High-order surface relaxation versus the Ehrlich Schwoebel effect

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1 INSTITUTE OF PHYSICS PUBLISHING Nonlinearity 19 (006) NONLINEARITY doi: / /19/11/005 High-order surface relaxation versus the Ehrlich Schwoebel effect Bo Li Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, Mail code: 011, La Jolla, CA , USA Received February 006, in final form 8 September 006 Published 8 September 006 Online at stacks.iop.org/non/19/581 Recommended by C Le Bris Abstract We consider a class of continuum models of epitaxial growth of thin films with two competing mechanisms: (1) the surface relaxation described by high-order gradients of the surface profile and () the Ehrlich Schwoebel (ES) effect which is the asymmetry in the adatom attachment and detachment to and from atomic steps. Mathematically, these models are gradient-flows of some effective freeenergy functionals for which large slopes are preferred for surfaces with low energy. We characterize the large-system asymptotics of the minimum energy and the magnitude of gradients of energy-minimizing surfaces. We also show that, in the large-system limit, the renormalized energy with an infinite ES barrier is the Ɣ-limit of those with a finite one, indicating the enhancement of the ES effect in a large system. Introducing λ-minimizers as energy minimizers among all candidates that are spatially λ-periodical, we show the existence of a sequence of such λ-minimizers that are in fact equilibriums. For the case of a finite ES effect, we prove the well-posedness of the initial-boundary-value problem of the continuum model and obtain bounds for the scaling laws of interface width, surface slope and energy, all of which characterize the surface coarsening during the film growth. We conclude with a discussion on the implications of our rigorous analysis. Mathematics Subject Classification: 34K6, 49J45, 74G65, 74K30 PACS numbers: Ct; Jk; Aa /06/ $ IOP Publishing Ltd and London Mathematical Society Printed in the UK 581

2 58 BLi 1. Introduction In the form of mass balance, a continuum model of epitaxial growth of thin films is given by t h + j = F, (1.1) where h = h(x, t) with x = (x 1,x ) is the coarse-grained height profile of the film surface at time t, j = j( h, h,...)is the surface current or flux which depends only on gradients of h but not explicitly on x due to the translational invariance in h and x, and F is the mean deposition flux which we assume to be a positive constant [4, 4]. Here, we neglect the noise in the deposition flux. The current j describes microscopic processes in the growth that determine macroscopic properties of films and growth scaling laws. It can include many different processes and mechanisms. In this work, we consider two such important processes and mechanisms that compete with each other. The first one is a high-order surface relaxation, i.e. surface relaxation described by highorder gradients of the height profile h. This process smoothens the surface in general. The current due to such relaxation is given by j RE = ( 1) m M m m 1 h, (1.) where m is an integer, M m the mobility which is taken to be a positive constant here, the gradient and the Laplacian. Here, we assume that the surface current is isotropic often an idealized situation in the growth of a crystalline surface. For m =, the current (1.) is the Herring Mullins term for the isotropic surface diffusion [11, 3]. It can be derived for some cases from a Burton Cabrera Frank (BCF) type model [6, 17, ]. The surface relaxation with m = 3in(1.), together with lower-order terms, has been suggested to model the homoepitaxy of Fe(001) at room temperature [33]. The physical origin of the relaxation with m 3 remains unclear and a satisfactory derivation of such a term seems to be challenging. The second one is the adatom (adsorbed atom) attachment detachment with the Ehrlich Schwoebel (ES) effect: in order to stick to an atomic step, an adatom from an upper terrace must overcome an energy barrier the ES barrier in addition to the diffusion barrier on an atomistically flat terrace [7, 9, 30]; cf figure 1. The ES effect generates an uphill current that destabilizes nominal surfaces (highsymmetry surfaces) but stabilizes vicinal surfaces (stepped surfaces that are in the vicinity of high-symmetry surfaces) with a large slope, preventing step bunching [7, 30, 36]. This is the origin of the Bales Zangwill instability, a diffusional instability of atomic steps [3]. It also affects the island nucleation [16]. With the ES effect, the film surface prefers a large slope. The competition between this large-slope preference and the surface relaxation determines the large-scale surface morphology and growth scaling laws [, 10, 1, 0, 5, 7, 31, 36]. The exact form of the current induced by an infinite ES barrier was first proposed in [36] and that by a finite ES barrier in [1]; see also [15, 6, 31]. These forms are F h for an infinite ES barrier, h j ES = FS c σ (1.3) h for a finite ES barrier, 1+α σ h where S c > 0 is a constant measuring the strength of the ES effect, σ>0 the nucleation length and α>0 an interpolation constant a fitting parameter [1]. For the case of an infinite ES barrier, the current is also given in [1] in a slightly altered form (without the factor 1/).

3 High-order surface relaxation versus the Ehrlich Schwoebel effect 583 V Figure 1. The Ehrlich Schwoebel barrier. Now, the total surface current j is the sum of j RE and j ES : j = j RE + j ES. (1.4) Using the co-moving frame which is equivalent to the change in variable h Ft h, we obtain from (1.1) (1.4) the growth equations with a high-order surface relaxation and the ES effect: ( ) F t h = ( 1) m 1 M m m h h for an infinite ES barrier, (1.5) h ( t h = ( 1) m 1 M m m FSc σ ) h h 1+α σ h for a finite ES barrier. (1.6) Setting H(X,T) = ηh(x, t), X = ξx, T = ζt, (1.7) with ( ) F 1/(m) ξ =, η = 1, ζ = M m ξ m, for an infinite ES barrier, M m ( FSc σ ) 1/(m ) ξ =, η = ασξ, ζ = M m ξ m, for a finite ES barrier, M m we obtain from (1.5) and (1.6) the following equations for the rescaled height H = H(X,T) but written using h = h(x, t) instead for convenience: ( ) h t h = ( 1) m 1 m h for an infinite ES barrier, (1.8) h ( ) t h = ( 1) m 1 m h h for a finite ES barrier. (1.9) 1+ h We shall consider these equations in d-dimensional Euclidean space R d for some d 1 with a periodical boundary condition. Let R d be the open periodical cell, an open cube in R d with its faces parallel to the coordinate planes. Denote by the closure of. If h = h(x, t) is smooth, -periodical in x, and satisfies (1.8) or(1.9), then d h(x, t) dx = 0, (1.10) dt i.e. the mass is conserved.

4 584 BLi Let us denote for any integer k 0 and any function u which has all the derivatives up to order k { k/ u if k is even, W k (u) = (1.11) (k 1)/ u if k is odd, where 0 u = u. We can verify that equations (1.8) and (1.9) with the -periodical boundary condition are formally the gradient flows of the following effective free-energy functionals, respectively: [ ] 1 I (h) = W m(h) log h dx for an infinite ES barrier, (1.1) [ 1 J (h) = W m(h) 1 ] log(1+ h ) dx for a finite ES barrier. (1.13) Here and below, u dx = 1 u dx (1.14) E E E denotes the mean value of a Lebesgue integrable function u : E R with E a d-dimensional Lebesgue measurable set with a finite Lebesgue measure E > 0. It is clear by (1.1) or(1.13) that a low-energy profile should have a large slope which is balanced by the relaxation term. The surface slope should thus increase with time during the dynamics governed by (1.8)or(1.9). Our main results are as follows. (1) For each of the energy functionals I and J, the infimum is attained in a suitable Sobolev space. Moreover, if L>0is the linear size of the periodical cell, then the minimum energy for I or J is (m 1) log L + O(1) as L, and for any energy minimizer h of I or J, k h dx = O(L m k ), k = 0,...,m, as L, where k represents all the derivatives of order k, cf (.1). See theorem.1 and corollary.1. () If L is the linear size of the periodical cell, then there exists an L/n-minimizer of the energy functional I or J for each integer n 1, and such an L/n- minimizer is in fact an equilibrium solution in the case of a finite ES barrier. Here, we define a λ- minimizer for λ>0to be an energy-minimizer among all admissible functions that are [0,λ] d -periodical. See definition.1 and theorem.. (3) As the system size increases, a new energy functional renormalized from I is the Ɣ-limit of those renormalized from J. See theorems 3.1 and 3.. (4) The initial-boundary-value problem of the growth equation (1.9) with a periodical boundary condition is well-posed. See theorem 4.1. (5) If h = h(x, t) is a smooth, -periodical solution of (1.9), then w h (t) Ct 1/, t k h(x, τ) dx dτ Ct (m k)/(m), k = 1,...,m, t 0 I (h(,t)) m 1 m log t + C,

5 High-order surface relaxation versus the Ehrlich Schwoebel effect 585 where w h (t) is the interface width of h (cf (5.1)) and C is a generic constant that is independent of h, and t. See theorem 5.1. If h = h(x, t) is a solution of (1.6) with the given parameters M m, F, S c, σ and α, then we find that for any t 0 0 and t>t 0 large enough FSc w h (t) t + C, α t FSc W k (h(x, τ)) dx dτ t 0 αmm k/(m) t m k m + C, k = 1,...,m, where C>0 is a constant independent of h, and t. See corollary 5.1. Part (1) and Part () above are parallel in part to the related results in [0]. They are obtained by studying the rescaled, singularly perturbed energy functionals; cf (.7) and (.8). Our results here, however, are valid for any integer m and for both the functionals I and J. These results are optimal and their proofs are much refined. Note that the term log h in the energy functional I defined in (1.1) has to be treated with care. The concept of λ- minimizers, though first introduced here, has been used implicitly in [0, 5] to predict scaling laws for the coarsening dynamics in epitaxial growth. See more discussions in section 6. It is important to treat the spacial case of an infinite ES barrier which is studied in [10] to obtain the scaling laws. Our Ɣ-convergence result in Part (3) indicates that the ES effect is enhanced in a large system. This is because the slope of a film surface can become large and hence the ES instability can be fully developed only in a large system. Consequently, a finite ES barrier can be regarded effectively as an infinite one in a large system. Some of the techniques used in proving the results in Parts (1) (3) are developed in our recent work [1]. The well-posedness in Part (4) provides a basis for any of the growth scaling laws to be mathematically meaningful. At this point, the well-posedness of the initial-boundary-value problem for the equation (1.8) is not obtained. Finally, our bounds in Part (5) are only one-sided. Two-sided bounds can never be valid for steady-state solutions and hence for all the solutions. Our method for proving the upper bounds in Part (5) is different from that in [13, 14]. In our case, there are two independent quantities: one is the surface height and the other the lateral size of mounds (or, independently, the surface slope). Moreover, the surface slope can increase and the energy can decrease unbounded with respect to the increase in the system size. Our argument is elementary and is based on observations on the relation between the height profile and its gradients. The rest of this paper is organized as follows: in section, we study the large-system asymptotics as well as λ-minimizers of the energy functionals (1.1) and (1.13); in section 3, we show the Ɣ-convergence of renormalized energies; in section 4, we prove the well-posedness of the initial-boundary-value problem of equation (1.9); in section 5, we give bounds on the interface width, gradients and energy for solutions of equation (1.9); finally, in section 6, we further discuss the result of our analysis.. Energy minimization In this section, we study the asymptotics of ground states and special equilibriums, the λ-minimizers. These variational properties can be used to predict scaling laws; cf section 6. Let us fix a cube R d as above. Denote by Cper () the set of all real-valued, - periodic, C -functions on R d. For any integer k 1, let Hper k () be the closure in the usual Sobolev space H k () = W k, () of the set of all functions in Cper () restricted onto [1,9].

6 586 BLi As usual, we denote H 0 () = L (). By (1.10), we can always assume that the constant mean-value of h over is in fact 0. Thus, we introduce { } H k per () = u Hper k () : u dx = 0. It is clear that H k per () is a closed subspace of H per k (). For any integer k 1 and any u : R that has all the weak derivatives of order k,we define k u by k u = β u, (.1) β =k where β in the sum is a d-dimensional index. As usual, 0 u = u for any function u. Note that for d. By the Poincaré inequality, there exist constants K 1 (k, ) > 0 and K (k, ) > 0 such that K 1 (k, ) u H k () k u L () K (k, ) u H k () u H k per (). (.) By integration by parts, we have (cf lemma 3.1 in [19]) u dx = u dx u Hper (). Thus, it follows from (1.11) and (.) that there exist constants K 3 (k, ) > 0 and K 4 (k, ) > 0 such that K 3 (k, ) u H k () W k (u) dx K 4 (k, ) u H k () u H k per (). (.3) Associated with the integral in (.3) is the bilinear form A : Hper k () H per k () R, defined by { A k (u, v) = k/ u k/ v dx if k is even u, v H (k 1)/ u (k 1)/ per k (). (.4) v dx if k is odd Clearly, A k (u, u) = W k (u) dx u Hper k (). (.5) To study the large-system asymptotics, we need to scale the underlying domain in the definition of the functionals I and J (cf (1.1) and (1.13)) to a fixed domain. Thus, assuming = (0,L) d and setting ε = 1/L, we rescale the energy functionals (1.1) and (1.13) toget I(ĥ) = I ε (h) and J(ĥ) = J ε (h) with h(x) = εĥ( ˆx) and x = ε ˆx, (.6) where [ ε (m 1) ] I ε (h) = W m (h) log h dx, (.7) 1 [ ε (m 1) J ε (h) = W m (h) 1 1 log ( 1+ h )] dx, (.8) and 1 = (0, 1) d is the unit cube in R d.

7 High-order surface relaxation versus the Ehrlich Schwoebel effect 587 Throughout the rest of the paper, we fix the integers d 1 and m and the constant L>0. We also use the notation = (0,L) d, 1 = (0, 1) d, H() = H m per (), H = m H per ( 1). (.9) The following theorem gives the optimal asymptotics of the minimum energy and magnitude of gradients of energy minimizers for both the singularly perturbed functionals (.7) and (.8). Theorem.1. (1) For any ε>0, the infimum of I ε : H R { } and that of J ε : H R are finite and attained. Moreover, for any ε (0, 1] (m 1) log ε + min u H J 1(u) min h H J ε(h) min h H I ε(h) = (m 1) log ε + min u H I 1(u). (.10) () There exist constants C 1 > 0, C > 0 and ε 0 (0, 1], all depending only on d and m, such that for any energy minimizer h H of I ε : H R { } or J ε : H R and any ε (0,ε 0 ], C 1 ε 1 m k h L ( 1 ) C ε 1 m, k = 0,...,m. To prove this theorem, we recall the following result (cf lemma 3.1 in [1]) that shows the lower semicontinuity of the logarithmic part of the energy functional (.7) and the finiteness of such energy of a limiting function. Lemma.1. Let E R d be Lebesgue measurable with 0 < E <. Suppose g j g in L 1 (E) and { E log g j dx} j=1 is bounded. Then, log g L1 (E) and ( ) lim inf log g j dx log g dx. (.11) j Proof of theorem.1. E (1) Fix ε>0. It follows from (.), (.3) and the fact that for m there exits a constant C 3 = C 3 (d, m) > 0 such that h dx C3 1 W m (h) dx 1 h H. (.1) Since (1/s) log(1+s) 0ass, there exists R = R(d, m, ε) 1 such that log(1+s) ε (m 1) s/(c3 ) s R. Consequently, we have by (.1) and (.3) that I ε (h) J ε (h) = 1 ε (m 1) ε(m 1) ( W m (h) dx E + {x 1 : h <R} W m (h) dx 1 1 log(1+r ) ε(m 1) {x 1 : h R} 4C 3 ) 1 log(1+ h ) dx 1 h dx ε(m 1) (K 3 (m, 1 )) h H 4 m ( 1 ) 1 log(1+r ) h H. (.13) Set µ ε = inf h H I ε (h) and ν ε = inf h H J ε (h). Clearly, both µ ε and ν ε are finite. Let {h j } j=1 and {g j } j=1 be infimizing sequences of I ε : H R { } and J ε : H R,

8 588 BLi respectively. It follows from (.13) that both these sequences are bounded in H. Thus, up to subsequences, h j h ε in H and h j h ε in H 1 ( 1 ) and g j g ε in H and g j g ε in H 1 ( 1 ), for some h ε H and g ε H, respectively, where the symbol and denote the weak and strong convergence, respectively. It is easy to see from (.3), (.7) and (.13) that { 1 log h j dx} j=1 is bounded. Thus, by the fact that W m(h) is quadratic and convex in h and by lemma.1, µ ε = lim inf j I ε(h j ) I(h ε ) µ ε. (.14) By the fact that log(1+s) s for all s 0 and the Cauchy Schwarz inequality, we have [log(1+ g j ) log(1+ g ε )]dx = log (1+ 1 g j g ε ) dx 1 1+ g ε g j g ε 1+ g ε dx ( g j L ( 1 ) + g ε L ( 1 )) g j g ε L ( 1 ) 0 1 as j. Consequently, ν ε = lim inf j J ε(g j ) J(g ε ) ν ε. (.15) The attainment of the infimum of I ε : H R { } and that of J ε : H R now follow from (.14) and (.15). Fix ε (0, 1]. For each h H, let u = ε m 1 h H. Then, we have by (.7) and (.8) that (m 1) log ε + J 1 (u) J ε (h) I ε (h) = (m 1) log ε + I 1 (u), leading to (.10). () Let first h H be a minimizer of I ε : H R { }. The function ξ(s) := I ε (h+sh), s ( 1/, 1/), is smooth and attains its minimum at s = 0. Thus, ξ (0) = 0, i.e. 1 ε (m 1) W m (h) dx = 1. This, together with (.3), implies that h H m ( 1 ) C 4 ε 1 m, (.16) where C 4 = 1/K 3 (m, 1 )>0. Now, applying Jensen s inequality to the convex function log( ) and using the upper bound in (.10) with C 5 := min u H I 1 (u) R, wehave C 5 + (m 1) log ε = I ε (h) 1 1 log h dx 1 ( ) log h dx. 1 This, together with an integration by parts, the Cauchy Schwarz inequality, (.16) and the fact that m, implies e C 5 ε (1 m) h dx = ( h) h dx 1 1 ( ) 1/ ( ) 1/ ( ) 1/ h dx h dx C 4 ε 1 m h dx. (.17) Therefore, h L ( 1 ) C 1 4 e C 5 ε 1 m. (.18)

9 High-order surface relaxation versus the Ehrlich Schwoebel effect 589 Let now g H be a minimizer of J ε : H R. Thus, the first variation of J ε at g vanishes: δj ε (g)(ĝ) = 0 for any ĝ H. In particular, δj ε (g)(g) = 0. Therefore, ε (m 1) 1 W m (g) dx = This, together with (.3), leads to 1 g g g H m ( 1 ) C 4 ε 1 m (.19) with the same constant C 4 as in (.16). Consequently, applying Jensen s inequality to log( ) and using the upper bound in (.10) with C 5 = min u H I 1 (u) R, we obtain C 5 + (m 1) log ε J ε (g) 1 1 log(1+ g ) dx 1 ( ) log 1+ g dx, 1 leading to g dx 1 1 e C 5 ε (1 m) if 0 <ε ε 0 := (e C 5 ) 1/((1 m)). By this and (.19), and by the argument similar to that in (.17) and (.18), we have g L ( 1 ) (C 4 ) 1 e C 5 ε 1 m ε (0,ε 0 ]. (.0) Now Part () of the theorem follows from (.16), (.18) (.0) and (.), with C 1 = (C 4 ) 1 e C 5 min(1,k 1 (1, 1 ),...,K 1 (m, 1 )) > 0, C = C 4 max(1,k (1, 1 ),...,K (m, 1 )) > 0. ED The following result on the ground states for the original functionals I : H() R { } and J : H() R is a direct consequence of theorem.1 and the change in variables (.6); cf (.9) for notation. Corollary.1. (1) The infimum of the energy functional I : H() R { } and that of J : H() R are finite and attained. Moreover, for any L 1, (m 1) log L + min u H J 1(u) min J (h) min h H() I (h) = (m 1) log L + min I 1(u). h H() u H () Let C 1, C and ε 0 be the same as in theorem.1. We have for any energy minimizer h H() of I : H() R { } or J : H() R and any L 1/ε 0 that ( 1/ C 1 L m k k h dx) C L m k, k = 0,...,m. We now give the definition of λ-minimizers and prove their existence for the functionals I : H() R { } and J : H() R. See section 6 for more discussions on the related result. Let n 1 be an integer, λ = L/n and λ = (0,λ) d. Define { H λ () = h H() : there exist φ k Cper ( λ), k = 1,..., such that φ k h in H m () }. Definition.1. A function h H() is a λ-minimizer of I : H() R { } (or J : H() R), if h H λ () and I (h) I(g) (or J (h) J(g)) g H λ ().

10 590 BLi Theorem.. For any integer n 1, there exist h n H() and g n H() that are L/n-minimizers of I : H() R { } and J : H() R, respectively. Moreover, g n C () and g n is an equilibrium solution of equation (1.9), i.e. ( ( 1) m m g n + g n 1+ g n ) = 0 in. (.1) Proof. Fix an integer n 1 and let λ = L/n. Define Jˆ n : H() R by [ Jˆ n (m 1) n (ĝ) = W m (ĝ) 1 log ( 1+ ĝ )] dx ĝ H(). By the proof of theorem.1, we see that there exists a global minimizer ĝ n H() of Jˆ n : H() R. Now, by the definition of the space H(), we can extend ĝ n to be almost everywhere -periodical on R d, in the sense that ĝ n (x + Le j ) =ĝ n (x) for a. e. x R d and for any unit coordinate vector e j (j = 1,...,d). Define g n (x) = (1/n)ĝ n (nx) for x R d.it is easy to see that g n H λ (). Moreover, for any g H λ (), we can extend g to be almost everywhere λ -periodical on R d and define ĝ by the relation g(x) = (1/n)ĝ(nx) for x R d. Clearly, ĝ H(), when ĝ is restricted onto. Further, we can verify that J(g) = ˆ J n (ĝ) ˆ J n (ĝ n ) = J(g n ). Thus, g n is an L/n-minimizer of J : H() R. A similar argument shows the existence of an L/n-minimizer for I : H() R { }. Note that ĝ n H() Hper m () is in fact a global minimizer of Jˆ n : Hper m () R, since J ˆ(h) = J(h ˆ h) and h h H() for any h Hper m (), where h is the mean value of h over, cf(1.14). Thus, by simple calculations, we see that ĝ n is a weak solution of the following equation: ( ) n (m 1) ( 1) m m ĝ n ĝ n + = 0 in. (.) 1+ ĝ n Equivalently, g n is a weak solution of (.1). Notice that k g n = ( k 1 g n ) for 1 k m. Thus, by the regularity theory of elliptic problems and the standard boot-strapping argument, we see that g n is smooth and satisfies equation (.1) pointwise. ED 3. Γ-convergence of renormalized energies In this section, we present our mathematical results using the notion of Ɣ-convergence. These results indicate that, for a large system, the energy functional for a finite ES barrier is close to that for an infinite ES barrier. We in fact prove a result that is stronger than the usual Ɣ- convergence: any sequence of energy minimizers has a subsequence that converges strongly to a minimizer of the Ɣ-limit functional; cf theorem 3.. We define for each ε>0the renormalized energy functionals I ε (u) = I ε (ε 1 m u) (m 1) log ε = J ε (u) = J ε (ε 1 m u) (m 1) log ε = 1 1 [ ] 1 W m(u) log u dx, (3.1) [ 1 W m(u) 1 log ( ε (m 1) + u ) ] dx. (3.) Note that I ε = I 1 is independent of ε. For convenience, we shall write I = I ε.

11 High-order surface relaxation versus the Ehrlich Schwoebel effect 591 Theorem 3.1. The energy functionals J ε : H R (0 <ε 1) Ɣ-converge to I : H R { } as ε 0 with respect to the weak convergence in H. The precise definition of the Ɣ-convergence in the theorem is as follows [5]: for any decreasing sequence {ε j } j=1 in (0, 1] such that lim j ε j = 0, the following hold true: (1) if u j uin H, then lim inf j J εj (u j ) I(u); (3.3) () for any v H, there exist v j H (j = 1,...)such that v j vin H and Proof. lim J εj (v j ) = I(v). (3.4) j (1) Let u j uin H. We may assume that lim inf j J εj (u j )<, for otherwise (3.3) holds trivially. We may further assume, up to a subsequence, that lim inf j J εj (u j ) = lim J εj (u j )<. (3.5) j Since 0 <ε j 1, J εj (u j ) J 1 (u j ) for all j 1. Thus, by (.13) with ε = 1 and (3.5), the sequence { J εj (u j )} j=1 is bounded and {u j } j=1 is bounded in H. Consequently, by (.3), { } { } log ε (m 1) 1 j + u j dx = 1 1 W m(u j ) dx J εj (u j ) (3.6) j=1 is bounded. Moreover, since {u j } j=1 is bounded in H and m, up to a further subsequence, u j u in H 1 ( 1 ). Thus, since ε (m 1) j + u j u = ε (m 1) j + u j + u u ε (m 1) j + u j ε (m 1) j + u j u j 1, we have ε (m 1) j + u j u in L ( 1 ) as j. Therefore, it follows from lemma.1 that log u L 1 ( 1 ) and ( ) lim inf log ε (m 1) j + u j dx log u dx. (3.7) j 1 1 Since W m ( ) is quadratic and convex, we also have by u j uin H that lim inf W m (u j ) dx W m (u) dx. (3.8) j 1 1 Now, (3.3) follows from (3.5), (3.7) and (3.8). () Let v H and v j = v for all integers j 1. It follows from lemma.1 that lim log ε (m 1) j + v dx = log v dx. j 1 1 This implies (3.4). ED j=1 Corollary 3.1. We have lim J ε (u) = min I(u). (3.9) min ε 0 u H u H

12 59 BLi Proof. Let {ε j } j=1 be any decreasing sequence in (0, 1] with lim j ε j = 0. For each j 1, let u j H be a minimizer of J εj : H R. The existence of such a minimizer follows from theorem.1. By (.8) and (3.), ε 1 m j u j is a minimizer of J εj : H R for each j 1. Thus, by Part () of theorem.1, {u j } j=1 is bounded in H. Hence, it has a subsequence, not relabelled, such that u j uin H for some u H. Now,by(3.3), we obtain that lim inf j min J εj (w) = lim inf J εj (u j ) I(u) min I(w). (3.10) w H j Let v H be a minimizer of I : H R { }. By theorem 3.1, there exist v j H (j = 1,...)that satisfy (3.4). Thus, lim sup j w H min J εj (w) lim sup J εj (v j ) = I(v) = min I(w). (3.11) w H j w H Now, (3.9) follows from (3.10), (3.11) and the arbitrariness of {ε j } j=1. ED Theorem 3.. Let {ε j } j=1 be a decreasing sequence in (0, 1] such that lim j ε j = 0. For each integer j 1, let u j H be a minimizer of J εj : H R. Then, there is a subsequence of {u j } j=1, not relabelled, and a minimizer u H of I : H R { } such that u j u (strong convergence) in H. Proof. It follows from (.8) and (3.) that ε 1 m j u j is a minimizer of J εj : H R for each integer j 1. Thus, by Part () of theorem.1, {u j } j=1 is bounded in H. Hence, it has a subsequence, not relabelled, such that u j uin H and u j u in H 1 ( 1 ) for some u H. Since J 1 (v) J εj (v) I 1 (v) for all v H, and since both min v H J 1 (v) and min v H I 1 (v) are finite by theorem.1, the sequence { J εj (u j )} j=1 = {min v H J εj (v)} j=1 is bounded. Therefore, the sequence in (3.6) is bounded. By the same argument as in the proof of theorem 3.1, we have that log u L 1 ( 1 ) and that (3.7) and (3.8) hold true. Consequently, by corollary 3.1, 0 = lim min j v H J εj (v) min I(v) v H lim inf J εj (u j ) I(u) j [ 1 lim inf j 1 [ ( + lim inf log j 1 0. W m(u j ) dx 1 1 W m(u) dx ε (m 1) j + u j dx ] ) ( )] log u dx 1 This implies that I(u) = min v H I(v), i.e. u H is a minimizer of I : H R { }. By (3.7), (3.8) and (3.1), we have lim inf W m (u j ) dx = W m (u) dx. j 1 1 (3.1) Therefore, up to a further subsequence of {u j } j=1, still not relabelled, we have lim W m (u j ) dx = W m (u) dx. (3.13) j 1 1

13 High-order surface relaxation versus the Ehrlich Schwoebel effect 593 Note by (1.11) and (.4) that W m (u j u) dx = W m (u j ) dx + W m (u) dx A m (u j, u), (3.14) where A m (, ) is the same as in (.4) but with k = m and replaced by 1. Thus, since u j uin H,wehaveby(3.13), (3.14), the definition of A m (, ) and (.5) that lim W m (u j u) dx = 0. j 1 This and (.3) imply that u j u in H. ED 4. Well-posedness We consider the initial-boundary-value problem of the d-dimensional growth equation (1.9) for h : R d [0,T] R that is -periodical: ( ) t h = ( 1) m 1 m h h in (0,T], (4.1) 1+ h h(,t)is -periodic for all t [0,T], (4.) h(x, 0) = h 0 (x) x, (4.3) where = (0,L) d as before, T > 0 and h 0 : R is a given function. We denote by Hper m() the dual space of H per m (). Clearly, the definition of a weak solution and theorem 4.1 below can be generalized to the equation (1.6) with the cube replaced by any d-dimensional parallelepiped with its faces parallel to the coordinate planes. Definition 4.1. A function h : [0,T] R is a weak solution of the initial-boundary-value problem (4.1) (4.3), if the following hold true: (1) h L (0,T; Hper m ()) and th L (0,T; Hper m()), () for any φ Hper m (), h φ, t h + A m (φ, h) φ, = 0 a.e. t (0,T), (4.4) 1+ h where, without confusion,, denotes the value of a linear functional at a function or the inner product of L (), and A m : Hper m () H per m () R is defined in (.4); (3) h(x, 0) = h 0 (x) for a.e. x. The following is the main result in this section. Theorem 4.1. Let h 0 Hper m (). Then, the initial-boundary-value problem (4.1) (4.3) has a unique weak solution h : [0,T] R. Moreover, if g : [0,T] R is the weak solution to (4.1) (4.3) with h 0 replaced by g 0 Hper m (), then g h L (0,T ;L ()) + g h L (0,T ;H m ()) C g 0 h 0 H m (), (4.5) where the constant C = C(m, d,, T ) > 0 is independent of g 0 and h 0.

14 594 BLi The proof of this theorem is similar to that in [19]. To be self-complete, we give here a shortened proof. We first need some preparations. Denote for each integer N 1 { ( ) ( ) } πk x πk x H N = span 1, cos, sin : 0 < k N, L L where k = (k 1,...,k d ), all k j 0 (j = 1,...,d) are integers and k = d j=1 k j. Notice that H N C per (). Denote also by P N : L () H N the L ()-projection onto H N, which is defined for any u L () by P N u H N and P N u u, φ =0 φ H N. We have for any integer k 0 that (cf lemma 3. of [19]) P N u H k () u H k () u Hper k (), N 1, (4.6) lim P Nu u H N k () = 0 u Hper k (). (4.7) In what follows, we denote by C a generic, positive constant that, unless otherwise stated, can depend on m, d,, T and h 0, but not on N. Lemma 4.1. Let h 0 H m per (). For each integer N 1, there exists a unique h N : [0,T] R such that (1) h N C ( [0,T]) and h N (,t) H N for any t [0,T]; () for any φ H N and any t (0,T], h N φ, t h N + A m (φ, h N ) φ, = 0; (4.8) 1+ h N (3) h N (, 0) = P N h 0 ; (4) t h N L (0,T ;L ()) + h N L (0,T ;H m ()) C. Proof. Let {φ j } r j=1 be an orthonormal basis of H N with respect to the inner product in L (), where r = dim H N. Let h N (x, t) = r j=1 a j (t)φ j (x) with all a j = a j (t) to be determined. Set φ = φ i in (4.8) and use the orthogonality of {φ j } r j=1 to obtain a i (t) = f i(a 1 (t),...,a r (t)), i = 1,...,r, (4.9) where all f i : R r R (1 i r) are smooth and locally Lipschitz. Set a i (0) = h 0,φ i, i = 1,...,r, (4.10) which is equivalent to Part (3). It follows from the theory for initial-value problems of ordinary differential equations that there exists T N > 0 such that the initial-value problem, (4.9) and (4.10), has a unique smooth solution (a 1 (t),...,a r (t)) for t [0,T N ]. Setting φ = h N (,t) H N in (4.8) and integrating against t, we get from (.5) and (4.6) that 1 t h N(,t) + W m (h N (x, τ)) dx dτ T N h 0 t [0,T N ]. (4.11) Here and below, we denote by the L ()-norm. By the orthogonality of {φ j } r j=1, we thus obtain that r [a j (t)] = h N (,t) T N + h 0 t [0,T N ]. j=1

15 High-order surface relaxation versus the Ehrlich Schwoebel effect 595 The solution (a 1 (t),...,a r (t)) of the initial-value problem, (4.9) and (4.10), is thus bounded on [0,T N ] and hence can be uniquely extended to a smooth solution over [0, ). Parts (1) (3) are proved. By (.3) and the Poincaré inequality, we have ) u H m () ( u C + W m (u) dx u Hper m (). (4.1) Thus, replacing T N by T in (4.11), we obtain that h N L (0,T ;L ()) + h N L (0,T ;H m ()) C. (4.13) Set now φ = t h N (,t)in (4.8) to get for any t [0,T] that t h N (,t) + d dt J(h N(,t))= 0, (4.14) where J( ) is defined in (1.13). Note by Part (3), (.3) and (4.6) that J(h N (, 0)) C P N h 0 H m () C h 0 H m (). (4.15) Consequently, integrating against t in (4.14), noting that ln(1+s ) s for all s 0, using (.3), (4.6) and (4.15) and applying Young s inequality, we obtain t 0 t h N (,τ) dτ + h N (,t) H m () CJ N (h N (, 0)) + C log(1+ h N (x, t) dx C + C h N (x, t) dx C + 1 This and (4.13) lead to Part (4). h N (x, t) dx t [0,T]. ED Proof of theorem 4.1. Let h N H N be defined as in lemma 4.1. Then, there exists a subsequence of {h N } N=1, not relabelled, and h L (0,T; Hper m ()) with th L (0,T; L ()) such that h N h in L (0,T; H m ()), (4.16) t h N t h in L (0,T; L ()), (4.17) h N h in L (0,T; H 1 ()), (4.18) where the strong convergence (4.18) follows from the combination of the weak convergence h N hin L (0,T; H ()) which results from the weak- convergence (4.16) and the fact that m, the weak convergence (4.17), and a usual compactness result (cf [34, theorem.1, chapter III]). Clearly, Part (1) of definition 4.1 is satisfied. Let ψ Hper m () and η C[0,T]. For each N 1, setting φ = P Nψ in (4.8), multiplying both sides of the resulting identity by η(t) and integrating against t, we obtain that T 0 [ η(t) P N ψ, t h N + A m (P N ψ, h N ) 0 P N ψ, h N 1+ h N ] dt = 0. (4.19) Sending N, we get by (4.7), (4.16) and (4.18) that T [ ] h η(t) ψ, t h + A m (ψ, h) ψ, dt = 0. (4.0) 1+ h

16 596 BLi Since η C[0,T] is arbitrary, this implies (4.4) with φ replaced by ψ. Part () of definition 4.1 is satisfied. It follows from a standard argument (cf [9, theorem, section 5.9]) that, after a possible modification of h on a set of measure zero, we have h C([0,T]; L ( )). Moreover, h(t) = h(s) + t s h (τ) dτ for any s, t [0,T], where h(t) = h(,t) L ( ) and h (t) = t h(,t). Replacing η(t) in (4.19) and (4.0) byη T (t) = t/t + 1, integrating by parts against t for the first terms in (4.19) and (4.0) and repeating the argument for the passage from (4.19)to(4.0), we get ψ, h 0 = lim N P Nψ, h 0 = lim P Nψ, h N (, 0) N = lim N = +η T (t) T 0 { 1 0 T P Nψ, h N (,t) [ A m (P N ψ, h N (,t)) P N ψ, T { [ 1 T ψ, h(,t) + η T (t) A m (ψ, h(,t)) = ψ, h(, 0) ψ H m per (). h N (,t) 1+ h N (,t) ψ, ]} dt ]} h(,t) dt 1+ h(,t) Part (3) in definition 4.1 is satisfied. Thus, h is a weak solution. Let now f = g h. Since g and h are two weak solutions, we have for any ψ Hper m () that h ψ, t f + A m (ψ, f ) ψ, 1+ h g = 0 for a.e. t (0,T). 1+ g Since f L (0,T; Hper m ()) and tf L (0,T; Hper m()), the mapping t f(,t) is absolutely continuous and d/dt f, f = f, t f ;cf[9, theorem 3, section 5.9], with H0 1(U) and H 1 (U) replaced by Hper m m () and Hper (), respectively. Setting ψ = f(,t),wehave 1 d dt f + A m (f, f ) = f h h g, 1+ h h g 1+ g g (4.1) for a.e. t (0,T). It is easy to verify for any vectors a,b R d that (cf [19]) ( a ) ( a (a b) 1+ a b b a b ) ( a ) = ( 1+ b )( 1+ a 1+ b ) + a b 1+ a + b. 1+ b Setting a = gand b = hin (4.1), we then deduce by an integration by parts, the Cauchy Schwarz inequality and Young s inequality that 1 d dt f + A m (f, f ) f = f f dx f f H m () K 3 f + K 3 f H m (), where K 3 = K 3 (m, ) is the same as in (.3). This, together with (.5), (.3) and the Gronwall inequality, leads to (4.5). ED

17 High-order surface relaxation versus the Ehrlich Schwoebel effect Bounds for scaling laws For any h :[0,T] L (), we define its interface width for any t [0,T]tobe w h (t) = h(x, t) h(t) dx with h(t) = h(x, t) dx. (5.1) The interface width is readily measurable in laboratory. It describes the fluctuation of surface height. Often it obeys a scaling law w h (t) t β for some constant β > 0 called the growth exponent [4, 4]. Another quantity that is also of experimentally interest is the characteristic lateral size of mounds formed during the growth of crystalline surfaces. This length increases with time and the system thus coarsens. Intuitively, the lateral size is determined by the surface slope and interface width. It is important to understand these scaling laws, since they are distinguished by microscopic properties of an underlying growth environment. While proving a strict scaling seems to be impossible (see the discussion in section 1), in this section we provide one-sided bounds of some of the scaling laws for the coarsening dynamics predicted by the underlying models. More discussions on our results are provided in section 6. Theorem 5.1. Let h :[0, ) Hper m () be a weak solution of the initial-boundary-value problem (4.1) (4.3)on[0,T] for any T>0. Let t 0 0. We have w h (t) (t t 0 ) +[w h (t 0 )] t t 0, (5.) t W k (h(x, τ)) dxdτ (1+ [w h(t 0 )] ) k/m (t t 0 +[w h (t 0 )] ) (m k)/m (t t 0 ) t 0 t >t 0, k = 1,...,m, (5.3) t E(h(τ)) dτ 1 t 0 log(1+(m )/m 3 1/m (t t 0 ) (m 1)/m ) t >t 0 +[w h (t 0 )]. (5.4) To prove theorem 5.1, we need the following lemma. Lemma 5.1. Let n 1 be an integer and A 0,A 1,...,A n be n +1positive real numbers. Assume A k A k pa k+p for p = 0,...,min(k, n k) and k = 0,...,n. (5.5) Then A n k An k 0 A k n for k = 0,...,n. (5.6) Proof. We prove (5.6) by induction on n. Clearly, (5.6) is true for both n = 1 and n =. Assume that n 3 and that (5.6) is true when n is replaced by any l with 1 l n 1, i.e. By (5.5), Thus, A l k Al k 0 A k l for k = 0,...,l and l = 1,...,n 1. (5.7) A n 1 A n A n. A (n 1) n 1 A n 1 n An 1 n.

18 598 BLi This, together with the inequality in (5.7) with l = n 1 and k = n, leads to Hence, A (n 1) n 1 A 0 A n n 1 An 1 n. A n n 1 A 0A n 1 n. (5.8) By the inequality in (5.7) with l = n 1, we have A n(n 1) k A n(n 1 k) 0 A nk n 1 for k = 0,...,n 1. This and (5.8) imply the inequality in (5.6) for 0 k n 1. For k = n, the inequality in (5.6) is trivially true. ED Proof of theorem 5.1. It follows from (1.10) and (5.1) that the spatial mean of the solution is in fact a constant: h = constant. Thus, by definition 4.1, h h is also a weak solution (with a different initial value) of the initial-boundary-value problem (4.1) (4.3) on[0,t] for any T > 0. Consequently, it follows from (5.1), Part () of definition 4.1, and (.5) that (see [0, 7]) d dt [w h(t)] [ = h(x, t) h ] [ h(x, t) h ] dx t = W m (h) dx + h dx 1+ h t>0. (5.9) Integrating from t 0 to t>t 0 and taking the square root, we then obtain (5.). Set for any t>t 0 t A k = W k (h(x, τ)) dx dτ, k = 0,...,m. t 0 If A k = 0 for some k with 0 k m, then by (.3), h(x, τ) = 0 for a. e. τ (t 0,t), and hence all A j = 0 (0 j m). In this case, (5.3) holds true trivially. Assume now all A k > 0 (0 k m). By(5.1) and (5.), we have A 0 = t t [w h (τ)] [ dτ (τ t0 ) +[w h (t 0 )] ] dτ t t 0 +[w h (t 0 )]. (5.10) t 0 t 0 Moreover, it follows from (5.9) that 1 d dt [w h(t)] + W m (h(x, t)) dx = t 0 h dx h Thus, we have for any t>t 0 that t A m = 1 ( W m (h(x, τ)) dxdτ 1+ [wh (t 0 )] [w h (t)] ). (5.11) (t t 0 ) Fix integers k and p with 1 k m and 1 p min(k, m k). By (1.11) and integration by parts, we have the following: if k is even, then t A k = k/ h(x, τ) dxdτ t 0 { ( 1) p t t = 0 (k p)/ h(x,τ) (k+p)/ h(x, τ) dxdτ if p is even, ( 1) p t (k p 1)/ h(x, τ) (k+p 1)/ h(x, τ) dxdτ if p is odd; t 0

19 High-order surface relaxation versus the Ehrlich Schwoebel effect 599 if k is odd, then t A k = (k 1)/ h(x, τ) dxdτ t 0 { ( 1) p t t = 0 (k p 1)/ h(x, τ) (k+p 1)/ h(x, τ) dxdτ if p is even, ( 1) p t t 0 (k p)/ h(x,τ) (k+p)/ h(x, τ) dx dτ if p is odd. In both cases, we have by the Cauchy Schwarz inequality that A k A k pa k+p. Consequently, by lemma 5.1,(5.10) and (5.11), we obtain (5.3). If t>t 0 +[w h (t 0 )], then [w h (t 0 )] (t t 0 ) 1 and t t 0 +[w h (t 0 )] (t t 0 ). (5.1) Since log( ) is a convex function, we obtain by Jensen s inequality, (5.3), (5.1) and the fact that for m t E(h(τ)) dτ 1 ( t ) log 1+ h(x, τ) dx dτ t 0 proving (5.4). t 0 (1+ 1 (1+ log [w h(t 0 )] ) 1/m ) (t t 0 +[w h (t 0 )] ) (m 1)/m (t t 0 ) 1 log(1+(m )/m 3 1/m (t t 0 ) (m 1)/m ), ED The following is a direct consequence of theorem 5.1 and the change in variables (1.7); it gives the precise dependence of the upper bounds on the material parameters m, F, M m, S c and σ ; see section 6 for more discussions on this result. Corollary 5.1. Let h :[0, ) Hper m () be a weak solution of the initial-boundary-value problem (1.6), (4.) and (4.3)on[0,T] for any T>0. Let t 0 0. We have FSc w h (t) (t t α 0 ) +[w h (t 0 )] t t 0, (5.13) t FSc (1+ α [w h (t 0 )] ) k/m t 0 6. Discussions W k (h(x, τ)) dx dτ αm k/(m) m FS c (t t 0 ) (t t 0 + α [w h (t 0 )] ) (m k)/m t>t 0, k = 1,...,m. (5.14) FS c We first discuss two aspects of our analysis: the energy minimization and bounds for scaling laws. These are in fact general issues in the understanding of coarsening dynamics of energydriven systems. 1. Energy minimization. The characterization of minimum energy and the magnitude of minimizers can help predict the dynamic scaling for the saturation of interface width. Assume

20 600 BLi in general an energy-driven, coarsening system saturates when a global energy-minimizer is reached. Then, it follows from corollary.1 and (1.7) that the saturation interface width w s = w s (l) with l being the linear size of the system is given by the scaling w s (l) 1 FS c l m (6.1) α M m with the roughness exponent m. More generally, the saturated, time and space averaged, kth gradient of the surface height scales as 1 FS c l m k (1 k m). α M m Notice that the prefactor in these scaling laws is the square root of the ratio of FS c and M m. Thus, the surface roughness is uniquely determined by m and the competition of the deposition, ES effect and surface relaxation. Both the deposition and ES effect make a surface rough and the relaxation makes a surface smooth. Now, the saturation time t s = t s (l) can be regarded as the time at which a global minimizer is reached in the dynamics. Thus, the interface width at t s is equal to that of a global minimizer h which in turn is equal to the saturation interface width: w h (t s ) = w s (l). This, together with the upper bound (5.13) and the scaling (6.1), leads to a lower bound for t s in the dynamic scaling t s (l) 1 l m (6.) M m with the dynamics exponent m. This indicates that the dynamic scaling is determined only by m and the mobility M m. The concept of λ-minimizers arises naturally from the following simple scenario of the coarsening dynamics of an energy-driven system. (1) There exists a sequence of λ-minimizers that are equilibriums. The wavelength of these minimizers, λ N,...,λ 1, increases, and the corresponding energy decreases. The largest wavelength λ 1 is the characteristic wavelength of a global minimizer. It may or may not be of the order of the linear size of an underlying system. () The system is always near one of such equilibrium. (3) The system moves to the next equilibrium with larger wavelength to reduce the energy. Our analysis in section shows the existence of a sequence of λ-minimizer with the desired properties for our underlying system. In general, it remains challenging to understand mathematically how stable a λ-minimizer is and how much time is needed for a system to move from one λ-minimizer to another.. Upper bounds for scaling laws. By corollary 5.1, our upper bound for the interface width always scales as t 1/, independent of m. This agrees with the early analysis in [10]. Our result also indicates that the prefactor of this scaling depends only on FS c,cf(5.13). Taking k = 1in(5.14), we see that the surface slope scales as FSc αmm 1/(m) t (m 1)/(m). This and (5.13) indicate that the characteristic lateral size of mounds λ(t) should scale as λ(t) Mm 1/(m) t 1/(m) (6.3) with the coarsening exponent 1/(m). Notice that the prefactor in this scaling only depends on the mobility M m.

21 High-order surface relaxation versus the Ehrlich Schwoebel effect 601 Further studies are needed to obtain an upper bound for the scaling law (6.3) ofλ(t), the characteristic lateral size of mounds; and to show the optimality of all bounds. We remark that our studies on the growth scaling laws agree with previous studies, both analytical and numerical [10, 1, 5, 7, 8, 31]. In particular, we recover the analytical results in [10] for all m that are obtained under a strong assumption on scaling (cf equation (8) of [10]). We now compare our results with experiments. For m =, the case of surface diffusion, our predictions of the scaling laws agree well with the following reported experiments. (1) The growth of the Cu film at temperature 00 K for which the ES effect is believed to be strong. This is reported in [8] (cf figure 1 and paragraph 1 of page 3 in [8]). () The epitaxial growth of Fe(001) films on the Mg(001) substrate at the substrate temperate K in which the ES effect gives rise to a pyramid-like surface structure. This is reported in [35] (cf figure 3 and the discussion at the end of paragraph of page 3 in [35]). Experiments reported in [33] on the homoepitaxy of Fe(001) at room temperature show that the coarsening exponent is 0.16 ± 0.04, close to 1/6. This is predicted by our analysis with the case m = 3, cf (6.3) with m = 3. In [33], a coarse-grained model is proposed (cf equation () in [33]), and numerical calculations based on this model are also reported. These calculations reproduce the t 1/6 scaling of coarsening from the experiment. In this model, the high-order relaxation term is exactly the term (1.) with m = 3. The next two terms in this model describe the ES effect when the surface gradient is small. This can be seen through an expansion of the low-order term in the effective energy: log(1+ h ) = 1 ( h 1) 1 + O( h 6 ) if h 1. The last term in the model describes the up down asymmetry. As pointed out in [33] (cf paragraph of the last page of [33]), the agreement between numerical calculations and experiment on the coarsening rate is insensitive to the presence or absence of the symmetrybreaking term. Therefore, the experimentally observed coarsening rate results expectedly from the competition between the high-order surface relaxation (1.) with m = 3 and the ES effect. It remains challenging to understand the kinetic origin of such a relaxation mechanism. We wonder if our analysis can provide some insight into experiments. For instance, is it possible experimentally to measure the physical parameters such as the mobility M m or the ES parameter S c, using our predicted scaling laws such as (5.13), (5.14), (6.1) and (6.)? Finally, we discuss a natural extension of the current (1.) to a linear combination of several high-order relaxation terms p j RE = ( 1) m M m m 1 h m= for some integer p 3. Models with terms of high-order derivatives such as this have been used in small-slope approximations of anisotropic surface free-energy density [18, 3]. Our methods can be used to analyse the corresponding effective energy functionals. In particular, for the corresponding re-scaled, singularly perturbed functionals (cf (.7) and (.8)), global minimizers exist and the minimum energy scales as O(log ε). But, the exact constant in this asymptotics is not immediately clear. Similarly, the gradients of any global minimizer are inversely proportional to ε. But the exact two-sided bounds for all the gradients, as in Part () of theorem.1, may no longer hold true with a single parameter p. Our argument (5.) can be used directly to obtain an upper bound for the t 1/ of interface width. Since all the derivative terms can be controlled by the terms with the highest order derivatives, the energy method we use in obtaining bounds for gradients can be applied directly to the extended model. In particular, we expect the estimates (5.3), with m replaced by p, to

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