Stable phase interfaces in the van der Waals--Cahn--Hilliard theory
aa r X i v : . [ m a t h . DG ] J u l STABLE PHASE INTERFACES IN THEVAN DER WAALS–CAHN–HILLIARD THEORY
YOSHIHIRO TONEGAWA AND NESHAN WICKRAMASEKERA
Abstract.
We prove that any limit-interface corresponding to a locally uniformly bounded,locally energy-bounded sequence of stable critical points of the van der Waals–Cahn–Hilliardenergy functionals with perturbation parameter → + is supported by an embedded smoothstable minimal hypersurface in low dimensions and an embedded smooth stable minimalhypersurface away from a closed singular set of co-dimension ≥ Introduction
Let Ω ⊂ R n ( n ≥
2) be a bounded domain and consider the family of energy functionals E ε , ε ∈ (0 , E ε ( u ) = Z Ω ε |∇ u | W ( u ) ε dx, where u : Ω → R belongs to the Sobolev space H (Ω) = { u ∈ L (Ω) : ∇ u ∈ L (Ω) } and W : R → R + ∪ { } is a given C double-well potential function with (precisely two)strict minima at ± W ( ±
1) = 0. When ε → + with E ε ( u ε ) remaining boundedindependently of ε , it is clear (from the bound on the second term of the integral above)that u ε must stay close to ± { u ε ≈ } and { u ε ≈ − } each has measure ≥ a fixed proportion of the measure of Ω ) thereis a transition layer of thickness O ( ε ) , which we may call an “interface region” or a “diffusedinterface”.In the past few decades it has been established that in the presence of a uniform boundon the energy E ε ( u ε ) and under natural variational hypotheses on u ε of varying degrees ofgenerality, for small ε >
0, the interface region corresponding to u ε is close to a generalizedminimal hypersurface V of Ω (the “limit-interface” as ε → + ) and that E ε ( u ε ) approxi-mates a fixed multiple of the ( n − u ε satisfying a uniform volume constraint; they proved that in this case,the limit-interface V is area minimizing in an appropriate class. R. Kohn and P. Sternberg Key words and phrases. stability, minimal hypersurface, varifold, diffused interface, van der Walls–Cahn–Hilliard theory.Y.T. is partially supported by JSPS Grant-in-aid for scientific research 21340033. Both authors wish tothank Mathematisches Forschungsinstitut Oberwolfach for providing the opportunity to initiate this jointresearch. ([9]) studied the locally energy minimizing case, again in the context of Γ-convergence. Morerecently, J. Hutchinson and the first author ([8]) showed that V is a stationary integral vari-fold if u ε are assumed to be merely volume-unconstrained critical points of E ε (Theorem 2.2below), and that V is an integral varifold with constant generalized mean curvature whenthe u ε are critical points subject to a volume constraint (see also [16, Theorem 7.1]). Subse-quently, the first author ([20]) showed that whenever the u ε are unconstrained stable criticalpoints of E ε , the limit stationary integral varifold V is stable in the sense that V admits ageneralized second fundamental form which satisfies the stability inequality (Theorem 2.4below).With regard to smoothness of V in the absence of an energy minimizing hypothesis, littlehas been known beyond the following theorem of the first author ([20]): Suppose that n = 2 , ε i → + as i → ∞ and that for each i = 1 , , , . . . , u ε i ∈ H (Ω) is a stable critical point of E ε i with sup Ω | u ε i | + E ε i ( u ε i ) ≤ c for some c > independent of i . Then there exists a locallyfinite union L of non-intersecting lines of Ω such that after passing to a subsequence of { ε i } without changing notation, for any < s < , the sequence of sets { x ∈ Ω : | u ε i ( x ) | ≤ s } converges locally in Hausdorff distance to L . Thus in case n = 2 , any stable diffused interfacemust be close to non-intersecting lines for sufficiently small positive values of the parameter ε . It has remained an open question whether one can make analogous conclusions in di-mensions n >
2. Here we give an affirmative answer to this question in all dimensions.Specifically, we prove (in Theorem 2.1 below) that if u ε i are uniformly bounded stable crit-ical points of E ε i with no volume constraint and with uniformly bounded energy, then for ≤ n ≤ , there exists an embedded smooth stable minimal hypersurface M of Ω such thatafter passing to a subsequence of { ε i } without changing notation, for each fixed s ∈ (0 , ,the sequence of interface regions { x ∈ Ω : | u ε i ( x ) | < s } converges locally in Hausdorffdistance to M ; for n ≥ , the limit stable minimal hypersurface M may carry an interiorsingular set, which is discrete if n = 8 and has Hausdorff dimension at most n − if n ≥ . This regularity result was known for the limit-interfaces corresponding to sequences { u ε i } of energy minimizers since in that case the limit-interfaces are area-minimizing and thewell known regularity theory for locally area minimizing currents is applicable. The newresult in this paper is that the stability hypothesis, which is much weaker than any energyminimizing hypothesis, suffices to guarantee the same regularity for the limit-interfaces.The main reason why, in [20], the interface regularity was established only in case n = 2and not for n > n = 2 (i.e. when the interface is a 1-dimensionalvarifold), the structure theorem (due to W. Allard and F. Almgren [2]) for stationary 1-dimensional varifolds is applicable to the limit-interface, there was no sufficiently generalregularity theory available at the time for higher dimensional stable integral varifolds. Incontrast to limit-interfaces corresponding to sequences of locally energy minimizing criticalpoints of E ε , a general stable limit-interface may develop higher multiplicity, a priori variableeven locally. This fact gives rise to significant difficulties that need to be overcome inunderstanding smoothness properties of a stable limit-interface, and is the reason why theregularity question for stable limit-interfaces in arbitrary dimension remained unresolvedprior to the present work. Note that the Schoen–Simon regularity theory ([17]), which wasthe most general theory available for stable hypersurfaces at the time when work in [20]was carried out, requires knowing a priori that the singular set (in particular the set ofthose singular points where the varifold has tangent hyperplanes of multiplicity ≥
2) issufficiently small, a hypothesis which appears to be difficult to verify directly for a stable
TABLE PHASE INTERFACES IN THE VAN DER WAALS–CAHN–HILLIARD THEORY 3 limit-interface. The key new input to this problem is the recent work of the second author([21]), which gives a necessary and sufficient geometric structural condition for a generalstable codimension 1 integral varifold to be regular (Theorem 3.1 below). Here we showthat the limit-interface in question satisfies precisely this structural condition; its regularitythen follows directly from the general theory of [21].While the present work as well as the series of works mentioned above ([11, 19, 8, 16, 20])investigate the general character of limit-interfaces, there have been a number of articleswhich address the question of existence of critical points of (1.1) whose interface regions con-verge to a given minimal hypersurface. In this direction we mention the work by F. Pacardand R. Ritor´e ([14]), M. Kowalczyk ([10]) and a number of recent joint works by M. del Pino,M. Kowalczyk, F. Pacard, J. Wei and J. Yang (see for example [4]). We refer the reader tothe recent survey paper by Pacard [13] for a complete list of references.2.
Hypotheses and the main results
In this section, we state the hypotheses on W and u ε , state our main theorem (Theo-rem 2.1) and recall some definitions and known results needed for its proof. We will givethe proof of Theorem 2.1 in Sections 3 and 4.We assume:(A1) W ∈ C ( R ), W ≥ W has precisely three critical points, two of which areminima at ± W ( ±
1) = 0 and W ′′ ( ± >
0, and the third a local maximumbetween ± ε , ε , ε , . . . are positive numbers with lim i →∞ ε i = 0 , the constants c , c are posi-tive and for each i = 1 , , , . . . , the function u ε i ∈ H (Ω) and satisfies E ε i ( u ε i ) ≤ c and sup Ω | u ε i | ≤ c .(A3) u ε i is a stable critical point of E ε i for each i = 1 , , , . . . Thus u ε i solves, weakly,(2.1) − ε i ∆ u ε i + W ′ ( u ε i ) ε i = 0 on Ωand satisfies(2.2) Z Ω ε i |∇ φ | + W ′′ ( u ε i ) ε i φ ≥ φ ∈ C c (Ω) . Remarks: (1)
Hypotheses (2.1) and (2.2) are equivalent, respectively, to the conditions ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 E ε i ( u ε i + tφ ) = 0 and d dt (cid:12)(cid:12)(cid:12)(cid:12) t =0 E ε i ( u ε i + tφ ) ≥ φ ∈ C c (Ω) . (2) Since u ε i is bounded (by hypothesis (A2)), it follows from standard elliptic theory that u ε i ∈ C (Ω) with (2.1) satisfied pointwise in Ω . We shall use the following notation throughout the paper:A general point in R n is denoted by x = ( x , · · · , x n ) or by ( y, z ) with y = ( y , y ) ∈ R and z = ( z , · · · , z n − ) ∈ R n − . For x ∈ R n and r >
0, we let B r ( x ) = { ˜ x ∈ R n : | ˜ x − x | < r } Y. TONEGAWA AND N. WICKRAMASEKERA and abbreviate B r (0) as B r . For k = n , w ∈ R k and r >
0, we let B kr ( w ) = { ˜ w ∈ R k : | ˜ w − w | < r } and abbreviate B kr (0) as B kr . The k -dimensional Lebesgue measure on R k will be denoted by L k and ω k = L k ( B k ). The k -dimensional Hausdorff measure on R n isdenoted by H k .In order to characterize the limit-interfaces, we use the notion of varifolds ([1, 18]). Let G n − (Ω) be the product space Ω × G ( n, n −
1) with the product topology, where G ( n, n −
1) isthe space of ( n − R n . We identify each element S ∈ G ( n, n − n × n matrix corresponding to the orthogonal projection of R n onto S . A Radonmeasure V on G n − (Ω) is called an ( n − -varifold (henceforth just called a varifold ). Fora varifold V , k V k shall denote the associated weight measure on Ω, defined by k V k ( φ ) = Z G n − (Ω) φ ( x ) dV ( x, S ) for φ ∈ C c (Ω).The support of the measure k V k in Ω is denoted by spt k V k . We say that V is integral ifthere exists a countably ( n − M of Ω and an H n − measurable, positiveinteger valued function θ : M → Z + such that V is given by(2.3) V ( φ ) = Z M φ ( x, T x M ) θ ( x ) d H n − ( x ) for φ ∈ G n − (Ω) , where T x M is the approximate tangent space of M at x. The function θ is called the multiplicity of V .For an ( n − C submanifold M in Ω, | M | denotes the multiplicity 1 integralvarifold associated with M, as in (2.3) with θ ≡ V is stationary if V has zero first variation with respect to area underdeformation by any C vector field of Ω with compact support, namely (see [18]), if Z G n − (Ω) tr ( ∇ g ( x ) · S ) dV ( x, S ) = 0 for all g ∈ C c (Ω; R n ) . Here · is the usual matrix multiplication and tr is the trace operator.For V ∈ G n − (Ω), let reg V ⊂ Ω be the set of regular points of spt k V k ∩ Ω. Thus, x ∈ reg V if and only if x ∈ spt k V k and there exists some open ball B r ( x ) ⊂ Ω such thatspt k V k ∩ B r ( x ) is a compact, connected, embedded smooth hypersurface with boundarycontained in ∂B r ( x ). The interior singular set sing V of V is defined bysing V = (spt k V k \ reg V ) ∩ Ω . By definition, sing V is closed in Ω.To each u ε i satisfying (A1)-(A3), we associate the varifold V ε i ∈ G n − (Ω) defined by V ε i ( φ ) = 1 σ Z {|∇ u εi | > } φ ( x, I − n ε i ( x ) ⊗ n ε i ( x )) ε i |∇ u ε i | dx ( ⋆ )for φ ∈ C c ( G n − (Ω)), where n ε i ( x ) = ∇ u εi ( x ) |∇ u εi ( x ) | , σ = R − p W ( s ) / ds , I is the n × n iden-tity matrix and ⊗ is the tensor product. Note that k V ε i k then corresponds simply to σ ε i |∇ u ε i | dx .As a consequence of hypothesis (A2), there exists a subsequence of { V ε i } ∞ i =1 convergingas varifolds on Ω (i.e. as Radon measures on G n − (Ω)) to some V ∈ G n − (Ω). Our mainresult, which concerns regularity of V , is the following: TABLE PHASE INTERFACES IN THE VAN DER WAALS–CAHN–HILLIARD THEORY 5
Theorem 2.1.
Let the hypotheses be as in (A1)-(A3) and let V ε i be defined by ( ⋆ ). Let V ∈ G n − (Ω) be such that lim i →∞ V ε i = V, where the convergence is as varifolds on Ω . Then sing V is empty if ≤ n ≤ , sing V is a discrete set of points if n = 8 and H n − γ (sing V ) =0 for each γ > if n ≥ ; furthermore, reg V = (spt k V k\ sing V ) ∩ Ω is an embedded smoothstable minimal hypersurface of Ω . Remarks: (1)
As just mentioned, if the hypotheses are as in (A1)-(A3), then after passingto a subsequence of { ε i } without changing notation, we obtain V ∈ G n − (Ω) such that V ε i → V as varifolds on Ω . It is of course possible that V = 0. (2) There exists u ∈ BV (Ω) such that after passing to a subsequence of { ε i } withoutchanging notation, u ε i → u in L (Ω); in fact u ( x ) = ± x ∈ Ω , and hence the sets { u = 1 } and { u = − } have finite perimeter in Ω (see the discussion in [8], pp. 51-52) andspt k ∂ { u = 1 }k ∩ Ω ⊂ spt k V k (see [8], Theorem 1). Thus in particular, V = 0 is impliedby the condition that u u
6≡ − . We now further discuss known results we shall need for the proof of Theorem 2.1.The following theorem, due to Hutchinson and the first author ([8]), says among otherthings that a limit varifold V corresponding to a sequence of critical points of E ε (i.e. V ∈ G n − (Ω) arising as the varifold limit of a sequence { V ε i } , where ε i → + and V ε i ∈ G n − (Ω)is defined by ( ⋆ ) with u ε i satisfying (2.1)) is a stationary integral varifold, and that spt k V k indeed is the limit-interface corresponding to { u ε i } in the sense made precise in part (3) ofthe theorem. Note that no stability hypothesis is necessary for this result. Theorem 2.2. ([8, Theorem 1 and Proposition 4.3])
Suppose that (A1), (A2) and (2.1) hold, and let V ∈ G n − (Ω) be such that V = lim i →∞ V ε i , where V ε i is as in ( ⋆ ) and theconvergence is as varifolds on Ω . Let U be an open subset of Ω such that the closure of U iscontained in Ω . Then (1) V is a stationary integral varifold on Ω . (2) lim i →∞ Z U (cid:12)(cid:12)(cid:12)(cid:12) ε i |∇ u ε i | − W ( u ε i ) ε i (cid:12)(cid:12)(cid:12)(cid:12) dx = 0 . (3) For each s ∈ (0 , , {| u ε i | ≤ s } ∩ U converges to spt k V k ∩ U in Hausdorff distance. In order to discuss the additional known results relevant to us concerning stable criticalpoints of E ε , it is convenient to introduce the following notation: For u ∈ C (Ω) , let B u bethe non-negative function defined by(2.4) B u = 1 |∇ u | n X i,j =1 u x i x j − |∇ u | n X i =1 n X j =1 u x j u x i x j ! on {|∇ u | > } and B u = 0 on {|∇ u | = 0 } . Here and subsequently, u x i , u x i x j denote thepartial derivatives ∂ u∂ x i , ∂ u∂x j ∂x i respectively. Note that the expression on the right hand sideof (2.4) is non-negative when ∇ u = 0 and is invariant under orthogonal transformations of R n .We have the following: Lemma 2.3. ([15])
Let ε ∈ (0 , , u ∈ C (Ω) and suppose that u is a stable critical pointof E ε in the sense that (2.1) and (2.2) are satisfied with ε in place of ε i and u in place of Y. TONEGAWA AND N. WICKRAMASEKERA u ε i . Then (2.5) Z Ω B u |∇ u | φ dx ≤ Z Ω |∇ φ | |∇ u | dx for each φ ∈ C c (Ω) . One proves (2.5) by taking |∇ u | φ in place of φ in the inequality (2.2) and utilizing equation(2.1). See [15] or [20] for details.Let the hypotheses be as in (A1)-(A3), and write B ε i = B u εi . In view of hypothesis (A2), Lemma 2.3 implies that the L -norm of ε i B ε i |∇ u ε i | is locallyuniformly bounded. Let ν be a subsequential limit (as Radon measures on Ω) of the sequence ε i B ε i |∇ u ε i | dx. Thus after re-indexing,(2.6) ν ( φ ) = lim i →∞ Z Ω ε i φ B ε i |∇ u ε i | dx for φ ∈ C c (Ω).The following crucial stability inequality is established in [20]: Theorem 2.4. ([20, Theorem 3] )
Let the hypotheses be as in (A1)-(A3) and let V ε i bedefined by ( ⋆ ). Let V ∈ G n − (Ω) be such that V = lim i →∞ V ε i , where the convergence isas varifolds on Ω . Then V has a generalized second fundamental form A with its length | A | satisfying (2.7) Z Ω | A | φ d k V k ≤ Z Ω |∇ φ | d k V k for all φ ∈ C c (Ω) . We refer the reader to [20, Section 2] for the definition of the generalized second funda-mental form of a varifold. See also [7] where the notion was defined originally.We end this section with the following elementary consequence of (2.1) which we shallneed later:
Lemma 2.5. If u ∈ C (Ω) and (2.1) holds with ε ∈ (0 , in place of ε i and u in place of u ε i , then (2.8) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ ε |∇ u | − W ( u ) ε !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε √ n − |∇ u | B u in Ω . Proof . For any n × n symmetric matrix M and any unit vector m ∈ R n , one has that | M · m − (tr M ) m | ≤ √ n − M ) − m t · M · m ) . TABLE PHASE INTERFACES IN THE VAN DER WAALS–CAHN–HILLIARD THEORY 7
Using this and (2.1), we see that on the set {∇ u = 0 } , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ ε |∇ u | − W ( u ) ε !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ε (cid:12)(cid:12) ∇ u · ∇ u − ∆ u ∇ u (cid:12)(cid:12) ≤ ε √ n − |∇ u | (cid:18) tr (( ∇ u ) ) − |∇ u | ∇ u t · ( ∇ u ) · ∇ u (cid:19) = ε √ n − |∇ u | B u . If ∇ u = 0, the inequality holds trivially. ✷ Regularity of stable codimension 1 integral varifolds and the proof ofthe main theorem
In this section we recall the main content (Theorem 3.1 below) of the regularity theoryof the second author ([21]) for stable codimension 1 integral varifolds and show how itimplies our main result (Theorem 2.1) concerning regularity of the limit-interfaces, moduloverification of a certain structural condition satisfied by the limit-interfaces. This structuralcondition is precisely given in Proposition 3.2 below, and is necessary to apply Theorem 3.1.We shall establish its validity in the next section.Fix an integer m ≥ α ∈ (0 , S α the collection of all integral m -varifolds V on the open unit ball B m +11 ⊂ R m +1 with 0 ∈ spt k V k , k V k ( B m +11 ) < ∞ and satisfyingthe following conditions:( S
1) (Stationarity) V is a critical point of the m -dimensional area functional in B m +11 ,viz. V is a stationary integral m -varifold on B m +11 .( S
2) (Stability) V satisfies(3.1) Z reg V | A | φ d H m ≤ Z reg V |∇ reg V φ | d H m for all φ ∈ C c (reg V ), where A denotes the (classical) second fundamental form ofreg V, | A | its length and ∇ reg V is the gradient operator on reg V .( S
3) ( α -Structural Hypothesis) For each x ∈ sing V , there exists no ρ > k V k ∩ B ρ ( x ) is equal to the union of a finite number of m -dimensional embedded C ,α submanifolds-with-boundary of B ρ ( x ) all having common boundary in B ρ ( x )equal to an ( m − C ,α submanifold of B ρ ( x ) containing x , and no two intersecting except along their common boundary.With these hypotheses, we have the following: Theorem 3.1. ([21, Theorem 3.1]) If V ∈ S α , then sing V ∩ B m +11 is empty if ≤ m ≤ , sing V ∩ B is a discrete set of points if m = 7 and H m − γ (sing V ∩ B ) = 0 for each γ > if m ≥ . An obvious yet extremely useful feature of Theorem 3.1 is that it suffices, when applyingthe theorem, to verify the α -Structural Hypothesis for points x ∈ sing V in the complementof a set Z having ( m − Z is closed. We rely on this fact in an essential way in the present application, inwhich the α -Structural Hypothesis is verified by way of the following proposition. We shallprove this proposition in the next section. Y. TONEGAWA AND N. WICKRAMASEKERA
Proposition 3.2.
Let V be as in Theorem 2.1. There exists a (possibly empty) Borel set Z ⊂ spt k V k ∩ Ω with H n − ( Z ) = 0 such that for each x ∈ (spt k V k \ Z ) ∩ Ω and eachtangent cone C to V at x , spt k C k is not equal to a union of three or more half-hyperplanesof R n meeting along an ( n − -dimensional affine subspace. Theorem 2.1 follows directly from Proposition 3.2 and Theorem 3.1.
Proof of Theorem 2.1 . Let V be as in Theorem 2.1, y ∈ spt k V k∩ Ω and ρ ∈ (0 , dist ( y, ∂ Ω)).In order to prove Theorem 2.1, it clearly suffices to establish its conclusions with e V = η y,ρ V in place of V , which we can achieve by Theorem 3.1 if we can show that e V ∈ S α for some α ∈ (0 , η y,ρ : R n → R n is the map defined by η y,ρ ( x ) = ρ − ( x − y )and η y,ρ denotes the push-forward of V by η y,ρ . It follows from Theorem 2.2 that e V satisfies ( S e V satisfies ( S V is constant oneach connected component of reg V so we can replace d k V k in (2.7) by d H n − whenever φ ∈ C c (Ω \ sing V ). Thus given φ ∈ C c (reg e V ), we may choose any extension e φ ∈ C c ( B \ sing e V )of φ such that ∇ e φ = ∇ reg V φ on reg V and use (2.7) with e φ ◦ η y, ρ in place of φ to deducethat e V satisfies ( S V (and hence also e V ) satisfies ( S
3) (for any α ∈ (0 , α ∈ (0 , x ∈ spt k V k ∩ Ω and ρ ∈ (0 , dist( x, ∂ Ω)) such thatspt k V k ∩ B ρ ( x ) is a union of three or more C ,α hypersurfaces-with-boundary meeting alonga common ( n − C ,α submanifold L of B ρ ( x ) with x ∈ L . It is standard tosee, with the help of the Hopf boundary point lemma for divergence form elliptic operators([5, Lemma 7], see also [6, Lemma 10.1]) that at every point along L , these hypersurfaces-with-boundary must meet transversely. Hence the unique tangent cone to V at any ˜ x ∈ L is supported by a union of three or more half-hyperplanes meeting along a common ( n − e x ∈ L \ Z , this directly contradicts Proposition 3.2, where Z is the set as in Proposition 3.2. Note that L \ Z = ∅ since H n − ( Z ) = 0 by Proposition 3.2.Thus V must satisfy ( S e V ∈ S α , and Theorem 2.1 follows from Theorem 3.1. ✷ Structural condition for the stable limit-interfaces
To complete the proof of Theorem 2.1, it only remains to give a proof of Proposition 3.2,which we shall do in this section.Let the hypotheses be as in (A1)-(A3) and let ν be the Radon measure on Ω defined by(2.6). Let V be as in Theorem 2.1, obtained possibly after passing to a suitable subsequenceof { ε i } and the corresponding subsequence of { u ε i } . Let Z = (cid:26) x ∈ spt k V k ∩ Ω : lim sup r → ν ( B r ( x )) r n − > (cid:27) . It is standard to see that H n − γ ( Z ) = 0 for each γ > H n − ( Z ) = 0. We will show that Proposition 3.2 holds with this Z. To obtain a contradiction assume that we have a point x ∈ spt k V k \ Z where a tangentcone C to V has the property that spt k C k is equal to a union of three or more half-hyperplanes meeting along a common ( n − S ( C ). Without loss of TABLE PHASE INTERFACES IN THE VAN DER WAALS–CAHN–HILLIARD THEORY 9 generality we may assume that x = 0 and that S ( C ) = { } × R n − . Thus we may writespt k C k = ∪ Nj =1 P j for some N ≥ , where for each j = 1 , , . . . , N,P j = { t p j : t ≥ } × R n − with p , · · · , p N ∈ R distinct vectors such that | p j | = for j = 1 , , . . . , N. By the definition of tangent cone, there exists a sequence r i → i →∞ η r i V = C . Here η r is the map x r − x. Since V ε i → V , we may choose a subsequence of { ε i } for which, after relabeling, we have that lim i →∞ η r i V ε i = C and lim i →∞ ε i r i = 0. Letting˜ ε i = ε i r i and defining ˜ u ˜ ε i (˜ x ) = u ε i ( r i ˜ x ), we then have that η r i V ε i ( φ ) = 1 σ Z {|∇ ˜ u ˜ εi | > } φ (˜ x, I − n ˜ ε i ⊗ n ˜ ε i ) ˜ ε i |∇ ˜ u ˜ ε i | d ˜ x for φ ∈ C c ( G n − ( r − i Ω)), where ˜ n ˜ ε i = ∇ ˜ u ˜ εi |∇ ˜ u ˜ εi | . Since lim i →∞ ν ( B ri )(2 r i ) n − = 0, we may choose afurther subsequence of { ε i } without changing notation such thatlim i →∞ r i ) n − Z B ri ε i B ε i |∇ u ε i | dx = 0 . With the change of variables as above, this is equivalent tolim i →∞ Z B ˜ ε i ˜ B ε i |∇ ˜ u ˜ ε i | d ˜ x = 0 , where ˜ B ˜ ε i is defined by (2.4) with ˜ u ˜ ε i in place of u .By [8, Prop. 3.4], for each open set U with U ⊂⊂ Ω, there exist constants c = c ( c , n, dist ( U, ∂
Ω)) and ε = ε ( c , n, dist ( U, ∂
Ω)) such that if B r ( x ) ⊂ U and s ∈ (0 , r ],then for all i sufficiently large to ensure ε i ≤ ε , r − n Z B r ( x ) (cid:18) ε i |∇ u ε i | W ( u ε i ) ε i (cid:19) dx − s − n Z B s ( x ) (cid:18) ε i |∇ u ε i | W ( u ε i ) ε i (cid:19) dx ≥ Z rs τ − n Z B τ ( x ) (cid:18) W ( u ε i ) ε i − ε i |∇ u ε i | (cid:19) + dx ! dτ − cr + ε i Z B r ( x ) \ B s ( x ) (( y − x ) · ∇ u ε i ) | y − x | n +1 dy. (4.1)This implies in particular that Z B ˜ ε i |∇ ˜ u ˜ ε i | + W (˜ u ˜ ε i )˜ ε i ≤ C for all sufficiently large i , where C is a positive constant depending only on n , c and c . Thus hypotheses (A1)-(A3) are satisfied with ˜ ε i in place of ε i , ˜ u ˜ ε i in place of u ε i and C inplace of c , so by replacing the original sequences { ε i } , { u ε i } with the new sequences { ˜ ε i } , { ˜ u ˜ ε i } and the constant c with C , we have that (A1)-(A3) hold with Ω = B , together withthe additional facts that(4.2) lim i →∞ V ε i = C where the convergence is as varifolds on B and that(4.3) lim i →∞ Z B ε i B ε i |∇ u ε i | = 0 . For the rest of the discussion we shall assume that W , { ε i } , { u ε i } satisfy (A1)-(A3) with Ω = B , as well as (4.2) and (4.3).Our goal is to derive a contradiction. Lemma 4.1.
Set c = q min | t |≤ W ( t ) > and let D ε i = (cid:26) z ∈ B n − : |∇ u ε i ( y, z ) | ≥ c ε i holds for all y ∈ B with | u ε i ( y, z ) | ≤ (cid:27) . Then we have that lim i →∞ L n − ( B n − \ D ε i ) = 0 . Remark:
Note that D ε i contains the set D ′ ε i of points z ∈ B n − where | u ε i ( y, z ) | > forall y ∈ B . We shall prove that D ′ ε i is small in Lemma 4.2 below. Proof . For each i = 1 , , , . . . , let { B n − ε i ( z i,j ) } J i j =1 be a maximal pairwise disjoint collectionof balls such that z i,j ∈ B n − \ D ε i for j = 1 , · · · , J i . Then B n − \ D ε i ⊂ ∪ J i j =1 B n − ε i ( z i,j ) andby the definition of D ε i , there exists, for each j = 1 , , . . . , J i , a point y i,j ∈ B such that | u ε i ( y i,j , z i,j ) | ≤ and |∇ u ε i ( y i,j , z i,j ) | < c ε i . By standard elliptic estimates we have thatsup B / |∇ u ε i | ≤ C sup B (cid:18) | u ε i | + | W ′ ( u ε i ) | ε i (cid:19) where C = C ( n ), whence in view of the hypothesis sup B | u ε i | ≤ c , there exists a fixednumber r ∈ (0 , , depending only on n , W and c , such that(4.4) | u ε i ( x ) | ≤
34 and |∇ u ε i ( x ) | < c ε i for each x ∈ B r ε i ( y i,j , z i,j ). On this ball we have(4.5) v ε i ≡ W ( u ε i ) ε i − ε i |∇ u ε i | ≥ ε i min | t |≤ W ( t ) − c ! ≥ c ε i . Since B r ε i ( y i,j ) × { z } ⊂ B r ε i ( y i,j , z i,j ) for each z ∈ B n − r ε i ( z i,j ), we have by (4.5) that(4.6) 2 c √ π r ≤ Z B r εi ( y i,j ) ( v ε i ( y, z )) dy ! TABLE PHASE INTERFACES IN THE VAN DER WAALS–CAHN–HILLIARD THEORY 11 for each z ∈ B n − r ε i ( z i,j ) . On the other hand, by the relevant 2-dimensional Sobolev inequalityand (2.8), Z B r εi ( y i,j ) ( v ε i ( y, z )) dy ! ≤ Z B r εi ( v ε i ( y, z )) dy ! ≤ C Z B r εi | v ε i ( y, z ) | + |∇ y v ε i ( y, z ) | dy ≤ C Z B r εi | v ε i ( y, z ) | dy + C √ n − ε i Z B r εi B ε i ( y, z ) |∇ u ε i ( y, z ) | dy (4.7)where C is the relevant Sobolev constant. Combining (4.6) and (4.7) we obtain that c √ πr ≤ C Z B r εi | v ε i ( y, z ) | dy + C √ n − ε i Z B r εi B ε i ( y, z ) |∇ u ε i ( y, z ) | dy for all z ∈ B n − r ε i ( z i,j ). Integrating this over B n − r ε i ( z i,j ) first, summing over j and using the factthat { B n − r ε i ( z i,j ) } J i j =1 are pairwise disjoint, we obtain with the help of the Cauchy-Schwarzinequality that c √ πr n − ω n − ε n − i J i ≤ C Z B r εi × B n − r εi | v ε i | dx + C √ n − ε i Z B r εi × B n − r εi B ε i |∇ u ε i | dx ≤ C Z B / | v ε i | dx + C √ n − Z B / ε i |∇ u ε i | dx ! / Z B / ε i B ε i |∇ u ε i | dx ! / ≤ C Z B / | v ε i | dx + Cc √ n − Z B / ε i B ε i |∇ u ε i | dx ! / . Since L n − ( B n − \ D ε i ) ≤ J i ω n − (2 ε i ) n − , the lemma follows from Theorem 2.2(2) and (4.3). ✷ Choose δ > C so that { B δ ( p i ) } Ni =1 are disjoint, and define Q ε i = (cid:8) z ∈ B n − : ∀ t ∈ [ − / , / , ∀ j ∈ { , · · · , N } , ∃ y ∈ B δ ( p j ) s.t. u ε i ( y, z ) = t (cid:9) . The next lemma is obtained by re-examining [8, Section 5].
Lemma 4.2.
With Q ε i as above, we have that lim i →∞ L n − ( B n − \ Q ε i ) = 0 . Proof . For j = 1 , , . . . , N , let Q ε i ,j = (cid:8) z ∈ B n − : ∀ t ∈ [ − / , / , ∃ y ∈ B δ ( p j ) s.t. u ε i ( y, z ) = t (cid:9) . It suffices to prove that lim i →∞ L n − ( B n − \ Q ε i ,j ) = 0 for each j = 1 , , . . . , N . Thus withoutloss of generality, we may assume j = 1, P = { y = 0 } ∩ { y ≥ } and p = (1 / , k C k ∩ B δ ( p , z ) = { y = 0 } ∩ B δ ( p , z ) for each z ∈ R n − . On B δ ( p , z ) with z ∈ B n − , V ε i converge to θ | P | as varifolds, where θ ∈ N is the multiplicity of C on P . By Theorem 2.2, the sets B δ ( p , z ) ∩ {| u ε i | ≤ } converge to P ∩ B δ ( p , z )in Hausdorff distance. Note that u ε i ( x ) converges to different values ( ±
1) uniformly on (cid:16) ∪ z ∈ B n − B δ ( p , z ) (cid:17) ∩ { y > δ } and (cid:16) ∪ z ∈ B n − B δ ( p , z ) (cid:17) ∩ { y < − δ } in case θ is odd, andto the same value if θ is even (see the discussion in [8, p. 78]). Hence if θ is odd, bycontinuity of u ε i , the function y u ε i ( , y , z ) as y ranges over [ − δ , δ ] takes all valuesbetween − and , so that in this case we see that B n − = Q ε i , for all sufficiently large i, proving the lemma.If θ is even, we need to utilize results in [8, Section 5]. Assume without loss of generalitythat u ε i converges to +1 on both sides of { y > } and { y < } on B δ ( p , z ). Letˆ B δ/ ( p , z ) = { (ˆ y , ˆ y , ˆ z ) : (ˆ y − / + | ˆ z − z | < ( δ/ , | ˆ y | < δ/ } and S i = n x ∈ B δ/ ( p , z ) ∩ P : ∃ t ∈ [ − / , / , with { u ε i = t } ∩ T − ( x ) ∩ ˆ B δ/ ( p , z ) = ∅ o . Here T is the orthogonal projection R n → { y = 0 } . By the continuity of the u ε i ’s andtheir local uniform convergence to +1 away from P , we have for any b ∈ (0 , /
2) that S i ⊂ S bi ∪ ˆ S bi , where S bi = n x ∈ B δ/ ( p , z ) ∩ P : { u ε i ≤ − b } ∩ T − ( x ) ∩ ˆ B δ/ ( p , z ) = ∅ o andˆ S bi = ( x ∈ B δ/ ( p , z ) ∩ P : inf T − ( x ) ∩ ˆ B δ/ ( p ,z ) u ε i ∈ [ − / , − b ] ) . We claim that for any given sufficiently small s > , we can choose small b = b ( s, W ) > i →∞ L n − ( S bi ) ≤ c ( σ, n, θ ) s. To see this, we argue as follows: Note first that for any given s ∈ (0 ,
1) we have the estimates(5.5)-(5.8) of [8] with B δ ( p , z ) in place of B , where b = b ( s, W ) > η = η ( s, W, δ, θ ) ∈ (0 ,
1) and L = L ( s, W ) ∈ (1 , ∞ ) as in [8, Prop. 5.5,5.6] with R = δ , N = θ , and define G i by G i = ˆ B δ/ ( p , z ) ∩ {| u ε i | ≤ − b }∩ (cid:26) x : Z B r ( x ) (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ε i |∇ u ε i | − W ( u ε i ) ε i (cid:12)(cid:12)(cid:12)(cid:12) + (1 − ν , i ) ε i |∇ u ε i | (cid:19) ≤ ηr n − if 4 ε i L ≤ r ≤ δ (cid:27) (4.9)where ν , i = |∇ u ε i | − ∂ u εi ∂y if ∇ u ε i = 0 and ν , i = 0 otherwise. With the help of the Besi-covitch covering theorem and (4.1) we see that(4.10) k V ε i k ( ˆ B δ/ ( p , z ) ∩ {| u ε i | ≤ − b } \ G i ) + L n − ( T ( ˆ B δ/ ( p , z ) ∩ {| u ε i | ≤ − b } \ G i )) → . We also note that there exists c = c ( b, s, W ) with the following property: for a.e. t ∈ [ − b, − b ] , the level set { u ε i = t } is an ( n − C surface and for any x ∈ G i with u ε i ( x ) = t , the set { u ε i = t } ∩ B Lε i ( x ) is a graph y = f ( y , z , · · · , z n − ) of a C function f : T ( { u ε i = t } ∩ B Lε i ( x )) → R with |∇ f | ≤ cη / ( n +1) on T ( { u ε i = t } ∩ B Lε i ( x )).This follows from the proof of [8, Prop. 5.6], which yields that for any x = ( y ⋆ , y ⋆ , z ⋆ ) ∈ G i ,the function u ε i in the neighborhood B Lε i ( x ) is C close to ± q (( y − y ⋆ ) /ε i ) , where q is the TABLE PHASE INTERFACES IN THE VAN DER WAALS–CAHN–HILLIARD THEORY 13 standing wave solution defined by the ODE q ′′ = W ′ ( q ) with q ( ±∞ ) = ±
1; specifically,letting ˜ u ε i (˜ y , ˜ y , ˜ z ) = u ε i ( ε i ˜ y + y ⋆ , ε i ˜ y + y ⋆ , ε i ˜ z + z ⋆ ) and ˜ q (˜ y , ˜ y , ˜ z ) = ± q (˜ y + c ) (sothat q ( c ) = t ), we have that k ˜ u ε i − ˜ q k C ( B L ) ≤ cη / ( n +1) . In particular, we choose η = η ( s, W, δ, θ ) ∈ (0 ,
1) so small that(4.11) p |∇ f | ≤ s. For x ∈ P ∩ B δ/ ( p , z ) and | t | ≤ − b , define Y ix ( t ) = T − ( x ) ∩ G i ∩ { u ε i = t } . We claimthat the cardinality Y ix ( t ) of Y ix ( t ) is ≤ θ . To see this, assume for a contradiction that Y ix ( t ) ≥ θ + 1, and let Y ′ be any subset of Y ix ( t ) such that Y ′ = θ + 1. Then we haveby [8, Prop. 5.6](4.12) I ≡ X ˜ x ∈ Y ′ ω n − ( Lε i ) n − Z B Lεi (˜ x ) ε i |∇ u ε i | W ( u ε i ) ε i ≥ ( Y ′ )(2 σ − s )while by [8, Prop. 5.5], I ≤ s + 1 + sω n − δ n − Z { ˜ x | dist ( Y ′ , ˜ x ) <δ } ε i |∇ u ε i | W ( u ε i ) ε i ≤ s + 1 + sω n − δ n − k V ε i k ( B δ + o (1) ( x )) + o (1)(4.13)where o (1) → i → ∞ uniformly in x ∈ B δ/ ( p , z ) ∩ P . Since(4.14) k V ε i k ( B δ + o (1) ( x )) ≤ σθ δ n − ω n − + o (1) , having Y ′ = θ + 1 would contradict (4.12)-(4.14) for all sufficiently large i, provided s > θ , n, δ and σ . We may of course assume that s > w i as in [8, page 52], we have by [8, (5.8)] and (4.10) that( δ/ n − ω n − θ σ = lim i →∞ Z ˆ B δ/ ( p ,z ) |∇ w i |≤ s + lim inf i →∞ Z G i |∇ u ε i | p W ( u ε i ) / . (4.15)Using the co-area formula, (4.11), the fact that Y ix ( t ) ≤ θ and (4.15), we see that( δ/ n − ω n − θ σ ≤ s + lim inf i →∞ Z − b − b H n − ( { u ε i = t } ∩ G i ) p W ( t ) / dt ≤ s + σθ (1 + s ) lim inf i →∞ L n − ( T ( G i )) . (4.16)Note that T ( G i ) ∩ S bi = ∅ by the definition of G i and S bi , and hence L n − ( T ( G i )) ≤ ω n − ( δ/ n − − L n − ( S bi ). In view of (4.16), this implies that lim sup i →∞ L n − ( S bi ) ≤ c ( σ, n, θ ) s , completing the proof of (4.8).We next verify that(4.17) ˆ S bi ⊂ T ( {| u ε i | ≤ − b } ∩ ˆ B δ/ ( p , z ) \ G i )as follows: For any x = (ˆ y , , ˆ z ) ∈ ˆ S bi , there exist ˆ y with | ˆ y | ≤ δ/ t ∈ [ − / , − b ]with u ε i (ˆ y , ˆ y , ˆ z ) = t . If (ˆ y , ˆ y , ˆ z ) ∈ G i , again as above we have by [8, Prop. 5.6] that u ε i is C close to q (( y − ˆ y ) /ε i ) in the Lε i -neighborhood of (ˆ y , ˆ y , ˆ z ). In particular, wewould then have T − ( x ) ∩ { u ε i = − / } ∩ ˆ B δ/ ( p , z ) = ∅ , contradicting the assumptionthat x ∈ ˆ S bi . Thus (ˆ y , ˆ y , ˆ z ) ∈ {| u ε i | ≤ − b } ∩ ˆ B δ/ ( p , z ) \ G i , proving (4.17). It follows from (4.17) and (4.10) that(4.18) lim i →∞ L n − ( ˆ S bi ) = 0 . Since S i ⊂ S bi ∪ ˆ S bi , it follows from (4.8), (4.18) and arbitrariness of s > i →∞ L n − ( S i ) = 0 . Now to complete the proof, assume contrary to the assertion of the lemma thatlim sup i →∞ L n − ( B n − \ Q ε i , ) > . Then for some z ∈ B n − , we must have(4.20) lim sup i →∞ L n − ( B n − δ/ ( z ) \ Q ε i , ) > . Take any z ′ ∈ B n − δ/ ( z ) \ Q ε i , . For any y with | y − / | < δ/
4, we have(4.21) x = ( y , , z ′ ) ∈ S i ;for if not, there would exist y with | y − / | < δ/ x = ( y , , z ′ ) / ∈ S i so that u ε i ( y , y , z ′ ) must take all values t ∈ [ − / , /
2] as y ranges over [ − δ/ , δ/ z ′ ∈ Q ε i , , contrary to our assumption. Thus the claim (4.21) holds, andsays that(4.22) Z i ≡ { ( y , , z ′ ) : z ′ ∈ B n − δ/ ( z ) \ Q ε i , , | y − / | < δ/ } ⊂ S i . But then since L n − ( Z i ) = δ L n − ( B n − δ/ ( z ) \ Q ε i , ), the statements (4.19), (4.20) and (4.22)are contradictory, completing the proof of the lemma. ✷ In Lemma 4.3 below we shall make hypotheses and use notation as follows: Let u ∈ C ( B × B n − ) and suppose that t is a regular value of u with M = u − ( t ) = ∅ , so that M is an ( n − C submanifold of B × B n − . Let B u be thefunction defined by (2.4). Let L be the set of points z ∈ B n − satisfying the following tworequirements: (i) T − ( z ) ∩ M = ∅ and (ii) z is a regular value of the map T | M : M → R n − ,where T : R n → R n − is the orthogonal projection. Then for each z ∈ L, ℓ z ≡ T − ( z ) ∩ M is a C z ∈ L, let κ z ( p ) denote the curvature of ℓ z at p ∈ ℓ z . Lemma 4.3.
Let the hypotheses and notation be as described in the preceding paragraph.Then we have for any Borel set G ⊂ L, Z G dz Z ℓ z | κ z | ds ≤ Z M ∩ T − ( G ) B u d H n − ! (cid:0) H n − ( M ∩ T − ( G )) (cid:1) . Proof . Since L is open in B n − , we may choose an increasing sequence of open sets L i ⊂⊂ L such that ∪ ∞ i =1 L i = L . Then M i = T − ( L i ) ∩ M is a C submanifold-with-boundary ∂M i = T − ( ∂L i ) ∩ M in B × B n − . On M i , ( u y , u y ) = (0 , x ∈ M i , there exists ρ x > M ∩ B ρ x ( x ) is the graph of a C function v definedover an open subset U of either the plane { y = 0 } or the plane { y = 0 } . We may cover M i with a finite number of such coordinate charts M i ∩ B ρ x ( x ). Let us now fix such a chart, andassume without loss of generality that U ⊂ { y = 0 } for that chart, so that v = v ( y , z ) for( y , z ) ∈ U and by the definition of M, v satisfies u ( y , v ( y , z ) , z ) = t for all ( y , z ) ∈ U. In TABLE PHASE INTERFACES IN THE VAN DER WAALS–CAHN–HILLIARD THEORY 15 particular, for each z ∈ T ( U ), we have that ℓ z ∩ B ρ ( x ) = { ( y , v ( y , z ) , z ) : y ∈ U ∩ T − ( z ) } and hence κ z = v y y (1 + v y ) − . Using the identity u ( y , v ( y , z ) , z ) ≡ t on U , this can beexpressed in terms of u as κ z = − u y y u y − u y y u y u y + u y y u y ( u y + u y ) . Since the length element ds is given by ds = q v y dy = p u y + u y | u y | dy , we have that | κ z | ds = | u y y u y − u y y u y u y + u y y u y || u y | ( u y + u y ) dy . Next for unit vector m ∈ R n with m ⊥ ∇ u and M = ( ∇ u ), we have( m t M m ) ≤ | M m | = m t M m ≤ tr( M ) − |∇ u | ( ∇ u ) t M ( ∇ u ) = |∇ u | | B u | . Note that here we used the non-negativity of the eigenvalues of M . Since ( u y , − u y , , · · · , ⊥∇ u , we deduce that | u y y u y − u y y u y u y + u y y u y | ≤ ( u y + u y ) B u |∇ u | , which implies that | u y y u y − u y y u y u y + u y y u y || u y | ( u y + u y ) ≤ B u |∇ u || u y | . We also note that q v y + v z + · · · + v z n − = |∇ u || u y | , so that | u y y u y − u y y u y u y + u y y u y || u y | ( u y + u y ) ≤ B u q v y + v z + · · · + v z n − , where both the expression on the left hand side and the function B u are evaluated at( y , v ( y , z ) , z ) . After integrating over ( y , z ) ∈ U ∩ { z ∈ L i ∩ G } and summing over thefinitely many coordinate charts employing a suitable partition of unity subordinate to thecoordinate charts, we obtain Z L i ∩ G dz Z ℓ z | κ z | ds ≤ Z M i ∩ T − ( G ) B u d H n − . By letting i → ∞ in this and using the Cauchy-Schwarz inequality on the right hand side,we deduce the desired estimate. ✷ We now proceed to derive the contradiction necessary to prove Proposition 3.2. Firstnote that by Lemmas 4.1 and 4.2 we have that(4.23) lim i →∞ L n − ( B n − \ ( D ε i ∩ Q ε i )) = 0 . For the rest of the proof let T : B × B n − → B n − be the orthogonal projection. By thedefining property of D ε i we have that Z T − ( D εi ∩ Q εi ) ∩{| u εi |≤ } B ε i |∇ u ε i | dx ≤ ε i c Z B × B n − B ε i |∇ u ε i | dx which by (4.3) tends to 0 as i → ∞ . By the co-area formula it then follows thatlim i →∞ Z − dt Z T − ( D εi ∩ Q εi ) ∩{ u εi = t } B ε i d H n − = 0 . Now choose a generic t ∈ ( − / , /
2) such that { u ε i = t } is a C hypersurface of B × B n − for each i = 1 , , , . . . , lim i →∞ Z T − ( D εi ∩ Q εi ) ∩{ u εi = t } B ε i d H n − = 0and lim inf i →∞ H n − ( T − ( D ε i ∩ Q ε i ) ∩ { u ε i = t } ) < ∞ . This last requirement can be met since by the co-area formula and Fatou’s lemma, Z / − / lim inf i →∞ H n − ( T − ( D ε i ∩ Q ε i ) ∩ { u ε i = t } ) dt ≤ lim sup i →∞ Z T − ( D εi ∩ Q εi ) ∩{| u εi |≤ } |∇ u ε i | dx ≤ lim sup i →∞ ε i c Z B |∇ u ε i | dx < ∞ , Applying Lemma 4.3 with u = u ε i and G = D ε i ∩ Q ε i , we see in view of (4.23) thatafter passing to a subsequence without changing notation, there is a point z i ∈ D ε i ∩ Q ε i ,i = 1 , , , . . . , such that(4.24) lim i →∞ Z ℓ izi | κ iz i | ds = 0 , where ℓ iz = T − ( z ) ∩ { u ε i = t } and κ iz is the curvature of the curve ℓ iz . Note that ℓ iz i is theunion of disjoint, connected, embedded planar curves having at least one point in each ofthe disjoint balls B δ ( p j ) × { z i } , j = 1 , , . . . , N, and no boundary point in B × B n − . Since N ≥
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Department of Mathematics, Hokkaido University, Sapporo 060-0810 Japan.
E-mail address : [email protected] (N. Wickramasekera) Department of Pure Mathematics and Mathematical Statistics, Uni-versity of Cambridge, Cambridge, CB3 0WB, United Kingdom.
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