Sharp Interface Limit of a Stokes/Cahn-Hilliard System, Part II: Approximate Solutions

We construct rigorously suitable approximate solutions to the Stokes/Cahn-Hilliard system by using the method of matched asymptotics expansions. This is a main step in the proof of convergence given in the first part of this contribution, where the rigorous sharp interface limit of a coupled Stokes/Cahn-Hilliard system in a two dimensional, bounded and smooth domain is shown. As a novelty compared to earlier works, we introduce fractional order terms, which are of significant importance, but share the problematic feature that they may not be uniformly estimated in $\epsilon$ in arbitrarily strong norms. As a consequence, gaining necessary estimates for the error, which occurs when considering the approximations in the Stokes/Cahn-Hilliard system, is rather involved.

This work is organized as follows: Section 2 gives a short overview over the needed mathematical tools, particularly existence results for parabolic equations on Γ and a short summary of the differential geometric properties that will be needed later on. Section 3 is based on the approaches in [1,5,6,9]; here we present results for the construction of inner, outer and boundary terms of arbitrarily high order of the asymptotic expansions for solutions of (1.1)-(1.7). Due to constraints to the length of this contribution, many details are left out, but can be found in [7]. In Subsection 3.2, we introduce the auxiliary function w ǫ 1 , which turns out in [3] to be a representation of the leading term of the error in the velocity v ǫ A − v ǫ . Subsection 3.3 is then concerned with constructing fractional order terms in the asymptotic expansion, which are defined with the help of solutions to a nonlinear evolution equation involving w ǫ 1 .
To rigorously justify that the "approximate solutions" constructed in the work really are a good approximation of solutions, it is necessary to estimate the remainder terms in Section 4, i.e., the functions r ǫ CH1 , r ǫ CH2 , r ǫ S and r ǫ div presented in Theorem 1.4. Thus, in Section 4, we analyze these terms in detail, starting with a proper definition of the involved approximate solutions and a subsequent structural representation of r ǫ CH1 , etc. The facts that the terms of fractional order may not be estimated uniformly in ǫ in arbitrarily strong norms and that there appear terms of relatively low orders of ǫ in the representations of the remainder, when discussing the region close to the interface, account for many technical difficulties. The involved estimates rely heavily on Lemma 3.19, which is a direct consequence of our construction scheme of the fractional order terms. The actual proof for Theorem 1.4 is given at the end of this article.

Differential-Geometric Background
The following overview was already given in [3] in more detail; due to the importance of the concepts in view of later considerations in this article and for the sake of completeness, we give a brief reminder.
We parameterize the curves (Γ t ) t∈[0,T 0 ] by choosing a family of smooth diffeomorphisms X 0 : Moreover, we define the tangent and normal vectors on Γ t at X 0 (s, t) as We choose X 0 (and thereby the orientation of Γ t ) such that n(., t) is the exterior normal with respect to Ω − (t). Thus, for a point p ∈ Γ t with p = X 0 (s, t) it holds n Γt (p) = n(s, t). Furthermore, V (s, t) := V Γt (X 0 (s, t)) and H(s, t) := H Γt (X 0 (s, t)) and V (s, t) = ∂ t X 0 (s, t) · n(s, t) for all (s, t) ∈ T 1 × [0, T 0 ] by definition of the normal velocity. We write for a function v : . Choosing δ > 0 small enough, the orthogonal projection Pr Γt : Γ t (3δ) → Γ t is well defined and smooth for all t ∈ [0, T 0 ] and the mapping φ t (x) = (d Γ (x, t), Pr Γt (x)) is a diffeomorphism from Γ (3δ) onto its image. Its inverse is given by φ −1 t (r, p) = p + rn Γt (p). Although Pr Γt and φ t are well defined in Γ t (3δ), almost all computations later on are performed in Γ t (2δ), which is why, for the sake of readability, we work on Γ t (2δ) in the following.
With these notations we have the decompositions Remark 2.1. If h : T 1 × [0, T 0 ] → R is a function that is independent of r ∈ (−2δ, 2δ), the functions ∂ Γ t h, ∇ Γ h and ∆ Γ h will nevertheless depend on r via the derivatives of S. To connect the presented concepts with the classical surface operators we introduce the following notations: Later in this work (from Subsection 3.1.2 on forward) we will often consider h(S(x, t), t) and thus will write for simplicity for (x, t) ∈ Γ(2δ). Using the definitions and notations from this section we gain the identity for (s, t) ∈ T 1 × [0, T 0 ] and (X 0 (s, t), t) = (x, t) ∈ Γ. This might seem cumbersome but turns out to be convenient throughout this work.
Proof. This follows from the chain rule, (2.4) and the notations introduced in Remark 2.1.
The following embedding was already remarked in [1, Subsection 2.5].
The following estimates will be frequently used: Then there are constants C 1 , C 2 > 0 independent of h, ǫ and t such that Proof. Ad 1.: With two changes of variables we obtain Here we used the uniform boundedness of |det(∇X 1 )| in (−2δ, 2δ) × T 1 × [0, T 0 ] in the second inequality. Ad 2.: This can be shown in the same way as the first statement.
For future use, we introduce the concept of remainder terms, similar to [1, Definition 2.5].

Parabolic Equations on Evolving Surfaces
We introduce the space Proposition 2.6. Let T ∈ (0, ∞). Then we have where the operator norm of the embedding is bounded independently of T ,  The following result on solvability of a linearized Mullins-Sekerka/Stokes system is shown in [7] and in a more general form in [2] and will be important for the construction of the approximate solution.

Spectral Theory
In order to be able to access the results from [3], Subsection 2.4, we will need to show that our approximate solution c ǫ A has certain properties. For the readability of presentation, we repeat these assumptions here. Assumption 2.9. Let ǫ ∈ (0, ǫ 0 ), T ∈ (0, T 0 ] and ξ be a cut-off function satisfying (1.22). We assume that c ǫ A : Ω T → R is a smooth function, which has the structure The occurring functions are supposed to be smooth and satisfy for some C * > 0 the following properties: θ 1 : R → R is a bounded function satisfyingˆR Additionally, we suppose that holds.

Construction of Approximate Solutions
In the following we use the method of matched asymptotic expansions to construct approximate solutions (c ǫ A , µ ǫ A , v ǫ A , p ǫ A ) of (1.1)-(1.7). Throughout this chapter the formalism "≈" will represent a formal asymptotic expansion ansatz, that is, writing u ǫ ≈ k≥0 ǫ k u k means that for every integer K ∈ N we have whereũ K+1 is uniformly bounded in ǫ.

The First M + 1 Terms
Many of the following steps are based on ideas taken from [5], [1] and [6]. In order to present the results for the construction of terms of arbitrarily high order in Lemmata 3.6 and 3.8, we devise an inductive scheme similar to the approach in [5]. However, in favor of the brevity of presentation, we did not include this scheme in this article and simply state the results. For the background of the construction and the proofs, see [7].

The Outer Expansion
We assume that in Ω ± T 0 \Γ(2δ) the solutions of (1.1)-(1.7) have the expansions where for fixed c ± 0 the functions f k are polynomials in (c ± 1 , . . . , c ± k ) and are the result of a Taylor expansion. Moreover, f k (c ± 0 , . . . , c ± k ) are chosen such that they do not depend on ǫ. Matching the O(ǫ −1 ) terms yields f ′ (c ± 0 ) = 0 and in view of the Dirichlet boundary data for c ǫ we set Comparing the higher order terms O ǫ k , where k ≥ 1, yields: Remark 3.1.
1. As we will only construct c ± 0 , . . . , c ± M +1 , we need to consider the remainder of the Taylor expansion of f ′ . In this case, we choose to expand f ′ up to order M + 2 and get , which may be of even higher order in ǫ and which are either multiplied by . For µ ± k and p ± k we may use any smooth extension. One possibility is to use the extension operator defined in [10, Part VI, k is as well, by (3.8).
For v ± k we employ the same extension operator and then use the Bogovskii operator to ensure that the extension is divergence free in Γ t (2δ). In particular we may construct a divergence free extension For later use we define

The Inner Expansion
Close to the interface Γ we introduce a stretched variable is a given smooth function and can heuristically be interpreted as the distance of the zero level set of c ǫ to Γ, see also [6,Chapter 4.2]. In the following, we will often drop the ǫ-dependence and write ρ(x, t) = ρ ǫ (x, t).
Now assume that, in Γ(2δ), the identities hold for the solutions of (1.1)-(1.7) and some smooth functionsc ǫ ,μ ǫ ,p ǫ : R×Γ(2δ) → R,ṽ ǫ : R× Γ(2δ) → R 2 . Furthermore, we assume that we have the expansions When referring toc,μ,p,ṽ and the expansion terms we write ∇ = ∇ x and In order to match the inner and outer expansions, we require that for all k the so-called inner-outer matching conditions where ϕ = c k , µ k , v k , p k and k ≥ 0 hold for constants α, C > 0 and all ρ > 0, m, n, l ≥ 0.
Remark 3.2. We will only use the matching conditions for m, n, l ∈ {0, 1, 2}. However, since the ordinary differential equations for (3.27), (3.29), (3.31), (3.33)) are dependent on derivatives of lower order terms, it is necessary and sufficient for the matching conditions to hold for m, n, l ∈ {0, . . . , C(M )} for some C(M ) ∈ N depending on the general number of terms in the expansion.
Similarly as in [5], we introduce auxiliary functions g ǫ (x, t), j ǫ (x, t) and l ǫ (x, t) as well as u ǫ (x, t) and q ǫ (x, t) for (x, t) ∈ Γ(2δ). As a rough guideline, the functions g ǫ , j ǫ , and q ǫ will enable us to fulfill the compatibility conditions in Γ(2δ)\Γ. l ǫ and u ǫ on the other hand are of importance when it comes to fulfilling the matching conditions in Γ(2δ)\Γ. Moreover we choose η : is satisfied. For later use we also define for an arbitrary constant C > 0 and ρ ∈ R. Now we may rewrite (1.1)-(1.4) as where the equalities are only assumed to hold in but we consider them as ordinary differential equations in ρ ∈ R, where (x, t) ∈ Γ(2δ) are seen as fixed parameters. Thus we assume from now on that (3.21)-(3.24) are fulfilled in R × Γ(2δ).
The terms U ± and W ± (cf. (3.13), (3.14)) are used here in order to ensure the exponential decay of the right hand sides; in this context C S > 0 is a constant which will be determined later on (see Remark 3.4). We assume that the auxiliary functions have expansions of the form for (x, t) ∈ Γ(2δ). Matching the ǫ-orders, we gain the following ordinary differential equations in ρ: From (3.21) and (3.22) we get Similaly, from (3.23) and (3.24) we get and Here we used and Here V k−2 , W k−2 , A k−2 , and B k−2 denote terms of order k − 2 or lower which are unimportant in the following -the detailed structure of these terms can be found in [7,Subsection 5.1.2]. In all of the above identities we used the following conventions: 1. All functions with negative index are supposed to be zero. In particular 2. We introduced the notation (3.36)) are terms from a Taylor expansion defined in the same way as in Remark 3.1. In particular, we will later on also use a remainder term f as discussed in Remark 3.1 for the inner solutions. Moreover, we use the convention f 0 (c 0 ) = 0.
We will see after the construction of the zeroth order terms that the term h k appearing on the right hand side of (3.29) is actually multiplied by 0.
Remark 3.4. Note that W ± and U ± , which we inserted in (3.21) and (3.24), are not multiplied by terms of the kind (d Γ − ǫ (ρ + h ǫ )). So we have to make sure they vanish on the set S ǫ . This is accomplished by choosing the constant C S > 0 in a suitable way.
In particular we set holds for all ǫ > 0 small enough. It turns out that h 1 does not depend on the term ǫ 2 (U + η C S ,+ + U − η C S ,− ) and ǫ 2 (W + η C S ,+ +W − η C S ,− ), so this choice of C S does not cause problems. Choosing C S in this way, it is possible to show (see [5,Remark 4 A similar statement holds when d Γ (x, t) < 0.
We assume that for (x, t) ∈ ∂ T Ω (δ) the identities hold for the solutions of (1.1)-(1.7) and smooth functions c ǫ As in the case of the inner expansion, we also assume that the outer-boundary matching conditions where the differential operator ∇ = ∇ x , div = div x , ∆ = ∆ x act only on the variable x and not on z. In the calculations we used |∇d Using (3.40) and equating same orders of ǫ, we get We used the convention that all terms with negative index are supposed to be zero, i.e., µ −2 = µ −1 = 0.
To ensure the Dirichlet boundary condition we suppose that Regarding the boundary condition of the Stokes system we calculate ǫ , x, t n ∂Ω (x) and thus impose

Existence of Expansion Terms
For the proofs of the statements in this subsection we refer to [7, Subsection 5.1.6].

55)
for all (ρ, x, t) ∈ R × Γ(2δ; T 0 ) and the terms of the boundary expansion As a consequence of (3.54), (3.52), the equation for µ ± 0 on Γ t (1.14) and which are bounded on their respective domains, such that for k-th order the outer equations Here v ± k , µ ± k , c ± k and p ± k are considered to be extended onto Remark 3.9. Let us remark upon the difficulties that would arise if we considered e.g. no-slip boundary conditions for v ǫ . In that case, we would demand for v ǫ A to also satisfy v ǫ A = 0 on ∂ T 0 Ω, which may be achieved by suitable changes to the presented boundary layer expansion. As a consequence, the outer solution would need to satisfy (among other equations) where a 1 , a 2 are smooth functions, depending only on lower order terms. As a consequence, the divergence theorem implies 0 =ˆΩ for t ∈ [0, T 0 ]. However, this equality does not have to be satisfied for arbitrary k. To avoid this difficulty, we restricted ourselves to the case of the boundary condition (1.6). Now we "glue" together the inner and outer expansions of c ǫ in order to get an approximate solution. We will repeat this later for approximate solutions of µ ǫ , v ǫ , p ǫ , cf. Definition 4.1. We define ) due to Proposition 2.6.2 and Sobolev embeddings. Furthermore, we setc for ρ ∈ R, (x, t) ∈ Γ(2δ; T ′ ) and For the outer part we set for (x, t) ∈ Ω T ′ and for the boundary part we define (1.22). We now define the approximate solution Later on, the family H will be replaced by the terms of correct order h ǫ M − 1 2 , which will then depend on ǫ. But in order to find those terms we need some preparations first, which will turn out to be more flexible and notationally consistent when they are done with an arbitrary family of functions H.

A First Estimate of the Error in the Velocity
Let the assumptions and notations of Definition 3.10 hold throughout this subsection. Moreover, we denote and calculate for (x, t) ∈ Ω T . We understand the right hand side of equation (3.62) as a functional in (V 0 ) ′ given by Proposition 3.11. Let ǫ 0 ∈ (0, 1) and T ′ ∈ (0, T 0 ] be fixed. Furthermore, let for a given family H = (h ǫ ) ǫ∈(0,ǫ 0 ) ⊂ X T ′ withh ǫ | t=0 = 0 the functionw ǫ,H 1 be defined as the weak solution to (3.62)-(3.64) for ǫ ∈ (0, ǫ 0 ). Then the following statements hold: 1. For all ǫ ∈ (0, ǫ 0 ), there exists a constant C(ǫ) > 0 such that Proof. Ad 1.: By [3, Theorem 2.1] there is a constant C > 0 such that Now in order to estimate the right hand side, we first note that with a constant C > 0 that does not depend on H. This can be deduced from the representation (3.66) and the fact that c B and its appearing derivatives are in L ∞ (∂ T 0 Ω(δ)), c O and its derivatives are in L ∞ (Ω T 0 ) andc I and its appearing derivatives are in L ∞ (R × Γ (2δ; T 0 )). So we obtain , where we used that c ǫ is a known function and thus Now in order to show the statement we first note that for all (x, t) ∈ Γ(2δ, T ′ ) and k, l ∈ {0, 1} due to Taylor's theorem. Here ξ : Γ(2δ, T ′ ) → R is a suitable function depending on H 1 and H 2 . Since all the terms which do not depend on H 1 , H 2 cancel, we may estimate

Constructing the M − 1 2 -th Terms
Our goal is to construct approximate solutions (v ǫ A , p ǫ A , c ǫ A , µ ǫ A ) which fulfill (1.29)-(1.32) in Ω T 0 , where r ǫ S , r ǫ div , r ǫ CH1 and r ǫ CH2 are suitable error terms, which will be discussed in detail in Chapter 4. In (1.31) we consider is the weak solution to (3.62)-(3.64). Moreover, we write and we use a suitable family H = (h ǫ ) ǫ∈(0,ǫ 0 ) ⊂ X T 0 . Due to this appearance of a noninteger order term, it is natural to also consider non-integer order terms in the expansion of (c ǫ , µ ǫ , v ǫ , p ǫ ). More precisely, we assume that terms

appear in the outer expansion and that terms
Moreover, we assume that there is a term ǫ M − and so forth. In the following, we only assume that the zeroth and first order terms have been constructed with the help of Lemmata 3.6 and 3.8.

The Outer Expansion
Using a Taylor expansion in (1.4) as before, we explicitly get in Ω ± which can be derived similarly to (3.8). From (1.1)-(1.2), we deduce that the equations have to hold, as ∇c ± In the following, we assume that v ± are smoothly extended to Ω ± T 0 ∪ Γ(2δ; T 0 ), as discussed in Remark 3.1 for the integer order terms.

The Inner Expansion
We assume that the matching conditions (3.

19) hold for the inner terms
. As these are the first terms of fractional order which we introduce, the following identities can be derived from (3.21)-(3.24): which we will not construct. These are given by Note the appearance of w ǫ 1 | Γ ·n∂ ρ c 0 in (3.82) which is due to the fact that we want to approximate (1.31). In the following corollary we use the notation for terms u ± k of the asymptotic expansion.
where σ is given as in (1.18) Proof. This can be shown by direct calculations.
Proof. It is important to be aware that w ǫ 1 depends on h ǫ It remains to prove the second statement. Let T ǫ > 0 be given for ǫ ∈ (0, ǫ 0 ) as in the assumptions. As a consequence of Theorem 2.8, we have and C 1 can be chosen independently of T ′ . Now we choose C(K) as in Lemma 3.14 (note that this constant is independent of the choice ofh ǫ in the lemma) and defineĉ(K) := 2C 1 C(K). Then we find that Using Lemma 3.14 again (with T ′ ǫ instead of T ǫ ), we get the existence of ǫ 1 ∈ (0, ǫ 0 ] such that for all ǫ ∈ (0, ǫ 1 ) with the same constant C(K) as above. Thus, by (3.99) we have for all ǫ ∈ (0, ǫ 1 ). By the definition of T ′ ǫ this already implies T ′ ǫ = T ǫ . Finally, (3.98) follows from (2.29) and (2.19) taken together with the embedding H Notation 3.18. For simplicity, we often write v M − 1 etc., especially if we consider fractional and integer expansion orders together, as in Section 4.1.
The following lemma is a key ingredient in order to estimate the remainder terms properly. Lemma 3.19. Let the M − 1 2 -th order terms be given as in Lemma 3.17, let the assumptions of Theorem 3.15.2 hold and let ǫ ∈ (0, ǫ 1 ).
we get the desired splitting on Γ (with L 2 = 4). It is clear by the properties of c 0 and η that all terms (3.20) and the fact that Since θ ′ 0 has exponential decay by (1.21) we get (3.106). Now note that by the definition in Remark 2.1 we have e.g.
where S is a smooth function Γ(2δ; T 0 ) with bounded derivatives. Thus, by (3.97) and Proposition 2.6.3 it follows · ∇ Γ h 1 L 6 (0,Tǫ;L 2 (Γt(2δ))) ≤ C(K) and the same estimate also holds true if we exchange Γ t (2δ) for Γ t . On the other hand, the L 6 (L 2 ) estimate for µ ±,ǫ Note in particular that the assumptions of Lemma 3.13 are satisfied in this situation. Addtitionally, there is some C > 0 such that for all ǫ ∈ (0, 1) small enough. This is the case since c ± 0 = ±1 in Ω ± T 0 (cf. (3.7)) and since c B 0 = −1 and c B 1 = c − 1 in ∂ T 0 Ω(δ) due to Corollary (3.5). Moreover, it holds there is a discrepancy between the present contribution and [3]. In [3], µ I and v I are defined without the appearance of fractional order terms and in the present context, we would define

The Inner Remainder Terms
In the following, let Assumption 4.2 hold and we work under the notations and assumptions of Definition 4.1. We now analyze up to which order in ǫ the equations (1.1)-(1.4) are fulfilled by the inner solutions c I , µ I , v I , p I . For this we use the ordinary differential equations satisfied by (c k , µ k , v k , p k−1 ) for k ∈ {0, . . . , M + 1} as constructed for the inner terms and evaluate them at for (x, t) ∈ Γ(2δ; T ǫ ) and ǫ ∈ (0, ǫ 1 ). Before we give the explicit formula, note that we can choose ǫ 1 so small that for all ǫ ∈ (0, ǫ 1 ) we have |h ǫ A − h 1 | ≤ 1 due to (3.97). Thus, (3.38) is satisfied and using Remark 3.4 we get for (x, t) ∈ ∂ T 0 Ω(δ) and ǫ ∈ (0, ǫ 1 ) and the outer equations as discussed
Step  in the last line. A similar estimate holds on Ω − Tǫ ∩ Γ(2δ; T ǫ ). Now (4.26) follows since all not considered terms may be treated by simply using Hölder's inequality and L ∞ bounds (for v i · ∇c j note ∇c 0 = 0 and apply (3.98) for the fractional order term).
Regarding (4.27), the same ideas may be applied with the sole difference that z is only L 2 in time and as a consequence we do not get the term T 1 2 ǫ in the estimates. Due to the many similarities, we only show three estimates in detail: where we used Lemma 2.4.1 in the first inequality and H 1 (Γ t (2δ)) ֒→ L 2,∞ (Γ t (2δ)) (cf. Lemma 2.3) as well as (3.98) for (x, t) ∈ Γ(δ; T ǫ ) with according estimates for Z, F R 1 , F R 2 . Using theses estimates we get When inspecting the remainder terms (4.11)-(4.14), one observes that the terms are multiplied by a lower power of ǫ than the rest. Gaining these missing powers of ǫ needs delicate work; the main ingredient for this is that we have intricate structural knowledge of A M − 1 2 etc. due to Lemma 3.19.
To treat J 2 1 we again use the fact that all terms of kind B 2,Γ k exhibit exponential decay and thus Then we use Lemma 3.19.3 and (3.109) to obtain e.g.
For the other terms, the same argumentation as before can be applied.
For I 1 1 we use the decomposition in Lemma 3.19.1 on R × Γ to conclude The estimate in (4.37), (4.38), the properties of A 2,Γ k as shown in (3.106) and the exponential decay of θ ′ 0 imply for ǫ > 0 small enough, where we used the estimate for Z and (3.105) for A 1,Γ k .
The only remaining term in (4.49) can be treated bŷ where we used Lemma 3.14 and ∇c O = O(ǫ) in L ∞ (Ω ± T 0 ). Thus, we need only consider r ǫ CH1 in ∂ Tǫ Ω(δ)\∂ Tǫ Ω( δ 2 ). Here we get a structure very similar to (4.49): The proof now follows in the same manner as the one for (4.49) using the already shown estimates for r ǫ CH1,O and r ǫ CH1,B as well as the estimates close to the boundary in Corollary 4.9. This shows (1.34). Proof of (4.45): We use a similar approach as before: In Γ(δ; T ǫ ) we have r ǫ CH2 = r ǫ CH2,I , where r ǫ CH2,I is defined in (4.12). For all terms in r ǫ CH2,I , which can be estimated in L ∞ (Γ(2δ; T ǫ )) (uniformly in ǫ), we may use Proposition 4.6 to show the claim. Noting (4.9), the only terms that may not be treated in this fashion are the ones involving ∆ Γ h ǫ where we used (3.98) and (1.28). As c ± i ∈ L ∞ (Ω ± T 0 ) for all i ∈ {0, . . . , M + 1}, a similar estimate follows by (1.28) for the remaining terms in r ǫ CH2,O (cf. Remark 3.1 for thef term). In ∂ Tǫ Ω( δ 2 ), it holds r ǫ CH2 = r ǫ CH2,B and we may proceed as in Ω Tǫ \ (Γ (2δ; T ǫ ) ∪ ∂ Tǫ Ω(δ)). In Γ(2δ; T ǫ )\Γ(δ; T ǫ ), we have (4.53) The estimate for the second line in (4.53) follows by similar arguments as in the proof of (1.34), by using Corollary 4.9.
Thus we have to show To this end we will use the same notations as discussed right at the beginning of the proof of Lemma 4.7. We will first consider 1 ǫ θ ′ 0 n instead of ∇c ǫ A . Using the fundamental theorem of calculus we have ψ(r, s) = ψ(0, s) +´r 0 ∂ n ψ(r, s)dr for (r, s, t) ∈ (−δ, δ) × T 1 and write ˆΓ t(δ) Γ θ ′ 0 ψ drds =: I 1 1 + I 2 1 + I 2 . By Lemma 3.19 (after choosing ǫ > 0 small enough such that (4.37) holds), we may estimate by Lemma 3.19. As ∂ ρ c 1 ∈ R α and all other terms appearing in ∇c ǫ A are already of higher order in ǫ (see (4.57)). This proves (4.58) and as a consequence also In view of (4.56), we still need to consider ´Ω \Γt(δ) r ǫ CH2 ∇c ǫ A ·ψdx . But this term may be treated with similar techniques as used in the proof of (1.35). This shows (1.37).
Finally, (1.38) follows immediately by noting that r ǫ CH1 = r ǫ CH1,B in ∂ T 0 Ω δ 2 , the form of the boundary remainder terms (4.19) and the fact that all occurring terms in those boundary remainders are either uniformly bounded in L ∞ (∂ T 0 Ω(δ)) or may be estimated in L 2 (Ω − Tǫ ) with the help of (3.98).