Maximal Regularity of Parabolic Transmission Problems

Linear reaction-diffusion equations with inhomogeneous boundary and transmission conditions are shown to possess the property of maximal Lp regularity. The new feature is the fact that the transmission interface is allowed to intersect the boundary of the domain.


Introduction
The emerging and understanding of the theory of maximal regularity for parabolic differential equations, which took place within the last three or so decades, has provided a firm basis for a successful handling of many demanding nonlinear problems. Among them, phase transition issues play a particularly prominent role. The impressive progress which has been made in this field with the help of maximal regularity techniques is well-documented in the book by J. Prüss and G. Simonett [26]. The reader may also consult the extensive list of references and the 'Bibliographic Comments' in [26] for works of other authors and historical developments.
The relevant mathematical setup is usually placed in the framework of parabolic equations in bounded Euclidean domains, the interface being modeled as a hypersurface. In all works known to the author, it is assumed that the interface lies in the interior of the domain. Thus the important case of membranes touching the boundary has not been treated so far. A noteworthy exception is the recent paper by Ph. Laurençot and Ch. Walker [25]. These authors establish the unique solvability in the strong L 2 sense of a two-dimensional stationary transmission problem taking advantage of a particularly favorable geometric setting.
In this paper we establish the maximal regularity of linear inhomogeneous parabolic transmission boundary value problems for the case where the interface intersects the boundary transversally. This is achieved by allowing the equations to degenerate near the intersection manifold and working in suitable weighted Sobolev spaces. We restrict ourselves to the simplest case of a fixed membrane and a single reaction-diffusion equation.
In a forthcoming publcation we shall use our present result to establish the local well-posedness of quasilinear equations with nonlinear boundary and transmission conditions.

The Main Result
Now we outline-in a slightly sketchy way-the main result of this paper. Precise definitions of notions, facts, and function spaces which we use here without further explanation, are given in the subsequent sections.
Let Ω be a bounded domain in R m , m ≥ 2, with a smooth boundary Γ lying locally on one side of Ω. By a membrane in Ω we mean a smooth oriented hypersurface S of (the manifold) Ω with a (possibly empty) boundary Σ such that S ∩ Γ = Σ. Thus S lies in Ω if Σ = ∅. Otherwise, Σ is an (m − 2)-dimensional oriented smooth submanifold of Γ. In this case it is assumed that S and Γ intersect transversally. Note that we do not require that S be connected. Hence, even if Σ = ∅, there may exist interior membranes. However, the focus in this paper is on membranes with boundary. Thus we assume until further notice that Σ = ∅.
We denote by ν the inner (unit) normal on Γ and by ν S the positive normal on S. Of concern in this paper are linear reaction-diffusion equations with nonhomogeneous boundary and transmission conditions of the following form.
Set Au := − div(a grad u), Bu := γa∂ ν u, where γ 0 is the trace operator at t = 0. We are interested in the strong L p solvability of (2.1), that is, in solutions possessing second order space derivatives in L p . However, since S intersects Γ, we cannot hope to get solutions which possess this regularity up to Σ. Instead, it is to be expected that the derivatives of u blow up as we approach Σ. For this reason we set up our problem in weighted Sobolev spaces where the weights control the behavior of ∂ α u for 0 ≤ |α| ≤ 2 in relation to the distance from Σ. This requires that the differential operator is adapted to such a setting, which means that the 'diffusion coefficient' a tends to zero near Σ. In other words: we will have to deal with parabolic problems which degenerate near Σ. To describe the situation precisely, we introduce curvilinear coordinates near Σ as follows.
To introduce weighted Sobolev spaces on U \S we put where ∇ Σ is the Levi-Civita connection on Σ for the metric g Σ . Moreover, 1 < p < ∞ and ThenW 2 p (U \S; r) is the completion ofC 2 (U \S) in L 1,loc (U \S) with respect to the norm · W 2 p (U\S;r) . The (global) weighted Sobolev space consists of all u ∈ L 1,loc (Ω\S) with u U ∈W 2 p (U \S; r) and u V ∈W 2 p (V \S). It is a Banach space with the norm , whose topology is independent of the specific choice of ε and ω. Similarly, the Lebesgue space X 0 p :=W 0 p (Ω\S; r) is obtained by replacing u in (2.11) by |u|. Moreover, where (·|·) θ,p is the real interpolation functor of exponent θ.
We also need time-dependent anisotropic spaces. For this we use the notation s/2 := (s, s/2), 0 ≤ s ≤ 2. Then Here theW s p (X \Σ; r) are trace spaces of X 2 p . Moreover, ByBC(Ω\S) we mean the space of bounded and continuous functions (with possible jumps across S), endowed with the maximum norm. ThenBC Furthermore, To indicate the nonautonomous structure of (2.1), we write a(t) := a(·, t) and, correspondingly, A(t), B(t), and C(t). Now we are ready to formulate the main result of this paper, the optimal solvability of linear reaction-diffusion transmission boundary value problems.

Corollary 2.2
Suppose a is independent of t, that is, a ∈BC 1 (Ω\S; r). Set and A r := A r |X 2 p,0 . Then −A r , considered as a linear operator in X 0 p with domain X 2 p,0 , generates on X 0 p a strongly continuous analytic semigroup.
Proof The theorem implies that A r has the property of maximal X 0 p regularity. This fact is well-known to imply the claim (e.g., [6,Capter III] or [21]). Theorem 2.1 is a special case of the much more general Theorems 7.1 and 14.1. They also include Dirichlet boundary conditions and apply to transmission problems in general Riemannian manifolds with boundary and bounded geometry.
The situation is considerably simpler if Σ = ∅, that is, if only interior transmission hypersurfaces are present. Of course, if S = ∅, then (2.13) reduces to a quasilinear reaction-diffusion equation with inhomogeneous boundary conditions. In these cases no degenerations do occur.
We refrain from considering operators (A r , B r ) with lower order terms. This case will be covered by the forthcoming quasilinear result.
In the case of an interior transmission surface (that is, Σ = ∅) and if a is independent of t, Theorem 2.1 is a special case of Theorem 6.5.1 in [26]. The latter theorem applies to systems and provides an L p -L q theory.
If Σ = ∅, then the basic difficulty in proving Theorem 2.1 stems from the fact that Ω\Σ and, consequently, S \Σ and Γ\Σ, are no longer compact. The fundamental observation which makes the proofs work is the fact that we can consider Ω\Σ as a (noncompact) Riemannian manifold with a metric g which coincides on U (ε/3) with the singular metric r −2 dν ⊗ dµ + g Σ and on V with the Euclidean metric. With respect to this metric, A r is then a uniformly elliptic operator. Theorems 4.3 and 5.1 show that (Ω\Σ, g) is a uniformly regular Riemannian manifold in the sense of [9]. Thus we are led to consider linear parabolic equations with boundary and transmission conditions on such manifolds. As in the compact case, by means of local coordinates the problem is reduced to Euclidean settings. However, since we have to deal with noncompact manifolds, we have to handle simultaneously infinitely many model problems. In order for this technique to work, we have to establish uniform estimates which are in a suitable sense independent of the specific local coordinates. In addition, special care has to be taken in 'gluing together the local model problems'. These are no points to worry about in the compact case.
In our earlier paper [11] we have established an optimal existence theory for linear parabolic equations on uniformly regular Riemannian manifolds without boundary. The present proof extends those arguments to the case of manifolds with boundary. The presence of boundary and transmission conditions adds considerably to the complexity of the problem and makes the paper rather heavy.
In Section 3 we collect the needed background information. In the subsequent two sections we establish the differential geometric foundation of transmission surfaces in uniformly regular and singular Riemannian manifolds.
After having introduced the relevant function spaces in Section 6, we present in Section 7 the basic maximal regularity theorem in anisotropic Sobolev spaces for linear non-autonomous reaction-diffusion equations with nonhomogeneous boundary and transmission conditions on general uniformly regular Riemannian manifolds. Its rather complex proof occupies the next five sections. Finally, in the last section it is shown that our general results apply to the Euclidean setting presented here.

Uniformly Regular Riemannian Manifolds
In this section we recall the definition of uniformly regular Riemannian manifolds and collect those properties of which we will make use. Details can be found in [8], [9], [10], and in the comprehensive presentation [14]. Thus we shall be rather brief.
We use standard notation from differential geometry and function space theory. In particular, an upper, resp. lower, asterisk on a symbol for a diffeomorphism denominates the corresponding pull-back, resp. push-forward (of tensors). By c, resp. c(α) etc., we denote constants ≥ 1 which can vary from occurrence to occurrence. Assume S is a nonempty set. On the cone of nonnegative functions on S we define an equivalence relation ∼ by f ∼ g iff f (s)/c ≤ g(s) ≤ cf (s), s ∈ S.
An m-dimensional manifold is a separable metrizable space equipped with an m-dimensional smooth structure. We always work in the smooth category.
Let M be an m-dimensional manifold with or without boundary. If κ is a local chart, then we use U κ for its domain, the coordinate patch associated with κ. The chart is normalized if κ(U κ ) = Q m κ , where Q m κ = (−1, 1) m if U ⊂M , the interior of M , and Q m κ = [0, 1) × (−1, 1) m−1 otherwise. An atlas K is normalized if it consists of normalized charts. It is shrinkable if it normalized and there exists r ∈ (0, 1) such that κ −1 (rQ m κ ) ; κ ∈ K is a covering of M . It has finite multiplicity if there exists k ∈ N such that any intersection of more than k coordinate patches is empty.
The atlas K is uniformly regular (ur) if (i) it is shrinkable and has finite multiplicity; Two ur atlases K and K are equivalent if (i) there exists k ∈ N such that each coordinate patch of K meets at most k coordinate patches of K, and vice versa; (ii) condition (3.1)(ii) holds for all (κ, κ) and ( κ, κ) belonging to K × K.
This defines an equivalence relation in the class of all ur atlases. An equivalence class thereof is a ur structure. By a ur manifold we mean a manifold equipped with a ur structure. A Riemannian metric g on a ur manifold M is ur if, given a ur atlas K, Here g m := (·|·) = dx 2 is the Euclidean metric 1 on R m and (i) is understood in the sense of quadratic forms. This concept is well-defined, independently of the specific K. A uniformly regular Riemannian (urR) manifold is a ur manifold, endowed with a urR metric.
Remarks 3.1 (a) Given a (nonempty) subset S of M and an atlas K, We say that K is normalized on S, resp. has finite multiplicity on S, resp. is shrinkable on S if K S possesses the respective properties. Moreover, K is ur on S if (3.1) applies with K replaced by K S . Similarly, two atlases K and K, which are ur on S, are equivalent on S if (3.2) holds with K and K replaced by K S and K S , respectively. This induces a ur structure on S. Finally, M is ur on S if it is equipped with a ur structure on S. (b) Suppose K is a ur atlas for M on S. Given any ε > 0, there exists a ur atlas K ′ on S such that diam g (U κ ) < ε for κ ∈ K ′ , where diam g is the diameter with respect to the Riemannian distance d g .
In the following examples we use the natural ur structure (e.g., the product ur structure in Example 3.2(c)) if nothing else is mentioned.
Examples 3.2 (a) Each compact Riemannian manifold is a urR manifold and its ur structure is unique.
(b) Let Ω be a bounded domain in R m with a smooth boundary such that Ω lies locally on one side of it. Then (Ω, g m ) is a urR manifold.
More generally, suppose that Ω is an unbounded open subset of R m whose boundary is ur in the sense of F.E. Browder [19] (also see [24, IV. §4]). Then (Ω, g m ) is a urR manifold. In particular, (R m , g m ) and (H m , g m ) are urR man- (c) If (M i , g i ), i = 1, 2, are urR manifolds and at most one of them has a nonempty boundary, then (M 1 × M 2 , g 1 + g 2 ) is a urR manifold.
(d) Assume M is a manifold and N a topological space. Let f : N → M be a homeomorphism. If K is an atlas for M , then f * K := { f * κ ; κ ∈ K } is an atlas for N which induces the smooth 'pull-back' structure on N . If K is ur, then f * K also is ur.
Let (M, g) be a urR manifold. Then f * (M, g) := (N, f * g) is a urR manifold and the map f : It follows from these examples, for instance, that the cylinders R × M 1 or R + × M 2 , where M i are compact Riemannian manifolds with ∂M 2 = ∅, are urR manifolds. More generally, Riemannian manifolds with cylindrical ends are urR manifolds (see [10] where more examples are discussed).
Without going into detail, we mention that a Riemannian manifold without boundary is a urR manifold iff it has bounded geometry (see [9] for one half of this assertion and [22] for the other half). Thus, for example, (H m , g m ) is not a urR manifold. A Riemannian manifold with boundary is a urR manifold iff it has bounded geometry in the sense of Th. Schick [27] (also see [16], [17], [18], [23] for related definitions). Detailed proofs of these equivalences will be found in [14].

Uniformly Regular Hypersurfaces
Let (M, g) be an oriented Riemannian manifold with (possibly empty) boundary Γ. If it is not empty, then there exists a unique inner (unit) normal vector field ν = ν Γ on Γ, that is, a smooth section of T Γ M , the restriction of the tangent bundle T M of M to Γ. Furthermore, Γ is oriented by the inner normal in the usual sense.
Suppose that S is an oriented hypersurface inM , an embedded submanifold of codimension 1. Then there is a unique positive (unit) normal vector field ν S on S, where 'positive' means that ν S (p), β 1 , . . . , β m−1 is a positive basis for This means that, given We say that N has a uniform normal geodesic tubular neighborhood of width ε if the following is true: ε > 0 and there exists an open neighborhood N (ε) of N in M such that is a diffeomorphism satisfying ϕ N (N ) = {0} × N . If N = Γ, then a uniform tubular neighborhood is a uniform collar.
Given any embedded submanifold C of M , with or without boundary, we denote by g C the pull-back metric ι * g, where ι : C ֒→ M is the natural embedding. Now we suppose that This means that there exists an oriented ur atlas for M . Let S be a hypersurface with boundary Σ such that Σ = S ∩ Γ. Thus S ⊂M if Σ = ∅. An atlas K for M is S-adapted if for each κ ∈ K S one of the following alternatives applies: Then S is a regularly embedded hypersurface in M , a membrane for short, if there exists an oriented ur atlas K for M which is S-adapted.
Let S be a membrane. Each S-adapted atlas for K induces (by restriction) a ur structure and a (natural) orientation on S. Moreover, the ur structure and the orientation of S are independent of the specific choice of K.
For the proof of all this and the following theorem we refer to [14].
Then N has a uniform tubular neighborhood has a uniform tubular neighborhood of width ε(ρ) inM .
Now we suppose that S is a membrane with Σ = ∅. It follows from (ii) and (iii) that Σ has a uniform tubular neighborhood ψ : where we assume without loss of generality that ϕ and ψ are of the same width. Then is an orientation preserving diffeomorphism, a tubular neighborhood of Σ in M of width ε.
We refer once more to [14] for the proof of the next theorem. Henceforth, Theorem 4.2 Assume (4.2) and S is a membrane with nonempty boundary Σ. Then . We now restrict the class of membranes under consideration by requiring that S intersects Γ uniformly transversally. This means the following: In general, a submanifold C of a manifold B intersects ∂B transversally if The following theorem furnishes an important large class of urR manifolds and membranes intersecting the boundary uniformly transversally. Theorem 4.3 Let (M, g) be a compact oriented Riemannian manifold. Assume S is an oriented hypersurface in M with nonempty boundary Σ ⊂ Γ and S intersects Γ transversally. Then (M, g) is a urR manifold and S is a membrane intersecting Γ uniformly transversally.
Proof Example 3.2(a) guarantees that (M, g) is an oriented urR manifold. Thus (Γ, g Γ ) is an oriented urR manifold by Theorem 4.1(i). Since S intersects Γ transversally, it is a well-known consequence of the implicit function theorem that Σ is a compact hypersurface in Γ without boundary. It is oriented, being the boundary of the oriented manifold S. Hence, invoking Example 3.2(a) once more, (Σ, g Σ ) is an oriented urR manifold. As it is compact, it has a uniform tubular neighborhood in Γ. Thus, Γ having a uniform collar, Σ has a uniform tubular neighborhood χ in M of some width ε.
Since S intersects Γ transversally, χ S ∩ Σ σ (ε) can be represented as the graph of a smooth function f σ : [0, ε) → (−ε, ε) with f σ (0) = 0, and f σ depends smoothly on σ ∈ Σ. The compactness of Σ implies that (4.4) is true. Hence S intersects Γ uniformly transversally. Now, due to the compactness of S, it is not difficult to see that S is a regularly embedded submanifold of M . The theorem is proved. (b) Suppose (M, g) is an oriented urR manifold and S a membrane without boundary. Then the fact that S has a uniform tubular neighborhood inM prevents S from either reaching Γ or 'collapsing' at infinity.

The Singular Manifold
In this section (M, g) is an oriented urR manifold and S a membrane with nonempty boundary Σ such that S intersects Γ uniformly transversally.
is an open neighborhood of Σ in M contained in Σ(ε). We put Furthermore, r and ρ are given by (2.4) and (2.5), respectively. Then we define a Riemannian metric g on M by Note that, see Theorem 4.2, Hence ( M , g) is a Riemannian manifold with a wedge singularity near Σ.
The following theorem is the basis for our approach. It implies that it suffices to study transmission problems for membranes without boundary on urR manifolds.
is an oriented urR manifold and S := S \Σ is a membrane in M without boundary.
Let κ = (x 1 , . . . , x m ) be a local coordinate system and set ∂ i := ∂/∂x i . Then are the Christoffel symbols. It follows that |∇u| 2 and |∇ 2 u| 2 For 1 ≤ q < ∞ we set Lq(M,g) 2 If V is a vector bundle over M , then C k (V ) denotes the vector space of C k sections of V .
. If k < s < k + 1, the Slobodeckii space W s q (V ) is obtained by real interpolation: We also need the time-dependent function spaces is finite, and BC := BC 0 . Then with the usual Hölder space C 1/2 . The following lemma shows that in the Euclidean setting these definitions return the classical spaces.
the standard Sobolev-Slobodeckii spaces, and Proof The second assertion is obvious.
Definitions analogous to (6.5) and (6.6) give the Banach spacesW  is well-defined. Note that a ± is the trace of a on S 'from the positive/negative side of S'.
Consequently, the jump of the normal derivative, is also well-defined.
As a rule, we often drop the index q if q = p. ThusW 2/2 =W 2/2 p , ∂W = ∂W p , etc. Finally, is the closed linear subspace of ∂W ⊕ ∂ S W ⊕ γ 0W consisting of all (ϕ, ψ, u 0 ) satisfying the compatibility conditions where ϕ = (ϕ 0 , ϕ 1 ) and ψ = (ψ 0 , ψ 1 ). It follows from the anisotropic trace theorem ([12, Example VIII.1.8.6]) that ∂ B,C W is well-defined. Given Banach spaces E and F , we write Lis(E, F ) for the set of all isomorphisms in L(E, F ), the Banach space of continuous linear maps from E into F . Now we can formulate the following maximal regularity theorem for problem (7.2). Its proof, which needs considerable preparation, is found in Section 13.

The Uniform Lopatinskii-Shapiro Condition
In the proof of Theorem 7.1 we need to consider systems of elliptic boundary value problems. For this we have to be precise on the concept of uniform ellipticity.
Here ∇ i u = (∇ i u 1 , . . . , ∇ i u n ) so that, for example, a q ∇ 2 u = (a 1 s q ∇ 2 u s , . . . , a n s q ∇ 2 u s ), where s is summed from 1 to n and q denotes complete contraction, that is, summation over all twice occurring indices in any local coordinate representation. The (principal) symbol sA of A is the (n × n)-matrix-valued function defined by sA(p, ξ) := a(p) q (ξ ⊗ ξ), p ∈ M, ξ ∈ T * p M. Then A is uniformly normally elliptic if there exists an 'ellipticity constant' α ∈ (0, 1) such that for all p ∈ M and ξ ∈ T * p M with |ξ| 2 g * (p) = 1, where σ(·) denotes the spectrum.
Suppose Γ = ∅ and B = (B 1 , . . . , B n ) is a linear boundary operator of order at most 1. More precisely, we assume that there is k ∈ {0, . . . , n} such that Then the (principal) symbol of B is the (n × n)-matrix-valued function sB given by Observe that X q ω = ω, X if X is a vector and ω a covector field. We denote by ν ♭ ∈ T * Γ M the inner conormal on Γ defined in local coordinates by ν ♭ = g ij ν j dx i . Given q ∈ Γ, we write B(q) for the set of all Then, if (ξ, λ) ∈ B(q), we introduce linear differential operators on R by A(∂; q, ξ, λ) := λ + sA q, ξ + i ν ♭ (q)∂ , B(∂; q, ξ, λ) := sB q, ξ + i ν ♭ (q)∂ , where i = √ −1. As usual, C 0 (R + , C n ) is the closed linear subspace of BC(R + , C n ) consisting of the functions that vanish at infinity.
The basic feature, which distinguishes the above definition from the usual form of the LS condition, is the requirement of the uniform bound (8.7). Without this demand the LS condition is much simpler to formulate (e.g., [4], [5], [20], [21], [26], for example) and to verify.
It is known that the LS condition is equivalent to the parameter-dependent version of the complementing condition of S. Agmon, A. Douglis, and L. Nirenberg [1] (see, for example, [24,VII §9] or [29, Section 10.1]). Using this version, it is possible to define a uniform complementing condition which is equivalent to (8.7) (see [2] and [3]). However, that condition is even more difficult to verify in concrete situations. We refer to [14] for a detailed exposition of all these facts. It should be noted that the uniformity condition (8.7) is fundamental for the following, since we will have to work with infinitely many linear model problems.

Model Cases
For the proof of Theorem 7.1 we have to understand the model cases to which problem (7.2) reduces in local coordinates.
Until further notice, it is assumed that • assumption (7.3) is in force.
• K is an S-adapted ur atlas for M.

Continuity
First we note that √ g κ ∼ 1, κ ∈ K, (10.1) and, given k ∈ N, with ∇ κ u := κ * ∇κ * u (cf. [9, Lemma 3.1] or [14]). We set where 0 ≤ s ≤ 2. For the sake of a unified presentation, It is a consequence of (10.1) and Using this, (6.5), and (6.6), we infer that Given a vector field By this and the above it is verified that Now we consider Sobolev-Slobodeckii spaces on ∂H m ≃ R m−1 . We set g q κ := g κ∂H m for κ ∈ K Γ . Then and Then where the BUC ρ are the usual Hölder spaces and b σ/2 ∞ is an anisotropic little Besov space. Indeed, the first embedding follows from the mean value theorem and by using the localized Hölder norm (cf. [ (10.13) We deduce from (3.3) and (10.2) that K Γ -unif. Now it follows from (10.12) and (10.13) that Since, trivially, γ ∈ L BC 1/2 (H m × J), BC 1/2 (∂H m × J) , it is now clear that In local coordinates Hence δ κ B κ u = b i κ γ∂ i u, where it follows from (3.3), (9.5), and a κ ∞ ≤ c that and, from (10.14), for κ ∈ K Γ . Thus it is a consequence of (10.14), (10.16), and the boundary operator retraction theorem [ Clearly, 'K Γ -uniform' means that there exists a coretraction B c κ such that B κ and B c κ are K Γ -uniformly bounded. Obviously, If we replace in the preceding arguments the boundary operator retraction argument by Theorem VIII.2.3.5 of [12], we find that It follows from (10.5) that For the boundary operator we find Hence sB 1 u(q, t, ξ) = a(q, t) ν ♭ (q) ξ g * (q) , q ∈ Γ, ξ ∈ T * q M, t ∈ J. (11.4) Clearly, these formulas apply to any oriented Riemannian manifold, thus to (∂H m , g κ ), κ ∈ K Γ . It follows from (9.5) and (11.2) that for x ∈ X κ , ξ ∈ T x X κ , t ∈ J, and κ ∈ K. Hence A κ is uniformly normally elliptic, unif. w.r.t. κ ∈ K and t ∈ J. (11.5) We begin with the full-space problem.
Proposition 11.1 It holds that is, Proof It is obvious from (10.10) and (10.23) that Due to (11.5), the assertion now follows from Corollary 9.7 in [15] and Theorem III.4.10.8 in [6] and (the proof of) Theorem 7.1 in [7]. (See [14] for a different demonstration.) Next we study the case where κ ∈ K Γ . For this we first establish the validity of the uniform LS condition. Henceforth, it is always assumed that ζ = (x, ξ, λ) with x ∈ ∂H m and (ξ, λ) ∈ B(x). (11.6) We fix any t ∈ J and omit it from the notation. The reader will easily check that all estimates are uniform with respect to t ∈ J. From (11.2) we see that the first equation in (8.4) where with the principal value of the square root.
Nonhomogeneous linear parabolic boundary value problems (of arbitrary order and in a Banach-space-valued setting) on Euclidean domains have been studied in [21]. It follows, in particular from Proposition 6.4 therein, that the isomorphism assertion is true for each κ ∈ K Γ . However, it is not obvious whether the K Γ -uniformity statement does also follow from the results in [21]. For this one would have to check carefully the dependence of all relevant estimates on the various parameters involved, which would be no easy task. (The same observation applies to Proposition 11.1.) In [14] we present an alternative proof which takes care of the needed uniform estimates.
Lastly, we assume that κ ∈ K S . We set, once more suppressing a fixed t ∈ J.

Proposition 11.3 It holds
Proof Set u(x) := u(x), u(−x) for x ∈ H m andW s κ :=W s κ ⊕W s κ etc. Then the assertion is true iff By the preceding considerations, the proof of Proposition 11.2 applies verbatim to the system for u. This proves the claim.
The following consequences of the preceding results are needed to establish Theorem 7.1.
The next lemma provides analogous estimates for the boundary and transmission operators.
Proof Let s, q, and ε be as in the preceding proof. Given any Banach space E, Hölder's inequality gives Hence, by interpolation (see [ Then we get from (12.23) It is clear from the structure of B κ and the mapping properties of γ that Since L s κ (τ ) ֒→ L 1 κ (τ ) K Γ -unif. and unif. w.r.t. τ , the first assertion now follows from (12.24), Lemma 12.1, and the fact that B κ has its image in the closed linear subspace ∂ 1 W κ (τ ) of ∂W κ (τ ).
The proof of the second claim is similar.
13 Proof of Theorem 7.1 Let E = α∈A E α , where each E α is a Banach space and A is a countable index set. Then ℓ p (E) is the Banach space of p-summable sequences in E.
From this, the analogous relation for C and C, and (13.4) we infer that it suffices to prove that L : E 2 0 → E 0 ⊕ [G 0 ] B,C is surjective and has a continuous inverse. Obvious modifications apply to p < 3.
Remarks 13.4 (a) Recall that either some of Γ 0 , Γ 1 , and S, or all of them, can be empty. In either of these cases the result is new. (b) Theorem 7.1 is true for systems u = (u 1 , . . . , u n ), provided the uniform Lopatinskii-Shapiro conditions apply. This is trivially the case if a is a diagonal matrix.

Membranes with Boundary
Now we turn to the case of membranes intersecting Γ transversally. This case is handled by reducing it to the situation studied in the preceding section. In preparation for the proof of Theorem 2.1 we derive rather explicit representations of (A, B, C) and the relevant function spaces in a tubular neighborhood of Σ = ∂S in M .