); hence ('; X(t)) is a solution to problem (C). We proceed to consider problem (C0), and denote by A1(s), A2(s), the coe cients in the asymptotic expansion of the solution about the tip X(s) = (x; f(s)). From Lemma 7.1, it is clear that J2 = tan P + g(A2=A1) (7.15) J1 1 tan P g(A2=A1) 44 6u 54 6u2 g(u) = 9 + u2 ; g0(u) = (9 + u2)2 : (7.16) If the curve in problem (C0) is given by (s; f(s)), then, at s, tan P = f 0(s): Substituting this into (7.15) we conclude that (7.12) is equivalent to A g A2 = 0 1 or, by (7.16), either A1 = 0 or A2 = 0: We shall henceforce assume that A2(0) = 0: (7.17) Since jJ(0)j > 0, A1(0) is necessarily 6= 0 and by continuity (assuming that jJ(s)j > 0) we get that A2(s) = 0; A1(s) 6= 0: (7.18) In particular: Theorem 7.3 ('; f(s)) is a solution to problem (C0) if and only if A2(s) 0: (7.19) Thus the crack problem is equivalent to the following: Problem (C1). Find a pair ('; f(s)) such that ' satis es (6.13){(6.16) with n replaced by n s , s as in (7.11), with (7.12) replaced by (7.19). Condition (7.19) implies that 3=2 3=2+ ' = rP A1(s)B1( P ) + O(rP ) (0 < < 1=2) in a neighborhood of the tip P = (s; f(s)). Consequently, as we approach P from n s along the tangent to s at P, ' K (K 6= 0); ' n ! 0 1=2 rP where n is the direction normal to , or, in terms of the stress , nn K (K 6= 0); n ! 0: (7.20) 1=2 rP This local behavior is used by some authors (e.g. [7; p. 433] [4]) to model the propagation of cracks developed by traction (and commonly called mode I, or opening mode ([16; p. 24]). Since, conversely, (7.20) implies (7.19), we have thus obtained a very interesting physical result: Theorem 7.4 In the modeling of the crack propagation problem, the conditions (7.13) and (7.20) are equivalent (assuming the crack is in C 1+ ). 45 where 8 Remarks on problem (C0) The results of xx3, 5 can be used to study the regularity of the coe cients Ai(s). As an example, we shall establish in this section the H lder continuity of A(s) = (A1(s); A2(s)). o We assume that (8.1) s : x2 = f(s); 1 6 s 6 ( > 0) is a C 1+ curve initiating on @ and contained in , with f(0) = 0; f 0(0) = 0; and set s = n s . Let (x; s) be the solution of 2 = 0 in s; = @ = 0 from both sides of s : @n = g; @ = h on @ @n where g; h are independent of s. By Theorem 3.1, if X(s) = (s; f(s)), 0 6 s 6 , (x; s) = jx X(s)j3=2A(s) B( arctan f 0(s)) + O(jx X(s)j2 ) for any such that + > 1=2. Set w(x; s) = (x; s) (x; 0) for s > 0: 1=p (8.2) (8.3) (8.4) (8.5) (8.6) 2 H 2( s ); (8.7) Lemma 8.1 For any su ciently large p, Z Proof. It is clear that, for any s > 0, j (x; s)j 6 Cjx (s; f(s))j3=2; jrx (x; s)j 6 Cjx (s; f(s))j1=2: It follows that 0 jw(x; s)jpdx 6 Cps: (8.8) jw(x; s)j 6 Cs3=2 for jx (s; 0)j 6 2s; jrxw(x; s)j 6 Cs1=2 for jx (s; 0)j 6 2s: Applying Theorem 2.3 in the domain 0 \fjx (s; 0)j > sg with replaced by fjx (s; 0)j = sg, we obtain the assertion (8.8). We shall use Lemma 8.1 and (8.7) to prove the following: 46 Theorem 8.2 Let for 0 < < 1=4. Then A(s) is H lder continuous: o kfkC1 6 F1; [f 0]C 6 F1; (8.9) jA(X(s)) A(X(b)) 6 C(s b) (0 6 b < s 6 ) s se s for any 0 < < =4 where C depends only on , F1 , F1; . e Proof. It su ces to take s = 0. Note that in (8.7), b jO(jx X(s)j2 )j 6 Cjx X(s)j2 where C depends on F1, Fp but is independent of s. It is also clear that jX(s)j 6 Cs. 1; p Therefore, for s 6 jxj 6 2 s, w(x) = A(X(0)) B( ) A(X(s)) B( arctan f 0(s)) + O(s ); = 1 1 (8.10) jxj3=2 22 We substitute (8.10) into (8.8) and then integrate over the region f s < jxj < 2 sg. By choosing p to be su ciently large, we conclude that, for any 2 (0; =2), e Z p p Cs =2 +Cs jA(X(s)) B( arctan f 0(s)) A(0) B( )jpd =2 1=p e 6 Cs : (8.11) Since can be chosen arbitrarily close to (but smaller than) =2, can be chosen arbitrarily e 0 (s)) B( )j 6 C[f 0] s , (8.9) close to (but smaller than) =4. Noting that jB( arctan f easily follows. A simple approach to solving problem (C1) is to introduce a family of curves Y = f(s) f(0) = f 0(0) = 0; [f 0]C [0;s0] 6 M1; n o where 0 < < 1=4. e For any f(s) 2 Y , let '(x; s) denote the solution of (6.13){(6.16) with n (t) replaced e e e e by Gs , where Gs is de ned as in (7.11) with f replaced by f. Writing 3=2+ '(x; s) = rP A1(s)B1( P ) + A2(s)B2( P ) + O(rP ) e 3=2 e h i (where 0 < < ), we introduce the functional M(f) = e Z s0 0 A2(s) ds; e 2 (8.12) (8.13) and consider the minimization problem: e min M(f) = M(f); f 2 Y: e f2Y 47 e Since the A2(s) are uniformly H lder continuous (by Theorem 8.2), a minimizing sequence o ee (fn; A2;n) has a uniformly convergent subsequence to a limit (f; A2). If the minimum in (8.13) is equal to zero, then A2 = 0 and so f is a solution to problem (C0). Thus we may view (8.13) as a relaxation of the crack propagation problem. It is not clear how to prove that the minimum in (8.13) is equal to zero. In a future paper, currently under preparation, we shall use the results obtained in the previous sections in order to rewrite the condition A2(s) = 0 as a relation between the curvature (s) at X(s) and leading coe cients in the expansions near X(s) of (x; s) and its tangential derivative. This relation should enable us to establish the existence of a solution of problem (C0). 9 Appendix: sub-Schauder estimates Let be a 2-dimensional bounded domain containing the origin, and j (j = 1; ; m) m [ 1+ arcs initiating at the origin and contained in . Set be C = j . For any small r denote by !(r) the largest arc on the circle jxj = r which is contained in n , and set ! = inf 0<r<r0 !(r) for some small r0 > 0, max( ; !) < ! 6 2 . It is easy to verify that there b b e is a unique solution = (e) of ! sin2(e ) = 2 sin2 ! such that 0 < ! (e) 6 ; (e) > 2 : ! e e! !1 Now let u be a solution of 2u = f in n ; u 2 H 2( n ); (9.1) u = @u = 0 from both sides of ; (9.2) @ where Z jfj 6 C0: (9.3) j=1 Theorem 9.1 (Kondrat v-Oleinik[14],[15]) If kukH 2( n ) 6 C1, then the solution u bee ! longs to C 1+ (e) in r0 -neighborhood of the origin, and ju(x)j 6 Cjxj1+ (!); jru(x)j 6 Cjxj (!); (9.4) (9.5) where the constant C depends only on C0 , C1 and . consists of a single C 1+ arc with one endpoint at the origin. Theorem 9.1 is then valid with (e) = 1=2. ! Example 2. is a C 1+ curve passing through the origin (and = 1 [ 2). In this case, ! can be taken arbitrarily close to (if r0 is small enough). Hence (e) can be taken e ! arbitrarily close to 1 and, in particular, for any " > 0, kukC2 "fjxj<r0g 6 C (9.6) 48 Example 1. if r0 is small enough; C depends only on the C 1+ norm of restricted to fjxj < r0g and on R bounds on jfj and kukH 2( n ). We shall establish a local version of this theorem whereby kukH 2( n ) is not assumed to be (uniformly) bounded by a constant C1 but, instead, kukL1( ) is (uniformly) bounded by a constant C1. Let is a 2-dimensional bounded domain, S an open C 1+ subarc of @ containing the origin in its interior, and 2u = f in ; u 2 H 2( ); @u u = @ = 0 on S; Z (9.7) (9.8) (9.9) jfj 6 C0; juj 6 C1 in : Let 0 be any subdomain of such that 0 [ S. Lemma 9.2 If (9.7){(9.9) hold, then kukH 2( 0) 6 C2 (9.10) where C2 is a constant depending only on 0; ; S; C0 and C1 , and the estimates (9.4) (9.5) hold for any (e) arbitrarily close to 1. ! Proof. For simplicity we may assume that the curve S is given by S : x2 = g(x1); 1 < x1 < 1; g(0) = 0; and = fx 2 B1; x2 > g(x1)g and 0 = B1=2 \ . For any 0 < r 6 1=4, let be a cuto function such that C = 1 for jxj < 1 2r; = 0 for jxj > 1 r; jr j 6 C ; jD2 j 6 r2 : r Since u 2 H 2( ), Theorem 9.1 implies that for any 0 < " < 1, u = O(d2 " ), Du = O(d1 " ) (and then also D2u = O(d " ) and D3 u = O(d 1 " ) for jxj < 1 r where d = d(x) = dist(x; S). We can therefore integrate by parts Z B1 uf = 4 Z = = Z B1 B1 B1 u u = 4 4 4 2 Z j uj j uj 2+ 2+ Z B1 ( 4u) u 4+2 2 Z Z Z B1 B1 Z u u u u(12 jr j B1 B1 uru r 4 Z Z 2 + 4 3 ) + 2 1 Z 4j uj2 >2 B1 u2(12jr j2 + 4 )2 49 B1 jruj2(8 jr j)2: B1 uru (4 3r ) Notice that Z B1 j uj = 4 2 X Z = X i;j i;j Z B1 B1 4uiiujj uij uij + 4 X Z >1 2 Z Z i;j Combining these two inequalities, and using also the fact that juj 6 C1, we obtain B1 B1 4jD2uj2 C (12 2 Z B1 uiiuj 4 3 j X Z i;j B1 uij uj 4 3 i B1 jruj2 2jr j2: 2 + 4 3 ) + jD uj 6 CC 4 2 2 2 1 Z CC12 + C Z jruj2 + C C : 6 r3 r2 10 B1 r Z B1 r nB1 2r jr j C Z B1 r jruj2 2jr j2 C Z u2; jruj 6 " jD uj " B1 r B1 r B1 r where the constant C is independent of ". Taking " = r=(32C ), we get Z Z Z 2 2 2 uj2 6 4 jD2 uj2 6 C (C1 + C2 ) + 1 jD2uj2 80 < r 6 1 ; jD r3 16 B1 r 4 B1 2r B1 which implies that Z 2 2 2 uj2 6 16C (C1 + C2 ) : jD r3 2 Z By embedding 2 2+ Taking r = 1=4 the proof of (9.10) is complete. The conclusions (9.4), (9.5) now follows from Theorem 9.1 (example 2) with = S. Remark 9.1. The proof of Lemma 9.2 extends to the case where S is replaced by as de ned at the beginning of this section; i. e., Theorem 9.1 is valid if the assumption kukH 2( n ) 6 C1 is replaced by the assumption juj 6 C1 in . The estimate (9.6) is a sub-Schauder estimates for C 1+ boundary. The next subSchauder estimates are for C 2+ boundary. B1 r where S is a C 2+ subarc of @ . Let u be a solution of Theorem 9.3 Let be a bounded domain and let 0 be a subdomain of with 0 [ S 2u = f in ; u 2 H 2( ); u = g; @u = h on S: @ (i) (C 2+ estimate) If kgkC2+ (S) < 1; khkC1+ (S) < 1; 50 Z jfjpdx < 1 (p > 1 1 ); then kuk C 2+ ( 0 ) 6 C kgk C 2+ (S) + khk C 1+ (S) + kfk Z Lp ( ) + c0 kuk L1 ( ) ; (ii) (C 1+ estimate) If kgkC1+ (S) < 1; khkC (S) < 1; then Z jfjdx < 1; kukC1+ ( ) 6 C kgkC1+ (S) + khkC (S) + jfjdx + c0kukL1( ) ; if 0 = then the constant c0 can be taken to be zero in both cases (i) and (ii). If S 2 C 4+ then the result is a consequence of [3; x9] (which is valid also for n-dimensional domains). Proof. By subtracting the special solution (2.4), we may assume without loss of generality that f(x) 0. Let y = (x) be the conformal mapping which attens the boundary S. Under our assumptions, 2 C 2+ up to the boundary. Setting w(y) = u(x), the equation 2u = 0 becomes y [k(y) y w(y)] = 0 (9.11) where k(y) = jrx (x)j2 is in C 1+ up to the boundary. Now apply [3; x9] to immediately conclude both (i) and (ii). Acknowledgment. The rst author is partially supported by National Science Foundation Grant DMS #9703842. The second and third authors are grateful for a partial support from the Institute for Mathematics and its Application during their visit there. The third author is partially supported by DGICYT Grant PB96-0614. References [1] S. Agmon, The Lp approach to the Dirichlet problem, I. Regularity theorems, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 405{448. [2] S. Agmon, Maximum theorems for solutions of higher order elliptic equations, Bull. AMS, 66 (1960), 77{80. [3] S. Agmon, A. Douglas and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial di erential equations satisfying general boundary conditions I, Comm. Pure Appl. Math., 12 (1959), 623{727. [4] G.J. Barenblatt, Fracture in solids as a free boundary problem, in Free Boundary Problems: Theory and Applications, I.I. Diaz, M.A. Herrero, A. Li~an and J.L. Vazquez n eds., Pitman Research Notes in Mathematics, No. 323, Longman, Essex, England (1995), pp. 20{39. 51 [5] M. Costabel and M. Dauge, Stable asymptotics for elliptic systems on plane domains with corners, Comm. PDE, 19 (1994), 1677{1726. [6] M. Dauge, Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics, No. 1341, Springer-Verlag, Berlin, 1988. [7] L.B. Freund, Dynamic Fracture Mechanics, Cambridge University Press, Cambridge, 1990. [8] A. Friedman, Partial Di erential Equations, Holt, Rinehart and Winston, New York, 1969 (Reprinted by Krieger). [9] A. Friedman and Y. Liu, Propagation of cracks in elastic media, Arch. Rational Mech. Anal., 136 (1996), 235{290. [10] A.A. Griffith, The phenomenon of rupture and ow in solid, Philos. Trans. Royal. Soc. London, A221 (1920), 163{198. [11] M. E. Gurtin, On the energy release rate in quasi-static elastic crack propagation, J. Elasticity, 9 (1979), 187{195. [12] B. Hu and L. Wang, A free boundary problem arising in electrophotography: solutions with connected toner region, SIAM J. Math. Anal., 23 (1992), 1439{1454. [13] V.A. Kondrat v, Boundary value problems for elliptic equations in domains with e conical and angular points, Trans. Moscow Math. Soc., 16 (1967), 209{292. [14] V.A. Kondrat v and O.A. Oleinik, Unimprovable estimates in H lder spaces for e o generalized solutions of the biharmonic equation, the Navier-Stokes system of equations, and the Von Karman system in non-smooth two-dimensional domains, Vestnik Moskovskogo Universiteta, Matematika, 38 (1983), 22{39. [15] V.A. Kondrat v and O.A. Oleinik, Boundary value problems for partial di erene tial equations in non-smooth domains, Uspekh, Mat. Nauk, 38 (1983), 3{76. [16] B. Lawn, Fracture of Brittle Solids, second edition, Cambridge University Press, Cambridge, 1993. [17] Maz'ya and B.A. Plameneveskii, Estimates in Lp and in H lder spaces and the o Miranda-Agmon maximum principle for solutions of elliptic boundary value problems with singular points at the boundary, Math. Nachrichten, 81 (1978), 25{82; English transl., Amer. Math. Soc. Transl. (2) 123 (1984), 1{56. [18] C. Miranda, Teorema del massimo modulo e teorema di esistenza e di unicita per it problema di Dirichlet in due variabili, Annali di Matematica (4), 46 (1958), 265{311. 52 [19] J.L. Rice, Mathematical analysis in mechanics of fracture, in Fracture, Vol. 2, ed/ H. Liebowitz, Academic Press, New York, (1968), 191{311. [20] L.I. Slepyan, Principle of maximum energy dissipation rate in crack dynamics, J. Mech. Phys. Solids, 41 (1993), 1019{1033. [21] H. Stump and K.C. Le, Variational formulation of crack problem for an elastoplastic body at nite strain, Z. Angew. Math. Mech., 72 (1992), 387{396. [22] D. Vasilopoulos, On the determination of higher order terms of singular elastic stress elds near corners, Numer. Math., 53 (1988), 51{95. 53
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Path: Minnesota >> IMA >> 96 Fall, 2008
Path: Minnesota >> IMA >> 96 Fall, 2008
Path: Minnesota >> IMA >> 96 Fall, 2008
Path: Minnesota >> IMA >> 96 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008
Path: Minnesota >> IMA >> 99 Fall, 2008