The edge algebra structure of the Zaremba problem

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The edge algebra structure of the Zaremba problem

Der-Chen Chang∗, N.Habal†, B.-W. Schulze ‡ June 17, 2013

Abstract

We study mixed boundary value problems, here mainly of Zaremba type for the Laplacian within an edge algebra of boundary value problems. The edge here is the interface of the jump from the Dirichlet to the Neumann condition. In contrast to earlier descriptions of mixed problems within such an edge calculus, cf. [16], we focus on new Mellin edge quantisations of the Dirichlet-to-Neumann operator on the Neumann side of the boundary and employ the pseudo-differential calculus of corresponding boundary value problems without the transmission property at the interface. This allows us to construct parametrices for the original mixed problem in a new and transparent way.

Introduction

This paper is aimed at studying mixed elliptic boundary value problems for elliptic dif-ferential operators in a domain Rn with smooth boundary Y . We mainly focus here on the Zaremba problem for the Laplace operator where Y is subdivided into submanifolds Y± with common smooth boundary Z of codimension 1, Y = Y−∪ Y+, Z = Y− ∩ Y+.

On int Y− we pose Dirichlet, on int Y+ Neumann conditions. The problem has been

in-vestigated for a very long time, see Zaremba [40] and the subsequent development. The problem of characterising solvability has attracted many mathematicians, motivated by applications in models of physics. A list of references can be found in the monographs [18] and [16]. Mixed elliptic problems are of interest also for more general elliptic oper-ators, e.g., Lam´e’s system and different elliptic boundary conditions with jumps along some interfaces. For applications it is also interesting to admit domains with non-smooth boundary, e.g., polyhedral domains, and also interfaces with singularities.

Mixed elliptic problems can be studied from the point of view of pseudo-differential anal-ysis with symbolic structures reflecting not only the operators in the domain and the boundary conditions but also the nature of the jump Z which is here interpreted as an edge embedded on the boundary Y . In the articles [15] and [8] the corresponding operators have been interpreted as boundary value problems on a corresponding manifold M with boundary Y and edge Z.

The singular analysis, outlined in various papers and monographs [25], [30], [32], suggests to point out a stratification

s(M ) = (s0(M ), s1(M ), s2(M )) (0.1)

of M , where M = s0(M ) ∪ s1(M ) ∪ s2(M ) for the strata s2(M ) := Z, s1(M ) := Y \ Z,

s0(M ) := M \ Y . Those are smooth manifolds of different dimension. According to the

general philosophy of pseudo-differential operators A on such spaces, especially, differential operators, we observe a principal symbolic hierarchy

σ(A) := (σ0(A), σ1(A), σ2(A)) (0.2)

that determines the ellipticity. Generalities on operators on stratified spaces M of some singularity order k ∈ N are developed in [33], [35]. Here k = 0 indicates smoothness, k = 1 conical or edge singularities, k = 2 corners of second order, etc.. The components of (0.2) are associated with the strata in (0.1). In particular, σ0(A) is the standard homogeneous

principal symbol of A over s0(M ), moreover, σ1(A) is the boundary symbol, in the present

case referring to int Y±, and σ2(A) is the edge symbol. For instance, if A is the Laplacian

Pn j=1 ∂ 2 ∂x2 j in R n

+= {x = (x1, . . . , xn) ∈ Rn: xn > 0}, we have σ0(A) = −|ξ|2 with ξ being

the covariable of x, σ1(A)(η) = −|η|2+ ∂2 ∂x2 n (0.3) where η is the covariable of y = (x1, . . . , xn−1) ∈ Rn−1 when we represent the operator in

local coordinates x = (y, xn) close to the boundary, with xn∈ R+being the local variable

normal to the boundary. We see that (0.3) is operator valued, and we take it as a family of operators Hs(R+) → Hs−2(R+) parametrised by η 6= 0 (and s ∈ R not too small, see

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defined by xn−1= 0, xn= 0 in x = (z, xn−1, xn) ∈ Rn+ with ζ ∈ Rn−2 being the covariable

of z ∈ Rn−2, then the symbol σ2(A)(ζ) is the family of operators

σ2(A)(ζ) = −|ζ|2+ ∂2 ∂x2 n−1 + ∂ 2 ∂x2 n (0.4) operating as Hs(R2+) → Hs−2(R2+), parametrised by ζ 6= 0 (and s ∈ R again not too

small), where

R2+= {(xn−1, xn) ∈ R2: xn−1∈ R, xn> 0}.

The idea of the analysis of elliptic boundary value problems is to add conditions (such as Dirichlet or Neumann conditions) over s1(M ) and to understand the nature of

paramet-rices. As soon as we have no jump of the boundary conditions and impose the Shapiro-Lopatinskij condition then we are in the framework of “smooth” elliptic boundary value problems. The boundary operators also have a symbolic structure, and there is a well-known pseudo-differential calculus of boundary value problems, cf. Boutet de Monvel [4], that contains the elliptic elements themselves together with their parametrices. The gen-eral scenario in this case is that we consider 2 × 2 block matrices

A K

T Q

(0.5) with A in the upper left corner, in general, together with so-called Green operators. More-over, T represents the boundary (or trace) condition while an additional potential operator K appears, and Q is a pseudo-differential operator on the boundary. The boundary sym-bolic map σ1 then also applies to the other entries in (0.5).

In the case of mixed problems after having fixed different elliptic conditions T±, K±, Q−−,

Q−+, etc. for A on int Y± the corresponding operator

A =   A K− K+ T− Q−− Q−+ T+ Q+− Q++   (0.6)

acquires from Z new symbolic data σ2(A) with σ2(A) as in (0.4). Analogously as in Boutet

de Monvel’s calculus those give rise to a larger block matrix of operators

A=A K

T Q

(0.7) for A as in (0.6) and additional trace and potential operators T and K, respectively, with respect to Z and pseudo-differential operators Q on the closed manifold Z, the edge. The idea of the present paper is to express parametrices of mixed problems by operators of the kind (0.7), using ellipticity with respect to an extension of σ2(A) to a

correspond-ing principal symbolic object σ2(A). The block matrix algebra of operators (0.7) is a

substructure of the algebra of edge pseudo-differential boundary value problems on a manifold M with boundary and edge on the boundary. In general the boundary is not necessarily smooth but may have Z as a “real” edge.

This kind of calculus also plays an important role in the present paper. However, here we combine the consideration with new elements from the pseudo-differential calculus of boundary value problems on Y+ with boundary Z for an operator without transmission

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INTRODUCTION 5 principal symbolic hierarchy which is much more transparent than that with three components. At the end we achieve ellipticity and parametrices within the algebra of operators (0.7) by means of ellipticity and parametrices within the algebra of boundary value problems over Y+.

Our paper is organised as follows.

In Section 1 we analyse the first order pseudo-differential operator R that is obtained by reducing the Neumann boundary condition to the boundary. The operator R, often re-ferred to as the Dirichlet-to-Neumann operator, has been widely studied by many authors. Here we look at Y+ and realise R by a specific edge quantisation as an element of the

edge pseudo-differential calculus with Z as the edge, operating in weighted edge spaces. We explicitly compute admissible weights together with adequate edge conditions of trace and potential type such that the corresponding block matrix continuously acts between weighted spaces, plus standard Sobolev spaces over Z. An important aspect is that the relation to the original mixed problem remains under control in the above-mentioned larger edge calculus of boundary value problems. This element of the calculus was the desirable though missing aspect in the papers [8], [15], and also in [16] where we found other ways to characterise ellipticity and Fredholmness in adequate scales of weighed spaces. Here we produce the information from the edge calculus over Y+, or, alternatively,

from the calculus of boundary value problems without the transmission property. The latter is a far reaching extension of Boutet de Monvel’s calculus [4] which is designed for operators with transmission property. In pseudo-differential operators the transmission property is generically violated. Further background information is given in [16]. A special invention in the edge calculus are the smoothing Mellin plus Green operators. They substitute the Green operators from Boutet de Monvel’s calculus. Recall that special such operators on the half-axis R+ for zero order pseudo-differential operators truncated from

R to R+ have been established in Eskin [10, §15]. Those have been builded up later on

in the higher-dimensional cone and edge algebras to corresponding ideals of smoothing Mellin plus Green operators, see [31] or [32].

Section 2 is devoted to a more detailed analysis of the Dirichlet-to-Neumann opera-tor from the point of view of quantisations coming from the edge calculus. There are many different ways of establishing quantisations, i.e., obtaining representations of op-erators modulo smoothing elements, such that the resulting opop-erators are continuous in weighted edge spaces. It is essential in the calculus to achieve quantisations that are particularly explicit and with “small remainders”. As we shall see our quantisation agrees with the truncation quantisation, obtained on the half-axis in [31] and on manifolds with boundary in [36] within the respective cone or edge calculus. Another important aspect are Mellin quantisations that produce holomorphic Mellin symbols. The development of efficient and iterative quantisation machineries is a necessary task of the singular analysis. Recent achievements are published in [6] and [13]. Furthermore, in Section 2 we choose additional boundary conditions on Z for the quantised Dirichlet-to-Neumann operator. This is possible since a typical topological obstruction vanishes in our case, analogously as that from Atiyah and Bott [3]. We obtain are Fredholm boundary value problems for the truncated operators R on Y+ in weighted edge Sobolev spaces. In this stage of calculus

the weights are taken sufficiently large, up to some exceptional discrete weights, and the smoothness is arbitrary, as is the case ought in edge operators where the base of the model cone is closed. In boundary value problems the model cone of the local wedges is R+.

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In Section 3 we establish necessary tools on pseudo-differential algebras. We give more information on weighted edge spaces, furthermore, parameter-dependent operators in Boutet de Monvel’s calculus, including examples that play a role here. Then we develop more details on how the “larger” edge calculus on M of boundary value problems is or-ganised, and we generalise the admitted orders in our block matrix operators by applying reductions of orders within the involved pseudo-differential structures. Let us finally note that the general ideas of our approach can be extended from the Zaremba problem to other mixed elliptic problems.

Acknowledgment. The first draft of this paper was initiated when the authors visited the National Center for Theoretical Sciences, Hsinchu, Taiwan during January, 2013. They would like to express their profound gratitude to the Director of NCTS, Professor Winnie Li for her invitation and for the warm hospitality extended to them during their stay in Taiwan.

1 The Dirichlet-to-Neumann operator

1.1 The standard reduction to the boundary

As noted in the introduction mixed problems for elliptic differential operators, especially for the Laplacian, may be studied by an extension of Boutet de Monvel’s calculus , cf. [4], [11], [24], to the case without the transmission property at the boundary. Let G be the closure of a smooth bounded domain in Rn. We reduce boundary problems in G to the boundary, first in terms of operators in Boutet de Monvel’s algebra and then by using tools from the calculus of BVPs on a manifold with conical or edge singularities. The ‘standard’ idea is as follows. Let Ai = t(A Ti), i = 0, 1, denote the row matrix

operators representing two elliptic BVPs for an elliptic operator A with trace (or boundary) operators representing boundary conditions satisfying the Shapiro-Lopatinskij condition. In our case A is a second order elliptic differential operator in Rnwith smooth coefficients, in the simplest the Laplacian and T0u := u|∂G and T1u := ∂νu|∂G are Dirichlet and

Neumann conditions, respectively, with ∂ν being the derivative normal to the boundary.

For convenience we start with the assumption that A0 is invertible, say, as an operator

A0 : C∞(G) →

C∞(G) ⊕ C∞(∂G)

(which is the case when A is the Laplacian and in numerous other cases) and by P₀ = (P0 K0)

we denote its inverse. In particular, we have AP0= 1, AK0 = 0. This yields

A1P0= 1 0 T1P0 T1K0 (1.1) where T1K0 is often called the Dirichlet-to-Neumann operator.

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1 THE DIRICHLET-TO-NEUMANN OPERATOR 7 To illustrate phenomena we consider the case A = ∆. (However, we still write A instead of ∆, since the ideas apply for more general operators A). We have

R := T1K0 ∈ L1cl(∂G), σψ(R)(η) = c|η| (1.2)

for a constant c; here σψ(·) denotes the homogeneous principal symbol of the respective

operator an η the covariable on ∂G (the absolute value refers to a Riemannian metric on the boundary). The operator R is elliptic, as we see from (1.2). In general the ellipticity of operators obtained by reducing an elliptic BVP to the boundary follows from the ellipticity of both factors in (1.1). Explicit computations for other concrete BVPs reduced to the boundary may be found in [16, page 26]. In more complicated situations below, i.e., when we replace A1 by a mixed boundary problem with jumps of the conditions along an

interface Z of codimension 1 on the boundary, the intention will be (similarly as in the smooth case) to express parametrices within a controlled operator algebra with symbolic structure. In the simplest case there is no jump at all, i.e., we have a reduction of A1 to

the boundary by means of A0 in the form (1.1). The idea is then to construct a parametrix

A(−1)₁ of A1 in terms of the known parametrix (or inverse) P0 of A0 and a parametrix

R(−1) of R. First we obtain a parametrix P1 of (1.1) as

(A1P0)(−1) = 1 0 T1P0 T1K0 (−1) = 1 0 −R(−1)_T 1P0 R(−1) .

Then A(−1)₁ itself follows in the form

A(−1)₁ := P0(A1P0)(−1)= (P0 K0) 1 0 −R(−1)_T 1P0 R(−1) = (P0− K0R(−1)T1P0 K0R(−1)) = (P1 K1). (1.3)

In the consideration below we interpret A0 and A1 as continuous operators

A0 = A T0 : Hs(int G) → Hs−2(int G) ⊕ Hs−1/2(∂G) , A1= A T1 : Hs(int G) → Hs−2(int G) ⊕ Hs−3/2(∂G) ,

first for s > 3/2. Recall from [4] or [24] that the operators Pi = (Pi Ki) belong to Boutet

de Monvel’s calculus of pseudo-differential BVPs with the transmission property at the boundary, more precisely, Pi = F + Gi with F = r+F e˜ + being the truncation of a

fun-damental solution (or a parametrix) ˜F of ∆ in Rn _{to int G. Here e}+ _{is the operator of}

extension by zero from int G to Rn and r+ the operator of restriction of distributions to int G. Moreover, Giis a Green operator and Ki a potential operator in Boutet de Monvel’s

calculus. The advantage of this viewpoint is that we have the principal symbolic structure of such operators, more precisely, the pair σ = (σ0, σ1) of symbols where σ0 is the standard

homogeneous principal symbol of operators over G (smooth up to the boundary) while σ1 represents the so-called principal boundary symbol. Moreover, we can freely compose

operators in Boutet de Monvel’s algebra; this was done in (1.1) as well as in (1.3), and the symbols are (componentwise) multiplicative. Concerning Ti and the other operators

in lower left corners, those are trace operators in Boutet de Monvel’s calculus, and they have boundary symbols as well. Parametrices Pi of Ai belong to the inverted symbolic

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In the construction of R = T1K0, the reduction of the Neumann condition to the

bound-ary by means of the solution of the Dirichlet problem, it is not essential that P0 is

the inverse of A0. We are mainly interested in Fredholm operators between the chosen

Sobolev spaces, and it suffices to employ P0 as a parametrix of A0. By

interchang-ing the role of A0 and A1 for a parametrix P1 of the Neumann problem we obtain

A0P1 =

1 0

T0P1 T0K1

modulo a smoothing operator in Boutet de Monvel’s calcu-lus. Let us ignore such remainders in the following compositions. Then the composition (A0P1)(A1P0) = 1 0 T0P1 T0K1 1 0 T1P0 T1K0

is the identity modulo a smoothing op-erator, which implies that R(−1) := T0K1 ∈ L−1_cl (∂G) is a parametrix of R = T1K0 ∈

L1_cl(∂G). Let us now replace A1 by the Zaremba problem

Am:=   A D− N+  : C ∞ (G) → C∞(G) ⊕ C∞(Y−) ⊕ C∞(Y+) , (1.4)

for D−u := (T0u)|int Y−, N+u := (T1u)|int Y+ where Y := ∂G is subdivided into

subman-ifolds Y+, Y− with commen boundary Z = Y+∩ Y−, in the simplest case assumed to be

smooth. Then by virtue of (1.1) together with A0P0 = idHs−2_(G) we obtain

D−K0= 1 on int Y−,

N+K0 = R on int Y+.

(1.5) Thus the reduction of the mixed condition gives rise to R on the manifold Y+with

bound-ary ∂Y+ =: Z and the identity on Y−. The relation (1.5) shows that in the reduction

of the mixed conditions T := t(D− N+) to the boundary the resulting operator on the

boundary has a jump; it is equal to the Dirichlet-to-Neumann operator R on Y+ and to

the identity on Y−. Apart from the identity one of the main tasks is to solve a boundary

value problem for R on Y+ with Z = ∂Y+ as the boundary. However, R fails to have

the transmission property at the boundary, it has in fact, the anti-transmission property, cf. [34] and it is hard to imagine that Boutet de Monvel’s calculus extends to this case. Clearly many authors studied such operators, cf. Vishik, Eskin [39] or Eskin [10], see also [23], [36], or [16]. In the present paper we develop an approach that is based on a Mellin quantisation, already indicated in Dines, Liu, and Schulze [7]. Here we employ this method for the construction of a parametrix of (1.4) within a variant of edge calculus where Z plays the role of an edge.

1.2 Cutting and pasting along an interface

In the following it will be convenient to fix Sobolev spaces where we realise the operators in consideration. In this approach we employ cutting and pasting operators that glue together spaces with respect to Y+ and Y− and encode the jumps of data across the interface Z.

For our approach it is essential to employ the formalism of the edge pseudo-differential calculus. This concerns both the involved spaces and the edge symbolic structure of the

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1 THE DIRICHLET-TO-NEUMANN OPERATOR 9 occurring operators.

For s > 1/2 and s − 1/2 /∈ N = {0, 1, . . .}, we set

H_θs(R+) := {u ∈ Hs(R+) : Dtju|t=0for all 0 ≤ j ≤ s − 1/2}.

This is a Hilbert space with group action κ = {κλ}λ∈R+ in the sense of the notation at

the beginning of Subsection 3.1. According to (3.14), (3.15) we have the spaces

Ws_(Rq, H_θs(R+)) ⊂ Hs(Rq× R+). (1.6)

The latter relation is a consequence of H_θs(R+) ⊂ Hs(R+) and

Ws_(Rq, Hs(R+)) = Hs(Rq× R+) = Hs(Rq× R)|Rq×R+.

On a manifold N with boundary ∂N we fix a collar neighbourhood ∂N × [0, 1) of the boundary with [0, 1) 3 t being identified with the corresponding interval on the half-axis. Lemma 1.1. (i) Let N be a smooth compact manifold with boundary ∂N , and let s ∈ R, s > 1/2, s − 1/2 /∈ N, L(s) := #{l ∈ N : l + 1/2 < s}. Then there is an isomorphism

(Es Ks) : H_θs(int N ) ⊕ Hs_{(∂N, C}L(s)₎ → Hs(int N ) (1.7) where

H_θs(int N ) := {u ∈ Hs(2N )|int N : Dαxu|∂N = 0 for all |α| < s − 1/2} (1.8)

with x denoting local coordinates close to ∂N , and Es : Hθs(int N ) ,→ Hs(int N )

is the canonical embedding, while Ks : Hs(∂N, CL(s)) → Hs(int N ) is a potential

operator of Boutet de Monvel’s calculus. The inverse of (1.7) Ps

Ts

:= (Es Ks)−1 (1.9)

consists of a projection Ps: Hs(int N ) → Hθs(int N ) along im Ks, and a vector Ts of

trace operators of Boutet de Monvel’s calculus.

(ii) Let Y be a smooth closed manifold written as Y = Y−∪ Y+ for smooth compact

manifolds Y± with common boundary Z = Y−∩ Y+, and let s ∈ R, s > 1/2, s − 1/2 /∈

N, and L(s) be as in (i). Then there is an isomorphism

(Es Ks) : H_θs(int Y−) ⊕ Hs θ(int Y+) ⊕ Hs(Z, CL(s)) → Hs(Y ) where Es : H_θs(int Y−) ⊕ H_θs(int Y+)

→ Hs_{(Y ) is the canonical embedding and K}

s is a potential

operator of analogous meaning as Ks in (i). In the inverse P_Ts_s := (Es Ks)−1 the

operator Ps: Hs(Y ) →

Hs θ(Y−)

⊕ H_θs(Y+)

is a projection along im Ks, and Ts is a vector of

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Let us set

H_θs(Y ) := im Es.

Lemma 1.2. Let Y = Y−∪ Y+ be as in Lemma 1.1, and let s ∈ R, s > 1/2, s − 1/2 /∈ N,

L(s) := #{l ∈ N : l + 1/2 < s}. Then we have an isomorphism r− _d − r+ d+ : Hs_{(Y )} ⊕ Hs(Z, CL(s)) → Hs(int Y−) ⊕ Hs(int Y+) (1.10)

where r∓: Hs(Y ) → Hs(int Y∓) are the respective restriction operators and d∓ are

poten-tial operators from Boutet de Monvel’s version of transmission problems. The proof of Lemma 1.2 will be given in the beginning of Subsection 3.1.

We apply Lemma 1.2 for s − 1/2. Then for s > 1, s − 1 /_{∈ N we write the spaces H}s−1/2_{(Y )}

and Hs−1/2(int Y∓) as direct sums

Hs−1/2(Y ) = H_θs−1/2(Y ) ⊕ Vs−1/2(Y ), Hs−1/2(int Y∓) = H_θs−1/2(int Y∓) ⊕ Vs−1/2(int Y∓)

(1.11) with canonical embeddings

e∓_θ : H_θs−1/2(int Y∓) → H_θs−1/2(Y ), (1.12)

cf. analogously Es−1/2 in Lemma 1.1 (ii), and

Vs−1/2(Y ) := Ks−1/2Hs(Z, CL(s−1/2)), Vs−1/2(int Y∓) := Vs−1/2(Y )|int Y∓. Then we have r− _d − r+ d+ −1 =e − d e + d c− c+ (1.13) where e∓_d and the trace operators c∓ are determined by (3.8) below. Also the operators

c∓ depend on the choice of d∓.

The spaces H_θs(int Y∓) as well as H_θs(Y ) will be identified with weighted edge spaces with

weights where Z is interpreted as the edge of Y∓ or Y , and we write

H_θs(int Y∓) =: Hs,s(Y∓), Hθs(Y ) =: Hs,s(Y ). (1.14)

The spaces on the right of (1.14) are global weighted edge Sobolev spaces of smoothness s and weight s. Details are elaborated in Subsection 3.1. The relation (1.14) means that the Sobolev smoothness s together with the flatness θ close to the edge plays a two-fold role, namely, as the Sobolev smoothness in edge spaces and at the same time as a weight. We will apply Lemma 1.1 below for s − 1/2 or s − 3/2 instead of s and over Y∓ rather

than N. Together with (1.14) we obtain the following isomorphisms

Λ∓_D := P ∓ s−1/2 T_s−1/2∓ ! : Hs−1/2(int Y∓) → Hs−1/2,s−1/2(Y∓) ⊕ Hs−1/2(Z, CL(s−1/2)) (1.15) for s > 1, s − 1 /∈ N, Λ∓_N := P ∓ s−3/2 T_s−3/2∓ ! : Hs−3/2(int Y∓) → Hs−3/2,s−3/2(Y∓) ⊕ Hs−3/2(Z, CL(s−3/2)) (1.16)

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1 THE DIRICHLET-TO-NEUMANN OPERATOR 11 for s > 2, s − 2 /_{∈ N. Then} (Λ∓_D)−1= (E_s−1/2∓ K_s−1/2∓ ), (Λ∓_N)−1 = (E_s−3/2∓ K_s−3/2∓ ). (1.17) Moreover, we set ΛD:= diag(Λ−D Λ + D), ΛN:= diag(Λ−N Λ + N), ΛM:= diag(Λ−D Λ + N). (1.18)

1.3 The reduction of mixed problems to the boundary We realise the Zaremba problem as a continuous operator

A_m=   A D− N+  : Hs(int G) → Hs−2(int G) ⊕ Hs−1/2(int Y−) ⊕ Hs−3/2(int Y+) (1.19)

for s > 3/2, D∓ := r∓T0, with r∓ : Hs−1/2(Y ) → Hs−1/2(Y∓) being the restriction

operators. Also Dirichlet and Neumann conditions independently of Y∓ give rise to mixed

problems, namely, A_d=   A D− D+  , A_n=   A N− N+  , (1.20)

for N∓:= r∓T1, regarded as continuous operators

A_d : Hs(int G) → Hs−2_{(int G)} ⊕ Hs−1/2(int Y−) ⊕ Hs−1/2(int Y+) , An: Hs(int G) → Hs−2_{(int G)} ⊕ Hs−3/2(int Y−) ⊕ Hs−3/2(int Y+) . (1.21)

We now reduce the operator Am to the boundary by means of the operator

A_d. In order to proceed in an analogous manner as in (1.1) we have to employ a relationship between A0 and Ad. Consider

˜ A0 :=   A 0 T0 0 0 id_Hs−1/2_(Z,Ck(s−1/2)₎  : Hs(int G) ⊕ Hs−1/2(Z, CL(s−1/2)) → Hs−2(int G) ⊕ Hs−1/2(Y ) ⊕ Hs−1/2(Z, CL(s−1/2)) (1.22)

which is an isomorphism, since A0 itself is an isomorphism. Then, according to Lemma

1.2 we have an isomorphism D0 :=   idHs−2_{(int G)} 0 0 0 r− d− 0 r+ d+   : Hs−2_{(int G)} ⊕ Hs−1/2(Y ) ⊕ Hs−1/2(Z, CL(s−1/2)) → Hs−2_{(int G)} ⊕ Hs−1/2(int Y−) ⊕ Hs−1/2(int Y+) (1.23)

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for s > 1, s − 1 /_{∈ N. For A}D:= D0A˜0 it follows that AD=   A 0 D− d− D+ d+   : Hs(int G) ⊕ Hs−1/2(Z, CL(s−1/2)) → Hs−2(int G) ⊕ Hs−1/2(int Y−) ⊕ Hs−1/2(int Y+) . (1.24)

The operator ˜A₀ = A0 ⊕ idHs−1/2_(Z,CL(s−1/2)₎ is essentially the Dirichlet problem. The

operator D0A˜0 contains Ad and the extra potential operators d±. Let us write

PD:= A−1_D =: P0 C− C+ 0 c− c+ =P0 K0e − d K0e + d 0 c− c+ , (1.25)

cf. (3.8), which follows from

A−1_D = ˜A−1₀ D₀−1=P0 K0 0 0 0 id_Hs−1/2_(Z,CL(s−1/2)₎   id_Hs−2_{(int G)} 0 0 0 e−_d e+_d 0 c− c+   ,

cf. the formula (1.13). In particular, we have continuous operators C∓= K0e∓_d : Hs−1/2(int Y∓) → Hs(int G). Thus A_DP_D=   AP0 AC− AC+ D−P0 D−C−+ d−c− D−C++ d−c+ D+P0 D+C−+ d+c− D+C++ d+c+   =   1 0 0 0 1 0 0 0 1   , (1.26) P_DA_D=P0A + C−D−+ C+D+ C−d−+ C+d+ c−D−+ c+D+ c−d−+ c+d+ =1 0 0 1 . (1.27)

Let us also rephrase Am, cf. (1.19), and form the operator

A_M:=   A 0 D− d− N+ 0   : Hs(int G) ⊕ Hs−1/2(Z, CL(s−1/2)) → Hs−2(int G) ⊕ Hs−1/2(int Y−) ⊕ Hs−3/2(int Y+) , (1.28)

for N+:= r+T1. The extra operator d−has the form of an edge condition of the final edge

formulation, with Z being the edge. It follows that

AMPD=   AP0 AC− AC+ D−P0 D−C−+ d−c− D−C++ d−c+ N+P0 N+C− N+C+  =   1 0 0 0 1 0 N+P0 N+C− N+C+  , (1.29)

cf. the formula (1.26). Here

RND:= N+C+ : Hs−1/2(int Y+) → Hs−3/2(int Y+) (1.30)

corresponds to the Dirichlet-to-Neumann operator on the Neumann-side Y+. Since C+

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1 THE DIRICHLET-TO-NEUMANN OPERATOR 13 1.4 Elliptic edge conditions

In elliptic boundary or edge problems for a (pseudo-)differential operator A the principal boundary or edge symbol of A induces a family of Fredholm operators

a : H → ˜H (1.31)

between Hilbert spaces with group action. By virtue of twisted homogeneity it suffices to consider (1.31) as an operator function parametrised by the points of the unit cosphere bundle induced by the cotangent bundle of the boundary or the edge. We give here some further comment on the general role of (1.31) for ellipticity of boundary or edge conditions. For convenience we mainly consider the case of boundary conditions. Analogous structures can be observed in the edge case. We illustrate here some phenomena belonging to the key words Green, trace and potential symbols, and the associated operators.

Assume that the boundary Y is smooth and compact. Then also the unit cosphere bundle S is compact. Let π : S → Y be the canonical projection. Recall that Vect(·), the set of all smooth complex vector bundles on the manifold in parenthesis, generates the K-group K(·) as the additive group of equivalence classes of pairs (E, F ) for E, F ∈ Vect(·). Here (E, F ) ∼ ( ˜E, ˜F ) ⇔ E ⊕ ˜F ⊕ G ∼= ˜E ⊕ F ⊕ G for some G ∈ Vect(·). Equivalently we can ask the condition E ⊕ L ∼= ˜E ⊕ M, F ⊕ L ∼= ˜F ⊕ M for some L, M ∈ Vect(·). The corresponding equivalence class is denoted by [E] − [F ]. The bundle pull back induces a homomorphism

π∗ : K(Y) → K(S). (1.32)

Lemma 1.3. Let (1.31) be a family of Fredholm operators parametrised by S, regarded as a bundle morphism S × H → S × ˜H between the respective trivial Hilbert bundles. Then there are elements G1, G2 ∈ Vect(S) and an isomorphism

a k t q : S × H ⊕ G1 −→ S ×H˜ ⊕ G2 (1.33)

for suitable bundle morphisms k : G1 → S × ˜H, t : S × H → G2, and q : G1→ G2.

This result is well-known in K-theory; an explicit proof can be found in [16, Subsection 3.3.4] which is also a reference for other material in the present subsection. Clearly there are many choices of G1, G2 and k, t, q such that(1.33) is an isomorphism. However,

ind_S(a) := [G2] − [G1] ∈ K(S) (1.34)

only depends on a. A typical and most elementary example for an operator function a is the Fredholm family a(η) := σ∂(∆)(η) considered in (2.19) for H := Hs(R+), ˜H := Hs−µ(R+)

where the covariable η belongs to the cotangent bundle of the boundary (here in local representation). By virtue of twisted homogeneity σ∂(∆)(δη) = δ2κδσ∂(∆)(η)κ−1δ , δ ∈ R+,

the operator family a(η) is completely determined by its values on the unit sphere |η| = 1. In this particular case the dependence on η even disappears. In any case, as observed for (2.20), we find a block matrix isomorphism of the kind (1.33), here a column matrix, since a(η) is surjective. This example is important also for another reason, namely, in contrast to (1.34) we have

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For general elliptic operators A, say, pseudo-differential operators with the transmission property at the boundary, the boundary symbol a(y, η) := σ∂(A)(y, η) is not necessarily

surjective, and both kernel and cokernel can be non-trivial. If (1.35) holds we also say that the Atiyah-Bott obstruction vanishes, cf. [3] for the case of differential operators, [4] for pseudo-differential operators with the transmission property, [29] for elliptic edge operators. The information is as follows.

Theorem 1.4. An elliptic operator A in one of the above-mentioned pseudo-differential algebras admits a 2 × 2 block matrix Fredholm operator A = {Aij}i,j=1,2 with A (modulo

a Green or a smoothing Mellin plus Green operator) as the upper left corner A11, if and

only if (1.35) holds.

Let us make the construction more explicit in the case of an elliptic pseudo-differential operator R ∈ Lµ_(cl)(int Y+) that is quantised as an element R ∈ Lµ(Y+, g), g = (γ, γ − µ)

for some weight γ ∈ R in the edge calculus on Y+, regarded as a manifold with edge Z. In

such a quantisation we have also R ∈ Lµ_(cl)(int Y+) and

R = R mod L−∞(int Y+).

Moreover, R is σψ-elliptic in the sense of Definition 3.40. This has the consequence that

σ∧(R)(z, ζ) : Ks,γ(R+) → Ks−µ,γ−µ(R+), (1.36)

ζ 6= 0, is Fredholm for any s ∈ R if and only if the subordinate conormal symbol σcσ∧(R)(z, w) : C → C

is bijective for all z ∈ Z, w ∈ Γ1/2−γ. It is well-known in this context that this is the case

for all w ∈ C \ D(z) for a discrete set D(z) in the complex plane. In other words, (1.36) is Fredholm if and only if Γ_1/2−γ ∩ D(z) = ∅ for all z ∈ Z. Now applying Lemma 1.3 to a := σ∧(R)(z, ζ), |ζ| = 1, we obtain a block matrix isomorphism

σ∧(R) k t q (z, ζ) : Ks,γ_(R +) ⊕ G1,(z,ζ) −→ Ks−µ,γ−µ_(R +) ⊕ G2,(z,ζ) (1.37)

for (z, ζ) in the unit cosphere bundle S∗Z. Under the condition (1.35) which is fulfilled here in the case of the Dirichlet-to-Neumann operator (where µ = 1) we find k, t, q as edge symbols in the edge calculus, i.e., G1, G2 are liftings of bundles J1, J2 over Z with

respect to the canonical projection π : S∗Z → Z. So far we looked at the symbols for |ζ| = 1, but extensions by twisted homogeneities of any desired orders to T∗Z \ 0 gives us the homogeneous symbols themselves.

1.5 The edge-quantised Dirichlet-to-Neumann operator Theorem 1.5. (i) We have

N+C+ = R|int Y+ modulo L

−∞_{(int Y}

+), (1.38)

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1 THE DIRICHLET-TO-NEUMANN OPERATOR 15 (ii) The operator

R₁₁:= P_s−3/2+ N+C+E_s−1/2+ : Hs−1/2,s−1/2(Y+) → Hs−3/2,s−3/2(Y+)

cf. (1.16), (1.17), belongs to the edge calculus L1(Y+, g) for the weight data g =

(s, s − 1) over Y+ as a manifold with edge Z, cf. the formula (3.126). We have

R11= r+Re+ modulo M + G + G

with r+Re+ being the truncated operator belonging to the Dirichlet-to-Neumann op-erator (1.2) on the plus side of the boundary, moreover, M + G ∈ L1_{M +G}(Y+, g), cf.

(3.127), and G is an irregular Green operator on Y+, i.e., an Green operator

refer-ring to other (possibly more singular) weight data than g, cf. the weight shifts in [32, Remark 2.3.67].

Proof. (i) Because of the second relation of (1.5) the restriction of the Dirichlet-to-Neumann operator R to int Y+ is equal to N+K0|int Y+. This allows us to write

R|int Y+ = N+K0e

+

d + N+K0G,

Here G has the form G = EG+ for a Green operator G+ in Boutet de Monvel’s calculus,

G+ : Hs−1/2(int Y+) → Hs−1/2(int Y+), composed with a specific extension operator E :

Hs−1/2(int Y+) → Hs−1/2(Y ). Both G+ and E are contained in the construction of e+_d,

cf. the observations below in Subsection 3.1. This entails N+K0G ∈ L−∞(int Y+) which

is just the relation (1.38). Since c|η| is the homogeneous principal symbol of R|int Y+, cf.

(1.2), we obtain the same for N+C+.

(ii) From [36, Theorem 3.3.3] we know that the truncated operator r+Re+ belongs to the edge pseudo-differential calculus, modulo a smoothing Mellin plus Green operator and an extra finite-dimensional Green operator referring to “irregular” weights relative to the prescribed ones, contained in g, cf. the lower right 2×2 block matrix structure in Definition 3.29 below. Using (1.15) and (1.17) we now transform (1.30) to the 2 × 2 block matrix operator RND:= Λ+_NN+C+(Λ+_D)−1= P_s−3/2+ N+C+E_s−1/2+ P_s−3/2+ N+C+K_s−1/2+ T_s−3/2+ N+C+E_s−1/2+ T_s−3/2+ N+C+K_s−1/2+ ! , R_ND : Hs−1/2,s−1/2(Y+) ⊕ Hs−1/2(Z, CL(s−1/2)) → Hs−3/2,s−3/2(Y+) ⊕ Hs−3/2(Z, CL(s−3/2)) . (1.39)

In all these constructions we assume s to be so small that L(s − 1/2) and L(s − 3/2) become minimal.

Remark 1.6. As noted before the Dirichlet-to-Neumann operator R has not the trans-mission property with respect to any interface on Y . Therefore, we cannot expect that the truncated operator r+Re+ induces continuous maps Hs(int Y+) → Hs−1(int Y+) (say, for

s > −1/2). Theorem 1.5 (i) shows that there are s-dependent quantisations N+C+ such

that (1.30) are continuous under natural conditions on s. This choice of quantisation is crucial for the rest of this paper.

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Set

Γβ := {w ∈ C : Re w = β}

for any β ∈ R. We will interpret RND as an operator in the edge calculus, cf. the notation

(3.126) below. In this framework, as explained in connection with (1.14), we interpret the second upper subscript in the spaces over Y+ as a weight and often write γ instead of

s. Then, as soon as we realise RND as a continuous operator in such weighted spaces we

write RND(γ). In particular, RND in (1.39) then has the meaning of RND(s). Moreover,

N+C+= (N+C+)(s) will be replaced by (N+C+)(γ) when s is used in the meaning of γ.

Theorem 1.7. (i) The Dirichlet-to-Neumann operator R = T1K0 reduces the

Neu-mann boundary condition on the plus-side Y+ of the boundary to a continuous

oper-ator RND(γ) : Hs−1/2,γ−1/2_(Y +) ⊕ Hs−1/2(Z, CL(γ−1/2)) → Hs−3/2,γ−3/2_(Y +) ⊕ Hs−3/2(Z, CL(γ−3/2)) , (1.40)

for every s, γ ∈ R with

γ > 2 and γ /∈ 1 + N. (1.41)

The operator RND(γ) = (Rij(γ))i,j=1,2 belongs to the edge pseudo-differential

calcu-lus over Y+, regarded as a manifold with edge Z, where

R₁₁(γ) = P_γ−3/2+ (N+C+)(γ)E_γ−1/2+ , R12(γ) = Pβ−1+ (N+C+)(γ)K_γ−1/2+ ,

R21(γ) = T_γ−3/2+ (N+C+)(γ)E_γ−1/2+ , R22(γ) = T_γ−3/2+ (N+C+)(γ)K_γ−1/2+ .

(ii) There exists a discrete set D ⊂ C such that the edge symbol σ∧(RND(γ))(z, ζ) defines

a family of Fredholm operators

σ∧(RND(γ))(z, ζ) : Ks−1/2,γ−1/2_(R +) ⊕ CL(γ−1/2) → Ks−3/2,γ−3/2_(R +) ⊕ CL(γ−3/2) , (1.42)

for all γ ∈ R with (1.41) and Γ1/2−γ ∩ D = ∅, and RND(γ) itself admits an elliptic

operator R_ND(γ) :=RND K T Q (γ) : Hs−1/2,γ−1/2(Y+) ⊕ Hs−1/2(Z, CL(γ−1/2)) ⊕ Hs−1/2(Z, Cd1₎ → Hs−3/2,γ−3/2(Y+) ⊕ Hs−3/2(Z, CL(γ−3/2)) ⊕ Hs−3/2(Z, Cd2₎ (1.43)

for certain d1, d2 depending on γ in the edge calculus with additional potential and

trace operators and some classical first order pseudo-differential operator Q on Z. Proof. (i) The continuity of the operator (1.40) for s = γ under the conditions (1.41) follows from (1.39) for s − 1/2 instead of γ − 1/2. Then we can assume the smoothness s to be arbitrary, since operators in the edge calculus are continuous in edge spaces of any smoothness. According to Theorem 1.5 (ii) the operator RND(γ) belongs to the edge

pseudo-differential calculus on Y+ which is a smooth manifold with boundary Z, and Z

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1 THE DIRICHLET-TO-NEUMANN OPERATOR 17 to the edge calculus with the weight data g = (s, s − 1) for the involved s, namely, s > 2, s /∈ 1 + N. A such it induces continuous operators (1.40) for weights (1.41) and arbitrary s, and RND(γ) belongs to the edge calculus for the weight data g = (γ, γ − 1).

(ii) From the discussion in the preceding subsection, specialised to the case of the operator RNDin the formula (1.40) we obtain the following result. The operator (1.40) for γ > 2, γ /∈

1 + N, and Γ1/2−γ∩ D = ∅ for a discrete set D in C admits an elliptic operator (1.43) with

additional potential operators K, trace operators T, and pseudo-differential operators Q, on Z. In other words, (1.43) in L1(Y+, (γ, γ − 1); w) for w = (L(γ − 1/2) + d1, L(γ − 3/2) + d2)

is a Fredholm operator. In fact, in the case of the operator R = RND the exceptional

set D does not depend on z ∈ Z. In addition, since the homogeneous principal symbol of the Dirichlet-to-Neumann operator R only depends on |ζ| and the other ingredients in the quantisation R ⇒ RND are only of smoothing Mellin plus Green type, the topological

obstruction for the existence of additional Shapiro-Lopatinskij edge condition vanishes, i.e., we find an isomorphism (1.37) which corresponds to σ∧(RND). Together with the

interior ellipticity this is equivalent to the Fredholm property of the operator (1.43). It will be convenient to consider the operator (1.43) for a while for γ = s and to return to the standard Sobolev spaces over Y+

R_ND:=RND K T Q : Hs−1/2(int Y+) ⊕ Hs−1/2(Z, Cd1₎ → Hs−3/2(int Y+) ⊕ Hs−3/2(Z, Cd2₎ . (1.44)

More precisely, the relationship between RND and RND(s) is given by

R_ND(s) = diag(Λ+_N id2) RNDdiag((Λ+_D)−1 id1) (1.45)

for

id1 := idHs−1/2_(Z,Cd1), id2:= idHs−3/2_(Z,Cd2). (1.46)

As pointed out before the Dirichlet-to-Neumann operator R over Y+has not the

transmis-sion property at Z but the operator RND(s) with the quantisation RND(s) in the upper

left corner is an elliptic boundary value problem on the Neumann side of Y . It is of block matrix form, similarly as in Boutet de Monvel’s calculus, with extra trace and potential operators T and K, respectively, and a d2× d1 block matrix over Z. In the following

con-sideration we will relate RND(γ) with an elliptic problem of the edge calculus of boundary

value problems, applied to the Zaremba problem Am over G, cf. (1.19), interpreted as a

manifold with edge Z and boundary Y.We shall see that this interpretation works in both directions, i.e., we can work in the larger block matrix algebra of such edge/boundary problems over Y+. This shows that the calculus of boundary value problems for operators

without the transmission property on Y+ is an interesting substructure of mixed elliptic

problems. It can be interpreted as a special case of the edge calculus on Y+, regarded as

a manifold with edge Z and R+, the inner normal, as the model cone of local wedges.

1.6 Parametrices of the Zaremba problem

In the following considerations when we talk about the Fredholm property we tacitly assume that s is chosen in such a way that in the interpretation as a weight γ as in Theorem 1.7 we just have Γ1/2−s∩ D = ∅.

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Write the operator (1.29) in the form J_MD:= AMPD= I 0 N R_ND : Hs−2(int G) ⊕ Hs−1/2(int Y−) ⊕ Hs−1/2_{(int Y} +) → Hs−2(int G) ⊕ Hs−1/2(int Y−) ⊕ Hs−3/2_{(int Y} +)

for I := id_Hs−2_{(int G)⊕H}s−1/2_{(int Y}₋₎, N := (N+P0 N+C−) :

Hs−2(int G) ⊕ Hs−1/2(int Y−)

→ Hs−3/2_{(int Y}

+), and RND, the operator in (1.30). Then

JMD := I 0 M R_ND : Hs−2(int G) ⊕ Hs−1/2(int Y−) ⊕ Hs−1/2(int Y+) ⊕ Hs−1/2(Z, Cd1₎ → Hs−2(int G) ⊕ Hs−1/2(int Y−) ⊕ Hs−3/2(int Y+) ⊕ Hs−3/2(Z, Cd2₎ (1.47) for M :=N 0 is Fredholm, J_MD(−1) = I 0 − R(−1)_NDM R(−1)_ND ! (1.48)

is a parametrix of (1.47) with R(−1)_ND being a parametrix of RND. Let us form

A_D:=AD 0 0 id1 , PD:= PD 0 0 id1 , cf. the formulas (1.25), (1.46), and pass to

A_M:= JMDAD: Hs(int G) ⊕ Hs−1/2(Z, CL(s−1/2)) ⊕ Hs−1/2_{(Z, C}d1₎ → Hs−2(int G) ⊕ Hs−1/2_{(int Y} −) ⊕ Hs−3/2(int Y+) ⊕ Hs−3/2(Z, Cd2₎ (1.49)

which is a Fredholm operator, since both JMDand AD are Fredholm between the

respec-tive spaces and a parametrix is obtained by

A(−1)_M = PDJ_MD(−1), (1.50)

cf. formula (1.48). Let R(−1)_ND =:S C

B E

.

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1 THE DIRICHLET-TO-NEUMANN OPERATOR 19 Theorem 1.8. The operator Am which represents the Zaremba problem can be completed

by additional conditions Tm of trace type and Km of potential type with respect to the

interface Z and a d2× d1 block matrix Qm of classical pseudo-differential operators over

Z to a Fredholm operator A_M =Am Km T_m Q_m : Hs(int G) ⊕ Hs−1/2(Z, CL(s−1/2)) ⊕ Hs−1/2(Z, Cd1₎ → Hs−2(int G) ⊕ Hs−1/2(int Y−) ⊕ Hs−3/2(int Y+) ⊕ Hs−3/2(Z, Cd2₎ , (1.51)

which coincides with (1.49), and

A(−1)_M :=   P0− C+SN+P0 C−− C+SN+C− C+S C+C − c₊SN₊P0 c−− c+SN+C− c+S c+C − BN+P0 − BN+C− B E   (1.52) is a parametrix of (1.51).

Proof. First we identify the operator (1.49) as an operator containing Am as upper left

corner. In fact, we have

AM= JMDAD =      1 0 0 0 0 1 0 0 N+P0 N+C− RND K 0 0 T Q           A 0 0 D− d− 0 D+ d+ 0 0 0 id1      =     A 0 0 D− d− 0 N+ N+C−d−+ N+C+d+ K T D₊ T d₊ Q     (1.53)

for the operators RND, T , K, Q from (1.44). We have D−P0 = 0 as we see from (1.26). From

R_ND= N+C+it follows that N+P0A+N+C−D−+RNDD+= N+(P0A+C−D−+C+D+) =

N+, since P0A + C−D−+ C+D+= idHs_{(int G)}, cf. the formula (1.27). In other words, we

obtain AM in the form (1.51), where

T_m:= T D+, Km:=   0 0 d− 0 N+C−d−+ N+C+d+ K   , Qm:= (T d+ Q). (1.54)

Let us now compute the parametrix A(−1)_M of AM. Applying (1.48) and (1.50) we obtain

(1.52), namely, A(−1)_M =    P0 C− C+ 0 0 c− c+ 0 0 0 0 id1         1 0 0 0 0 1 0 0 − SN₊P0 − SN+C− S C − BN₊P0 − BN+C− B E      .

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2 Mellin-edge quantisations

2.1 The Zaremba problem in edge spaces

In the construction of the Zaremba problem with additional interface conditions (1.51), (1.53) we employ the operator RND, defined in (1.43), with elliptic conditions defined by

(1.54) for T , K, Q from (1.44). Now we translate the operator (1.51) into an operator between weighted edge spaces over Y±. Here we employ Lemma 1.1.

Using ΛD in (1.18) we first rephrase the operator AD in (1.24) for s > 1, s − 1 /∈ N, into

id_Hs−2_{(int G)} 0 0 ΛD AD: Hs(int G) ⊕ Hs−1/2(Z, CL(s−1/2)) → Hs−2(int G) ⊕ Hs−1/2,s−1/2(Y−) ⊕ Hs−1/2(Z, CL(s−1/2)) ⊕ Hs−1/2,s−1/2(Y+) ⊕ Hs−1/2_{(Z, C}L(s−1/2)₎ , (2.1)

with the inverse PD

id_Hs−2_{(int G)} 0

0 Λ−1_D

for PD = A−1_D . In order to express AMPD in

(1.29) we first write down the transformed operator AM, analogously as (2.1), namely,

id_Hs−2_{(int G)} 0 0 ΛM AM: Hs(int G) ⊕ Hs−1/2(Z, CL(s−1/2)) → Hs−2(int G) ⊕ Hs−1/2,s−1/2(Y−) ⊕ Hs−1/2(Z, CL(s−1/2)) ⊕ Hs−3/2,s−3/2(Y+) ⊕ Hs−3/2(Z, CL(s−3/2)) .

This allows us to transform the operator (1.29) to id_Hs−2_{(int G)} 0 0 ΛM   1 0 0 0 1 0 N+P0 N+C− N+C+   id_Hs−2_{(int G)} 0 0 Λ−1_D = I 0 N RND

for I = diag(idHs−2_{(int G)}, id_Hs−1/2,s−1/2_(Y

−), idHs−1/2(Z,CL(s−1/2))), N:= P + s−3/2N+P0 P + s−3/2N+C−E − s−1/2 P + s−3/2N+C−K − s−1/2 T_s−3/2+ N+P0 T_s−3/2+ N+C−E_s−1/2− T_s−3/2+ N+C−K_s−1/2− ! ,

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2 MELLIN-EDGE QUANTISATIONS 21 , and the operator (1.39). Analogously as (1.47) we consider the Fredholm operator

˜ J_MD:= I 0 M R_ND : Hs−2(int G) ⊕ Hs−1/2,s−1/2(Y−) ⊕ Hs−1/2_{(Z, C}L(s−1/2)₎ ⊕ Hs−1/2,s−1/2(Y+) ⊕ Hs−1/2(Z, CL(s−1/2)) ⊕ Hs−1/2(Z, Cd1₎ → Hs−2(int G) ⊕ Hs−1/2,s−1/2(Y−) ⊕ Hs−1/2_{(Z, C}L(s−1/2)₎ ⊕ Hs−3/2,s−3/2(Y+) ⊕ Hs−3/2(Z, CL(s−3/2)) ⊕ Hs−3/2(Z, Cd2₎ , M := N 0 , and RND as in (1.43). By ˜J (−1) MD := I 0 − R(−1)_NDM R(−1)_ND ! we obtain a parametrix. Let us form

˜ A_D: Hs(int G) ⊕ Hs−1/2(Z, CL(s−1/2)) ⊕ Hs−1/2(Z, Cd1₎ → Hs−2(G) ⊕ Hs−1/2,s−1/2(Y−) ⊕ Hs−1/2(Z, CL(s−1/2)) ⊕ Hs−1/2,s−1/2(Y+) ⊕ Hs−1/2(Z, CL(s−1/2)) ⊕ Hs−1/2(Z, Cd1₎ , ˜

AD := diag(idHs−2_{(int G)} Λ_D id₁)diag(A_D id₁). Then ˜A_M := ˜J_MDA˜_D is an analogue of

(1.53), namely, a Fredholm operator

˜ AM: Hs(int G) ⊕ Hs−1/2_{(Z, C}L(s−1/2)₎ ⊕ Hs−1/2(Z, Cd1₎ → Hs−2(int G) ⊕ Hs−1/2,s−1/2_(Y −) ⊕ Hs−1/2(Z, CL(s−1/2)) ⊕ Hs−3/2,s−3/2(Y+) ⊕ Hs−3/2_{(Z, C}L(s−3/2)₎ ⊕ Hs−3/2(Z, Cd2₎ where ˜A(−1)_M = ˜P_DJ˜(−1)_MD is a parametrix.

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2.2 Operators with jumping conditions of Dirichlet or Neumann type We now identify operators connected with the Dirichlet and the Neumann problem with edge operators of the spaces studied in Subsections 3.4, 3.5.

Concerning s in the meaning of a weight we make here the same assumptions as at the beginning of the preceding subsection.

We apply Proposition 3.9 (ii) below to s and s − 2 and obtain isomorphisms

Γs: Hs(int G) → Hs,s(G) ⊕ Hs_{(Z, C}N (s)₎ , Γs−2: Hs−2(int G) → Hs−2,s−2(int G) ⊕ Hs−2_{(Z, C}N (s−2)₎ . (2.2)

Let us define the isomorphism

ΞD:= Γ−1_s 0 0 id_Hs−1/2_(Z,CL(s−1/2)₎ ! : Hs,s(G) ⊕ Hs_{(Z, C}N (s)₎ ⊕ Hs−1/2(Z, CL(s−1/2)) → Hs(int G) ⊕ Hs−1/2(Z, CL(s−1/2)) . For ΠD:= Γs−2 0 0 ΛD : Hs−2(int G) ⊕ Hs−1/2(int Y−) ⊕ Hs−1/2(int Y+) → Hs−2,s−2(G) ⊕ Hs−2(Z, CN (s−2)) ⊕ Hs−1/2,s−1/2(Y−) ⊕ Hs−1/2(Z, CL(s−1/2)) ⊕ Hs−1/2,s−1/2(Y+) ⊕ Hs−1/2(Z, CL(s−1/2)) we form the composition

AD:= ΠDADΞD: Hs,s(G) ⊕ Hs(Z, CN (s)) ⊕ Hs−1/2(Z, CL(s−1/2)) → Hs−2,s−2(G) ⊕ Hs−2(Z, CN (s−2)) ⊕ Hs−1/2,s−1/2(Y−) ⊕ Hs−1/2(Z, CL(s−1/2)) ⊕ Hs−1/2,s−1/2(Y+) ⊕ Hs−1/2(Z, CL(s−1/2)) . (2.3)

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2 MELLIN-EDGE QUANTISATIONS 23 Similarly as Ad we consider the operator An and the Neumann analogue ˜A1 of (1.22),

namely, ˜ A1 := _A 1 0 0 id_Hs−3/2_(Z,CL(s−3/2)₎ : Hs(int G) ⊕ Hs−3/2(Z, CL(s−3/2)) → Hs−2(int G) ⊕ Hs−3/2_{(Y )} ⊕ Hs−3/2(Z, CL(s−3/2)) . Let D1 :=   idHs−2_{(int G)} 0 0 0 r− n− 0 r+ n+   : Hs−2(int G) ⊕ Hs−3/2(Y ) ⊕ Hs−3/2(Z, CL(s−3/2)) → Hs−2(int G) ⊕ Hs−3/2(int Y−) ⊕ Hs−3/2(int Y+) for

s > 2, s − 2 /∈ N which is an isomorphism. For AN:= D1A˜1 we then obtain

AN=   A 0 N− n− N+ n+   : Hs(int G) ⊕ Hs−3/2(Z, CL(s−3/2)) → Hs−2(int G) ⊕ Hs−3/2(int Y−) ⊕ Hs−3/2(int Y+) (2.4)

which is again Fredholm of index 0 but not an isomorphism as (1.24). Analogously as (2.3) we consider the operator

A_N:= ΠNANΞN: Hs,s(G) ⊕ Hs(Z, CN (s)) ⊕ Hs−3/2(Z, CL(s−3/2)) → Hs−2,s−2(G) ⊕ Hs−2(Z, CN (s−2)) ⊕ Hs−3/2,s−3/2(Y−) ⊕ Hs−3/2(Z, CL(s−3/2)) ⊕ Hs−3/2,s−3/2(Y+) ⊕ Hs−3/2(Z, CL(s−3/2)) , (2.5) for ΠN := Γs−2 0 0 ΛN , ΞN := Γ−1_s 0 0 id_Hs−3/2_(Z,CL(s−3/2)₎ . Similarly as (1.1) we perform the composition

ANA−1_D (2.6)

which is a 6 × 6 block matrix operator. The lower 4 × 4 submatrix is the announced new representation of the Dirichlet-to-Neumann operator between the respective weighted edge

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spaces plus Sobolev spaces on the interface Z, namely, S: Hs−1/2,s−1/2(Y−) ⊕ Hs−1/2(Z, CL(s−1/2)) ⊕ Hs−1/2,s−1/2(Y+) ⊕ Hs−1/2_{(Z, C}L(s−1/2)₎ → Hs−3/2,s−3/2(Y−) ⊕ Hs−3/2(Z, CL(s−3/2)) ⊕ Hs−3/2,s−3/2(Y+) ⊕ Hs−3/2_{(Z, C}L(s−3/2)₎ . (2.7)

Note that the upper left 2 × 2 corner of (2.6) just represents the identity in the space Hs−2,s−2(G) ⊕ Hs−2(Z, CN (s−2)), while the lower left 2 × 4 corner is a corresponding reformulation of the operator T1P0 in (1.1).

Let us finally set

A_M:= ΠMAMΞD,

cf. the formula (1.28), for ΠM:=Γs−2 0

0 ΛM

. Analogously as (1.29) for we now form

A_MP_D= ΠMAMΞD

Ξ−1_D P_DΠ−1_D = ΠMAMPDΠ−1D (2.8)

for PD := A−1D which plays the role of the reduction to the boundary of the Zaremba

problem, now in the weighted edge spaces. It is again a 6 × 6 block matrix operator. Again the upper left 2 × 2 corner of (2.8) represents the identity operator in Hs−2,s−2(G) ⊕ Hs−2(Z, CN (s−2)) which is just the reformulated identity 1 = id_Hs−2_{(int G)}in the upper left

corner of (1.29) by means of Γs−2 in (2.2). The remaining 2 × 4 block matrix occurring in

the first two lines of AMPDis equal to zero; it corresponds to the zeros in the first line of

(1.29). The lower left 4 × 2 corner of (2.8) is not essential for the ellipticity. The 4 × 4 lower right corner of AMPD is a continuous map

SM: Hs−1/2,s−1/2(Y−) ⊕ Hs−1/2(Z, CL(s−1/2)) ⊕ Hs−1/2,s−1/2_(Y +) ⊕ Hs−1/2(Z, CL(s−1/2)) → Hs−1/2,s−1/2(Y−) ⊕ Hs−1/2(Z, CL(s−1/2)) ⊕ Hs−3/2,s−3/2_(Y +) ⊕ Hs−3/2(Z, CL(s−3/2)) ,

which has the form

S_M=idHs−1/2,s−1/2_(Y

−)⊕Hs−1/2(Z,CL(s−1/2)) 0

T R_ND

. (2.9)

The shape of the first 2 × 4 row of (2.9) comes from the transformation of (1 0) in the second row of (1.29). The operator T in (2.9) is again not essential for the algebraic constructions with ellipticity, because of the triangular form of (2.9). The lower right 2 × 2 corner RND of (2.9) just comes from the right lower corner N+C+in (1.29), namely,

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2 MELLIN-EDGE QUANTISATIONS 25 RND= Λ+_NN+C+(Λ+_D)−1for the isomorphisms Λ+_D and Λ+_N contained in (1.15) and (1.16),

respectively. In other words the operator

RND= S33 S34 S43 S44 : Hs−1/2,s−1/2(Y+) ⊕ Hs−1/2(Z, CL(s−1/2)) → Hs−3/2,s−3/2(Y+) ⊕ Hs−3/2(Z, CL(s−3/2)) (2.10)

coincides with (1.39), i.e., R11 = S33, R12 = S34 , R21 = S43, R22 = S44, cf. (1.39).

From the operator AMwhich is of the form (1.53) and defines a Fredholm operator (1.49)

we now pass to an operator AM by translating the Sobolev spaces over G and Y± into

weighted spaces via the isomorphisms (2.2) which gives us a Fredholm operator

AM = ΠM⊕ id2AM ΞD⊕ id1 : Hs,s(G) ⊕ Hs(Z, CN (s)) ⊕ Hs−1/2_{(Z, C}L(s−1/2)₎ ⊕ Hs−1/2(Z, Cd1₎ → Hs−2,s−2(G) ⊕ Hs−2(Z, CN (s−2)) ⊕ Hs−1/2,s−1/2_(Y −) ⊕ Hs−1/2(Z, CL(s−1/2)) ⊕ Hs−3/2,s−3/2(Y+) ⊕ Hs−3/2_{(Z, C}L(s−3/2)₎ ⊕ Hs−3/2(Z, Cd2₎ . (2.11) Note that AM will be interpreted as the Zaremba problem in the edge calculus.

2.3 The edge algebra structure of boundary value problems with respect to the interface

As noted before one of the essential issues of this paper is to identify the operators AD,

ANas elements of the edge pseudo-differential calculus of boundary value problems. To be

more precise, in Subsection 3.4 we see that

G is a manifold with edge Z and boundary (2.12) where

G \ Z is a non-compact manifold with smooth boundary ∂G \ Z, (2.13) and

∂G is a manifold with edge Z without boundary, (2.14) cf. Definition 3.23 below. The calculus

Bµ,d(G, g; w) (2.15)

over (2.12) is developed in Subsection 3.4 below for M := G, where g are weight data and w a tuple of involved dimensions, cf. Definition 3.34. The operators in (2.15) consist of

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3 × 3 block matrices A = (Aij)i,j=1,2,3where (Aij)i,j=1,2restricts to (2.13) as a subcalculus

of the operator spaces Bµ,d(G \ Z, v) with induced dimension data v, that we read off from (2.3) or (2.5), cf. Definition 3.16, here for N := G \ Z, and the case without parameters, i.e., l = 0. As such there is the principal symbolic hierarchy

(σψ(A), σ∂(A))

of interior and boundary symbols where σψ(A) = σψ(A11), A11 ∈ Lµ_cl(int G). In addition

for operators in (2.15) the edge Z contributes the principal edge symbol σ∧(A).

Both σ∂(A) and σ∧(A) take values in operators; σ∂(A) lives on T∗(∂G\Z)\0 and σ∧(A) on

T∗Z \ 0. The three symbolic components will contribute to the ellipticity of A. Moreover, let us announce that the lower right corners (Aij)i,j=2,3 =: A0 belong to the edge

pseudo-differential calculus over (2.14). As such we have a principal symbolic hierarchy

(σ0_ψ(A0), σ_∂0(A0)), (2.16)

here in the meaning of transmission symbols from the respective calculus of transmis-sion problems, since Z is of codimentransmis-sion 1 in ∂G. Finally, the operators A33 belong to

Lµ_cl(Z), more precisely, those are block matrices of such operators, according to the above-mentioned dimensions.

The fact that AD and AN are elements of (2.15) is a consequence of the nature of the

involved entries. The “upper left” corners simply consists of the Laplace operator, in the respective weighted edge Sobolev spaces, cf. [21, Theorem 4.2.2] or [7, Theorem 2.2.1]. Formally we obtain the latter relations by locally introducing cylindrical coordinates, ac-cording to corresponding splitting of variables locally near Z where the 2-dimensional normal (xn−1, xn)-plane to Z, intersected with G, is identified with R+× S+1 × Rq,

S1₊= S1∩ {xn≥ 0}, q = dim Z. (2.17)

In a similar manner the boundary conditions can be transformed into the framework, since they are compositions of differential operators with the operator of restriction r0 to the boundary.

These observations are valid not only for the Dirichlet or Neumann problem, rephrased with respect to the interface Z but for arbitrary differential operators A and boundary conditions r0B for a differential operator B. In the language of the symbolic structures for operators in (2.15) where the so-called type d is now equal to ord B + 1 the first two components σψ(A), σ∂(A) are elliptic. What concerns σ∧(A) (both for AD and AN) also

the edge symbol is elliptic. This comes from the fact, that the operators (2.3) and (2.5) are Fredholm, and this property entails the ellipticity (i.e., bijectivity) of the three symbolic components. The corresponding general theorem on the necessity of ellipticity for the Fredholm property in (2.15) is not yet published somewhere. However, in the present case we do not need this information; it is a consequence of the already known ellipticity of (σψ(A), σ∂(A)), cf. Theorem 3.44 in Subsection 3.4 below.

Summing up we have the following result.

Theorem 2.1. (i) The operator (2.7) belongs to the edge pseudo-differential calculus over (2.14) and represents in that way an edge quantisation with respect to Z of the Dirichlet-to-Neumann operator R = T1K0 occurring in (1.1).

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2 MELLIN-EDGE QUANTISATIONS 27 (ii) The operator (2.7) is elliptic with respect to both symbolic components in (2.16). Proof. (i) follows from the fact that the operators in the calculus of operators (2.15) can be composed within the structure and that both ANand A−1_D belong to the calculus. Then

the same is true of the submatrix (2.7).

(ii) The ellipticity of (2.7) is a consequence of the ellipticity of ANA−1_D itself and the

observation that the upper left 2 × 2 corner is the identity.

Remark 2.2. (i) The operator (2.9) belongs to the edge calculus over B. (ii) The operator (2.10) is elliptic with respect to σ0_ψ, cf. the notation in (2.16).

In fact, the operator (2.9) is simply a submatrix in the calculus of operators in (2.15), namely, AMA−1_D . Moreover, the operator (2.10) is the lower right 2 × 2 submatrix of (2.7).

Since S is σ0_ψ-elliptic and because the non-smoothing contributions to S come from

(Sij)i,j=1,2, (Sij)i,j=3,4, (2.18)

while the other elements are smoothing off Z (more precisely, they consists of smoothing Mellin plus Green operators in the transmission algebra over ∂G with respect to Z) the operators (2.18) are both elliptic and hence, also the operator (2.10).

The following result is formulated in terms of boundary value problems in the sense of Definition 3.34 below.

Theorem 2.3. (i) The isomorphism AD(γ), first given for γ = s by (2.3) and then

realised as an operator (3.131) in weighted spaces, belongs to BµD,1(G, g

D; wD) where

µ_D, g_D, are defined by (3.135), (3.136) and wD by (3.137). The operator AD(γ) is

(σψ, σ∂, σ∧)-elliptic, and we have PD(γ) = AD(γ)−1∈ B−µD,0(G, g−1D ; w −1 D ).

(ii) The Fredholm operator AN(γ), first given for γ = s by (2.5) and then realised as an

operator (3.133) in weighted spaces, belongs to BµN,2_{(G, g}

N; wN) where µN, gN, are

defined by (3.138), (3.139) and wN by (3.140). The operator AN(γ) is (σψ, σ∂, σ∧

)-elliptic, and for the parametrix we have PN(γ) ∈ B−µN,0(G, g−1_N ; w−1_N ).

Proof. (i) We start with the symbolic levels for the Dirichlet problem in Boutet de Mon-vel’s calculus. As noted in Subsection 3.2 the operator has a principal symbolic structure, namely,

σ(A0) = (σψ(A0), σ∂(A0))

that we express in suitable coordinates. First we have σψ(A0)(ξ) = −|ξ|2 every where

on G the Riemannian metric is induced by the ambient Euclidean space Rn. Concerning σ∂(A0) = _σσ∂(∆)

∂(T0) we refer to coordinates in the half-space R

n

+ 3 (y, t) and covariables

(η, τ ). Here σ∂(∆)(η) = −|η|2+ ∂2 ∂t2 : H s (R+) → Hs−2(R+) (2.19)

which we observe for s > −3/2; the choice of s unimportant in this case. Later on we specify s according to the steps in the construction of AD. Together with σ∂(T0) = γ0 with

γ0 being the restriction to t = 0 we then have an isomorphism

σ∂(A0)(η) = σ∂(∆) γ0 (η) : Hs(R+) → Hs−2_(R +) ⊕ C (2.20)

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for every η 6= 0. Next we look at ˜A0 which contains A0 as a submatrix, and we set

σψ( ˜A0)(ξ) = σψ(A0)(ξ) = σψ(∆)(ξ), σ∂( ˜A0)(η) = σ∂(A0)(η). (2.21)

In the next step we pass to AD, defined in the formula (1.24). On the level of the symbols

we write σψ(AD) = σψ( ˜A0) and σ∂(AD) = σ∂−(AD), σ∂+(AD) for σ∂±(AD)(η) = σ∂(∆)(η) σ∂(D±)

where σ∂(D±) refers to ± side. Here we choose the coordinates (y, xn) = (z, xn−1, xn)

where the interface Z (the edge) is locally described by xn= 0, xn−1= 0, z ∈ Rn−2.

From the representation ∆ = r−2(r∂r)2 + ∂ϕ2 + r2∆z

of the Laplacian in cylindrical coordinates we obtain σψ(∆)(r, ρ, ς, ζ) = r−2(−r2ρ2− ς2− r2|ζ|2), for r > 0, (ρ, ς, ζ) 6= 0, (2.22) ˜ σψ(∆)(r, ρ, ς, ζ) = (−ρ2− ς2− |ζ|2) up to r = 0, (ρ, ς, ζ) 6= 0. (2.23) Moreover, σ∂±(∆)(ρ, ζ) = r −2 (−r2ρ2+ ∂_ϕ2− r2|ζ|2) : Hs(R+) → Hs−2(R+), for r > 0, (ρ, ζ) 6= 0, (2.24) ˜ σ∂±(∆)(ρ, ζ) = −ρ 2_{+ ∂}2 ϕ− |ζ|2 : Hs(R+) → Hs−2(R+), up to r = 0, (ρ, ζ) 6= 0. (2.25)

Analogously as before ∂± indicates the boundary symbols over int Y±. This yields

isomor-phisms σ∂(AD)(r, ρ, ζ) = σ∂−(∆) γ0 ,σ∂+(∆) γ0 ! : Hs(R+) → Hs−2(R+) ⊕ C (2.26) for r > 0, (ρ, ζ) 6= 0, and ˜ σ∂(AD)(r, ρ, ζ) = ˜σ∂−(∆) γ0 , ˜σ∂+(∆) γ0 ! : Hs(R+) → Hs−2(R+) ⊕ C . (2.27) up to r = 0, (ρ, ζ) 6= 0.

It remains to recognise the principal edge symbol σ∧(AD)(z, ζ) and to verify that it defines

also isomorphisms. The orders in the Sobolev spaces over Z occurring in (2.3) show that our operator belongs to BµD,1_{(M, g}

D; wD). In the present concrete case it is not advisable

to change rows and columns in the 6 × 3 block matrix. The source of the edge symbol is nothing else than the parameter-dependent family of Dirichlet problems in the upper (xn−1, xn) half-plane coming from A0 locally expressed in Rn−2z × R

2 + 3 (z, xn−1, xn), namely, σ(∆)(ζ) := −|ζ|2+ ∆xn−1,xn : H s (R2+) → Hs−2(R2+), (2.28)

together with the trace operator t0 : Hs(R2+) → Hs−1/2(Rxn−1), R

2

+= {(xn−1, xn) ∈ R2 :

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2 MELLIN-EDGE QUANTISATIONS 29 Remark 2.4. The notation σ in (2.28) as well as in the subsequent considerations has an analogous meaning as edge symbols σ∧ in the operator spaces considered in

Defini-tion 3.34 below. The modified notaDefini-tion here is chosen since we did not yet translate the occurring standard Sobolev spaces into weighted spaces as we do later on by using Hs(R±) ∼= Ks,s(R∓) ⊕ CL(s) coming from Proposition 3.6 and the second relation in (3.10)

together with Vs(R∓) ∼= CL(s), and similarly for Hs(R), and Proposition 3.9 (ii).

The meaning of t0 is analogous to T0, indicating the Dirichlet condition, here in the case

of the parameter-dependent operator (2.28).

For the proof of the σ∧-ellipticity we need the following result.

Proposition 2.5. The operators

σ(Aj)(ζ) := σ(∆)(ζ) σ(Tj) : Hs(R2+) → Hs−2(R2+) ⊕ Hs−(j+1/2)(R) (2.29)

s > 3/2, σ(Tj) := γ0◦ ∂xjn, form a family of isomorphisms, for all ζ ∈ R

n−2_{\ {0}, j = 0, 1.}

Proof. We first consider the case j = 0 and show that the family of operators α2_{− ∂}2 xn γ0 : S(R+) → S(R+) ⊕ C (2.30)

for γ0u := u|xn=0 defines an isomorphism for every real α 6= 0. Equivalently,

α2_{− ∂}2 xn γ0 : Hs(R+) → Hs−2(R+) ⊕ C is an isomorphism for s > 3/2. We assume α > 0 and write

`+(ξn) := α + iξn, `−(ξn) := α − iξn, α2− ∂x2n = op

+_(`

−)op+(`+)

where op+(·) := r+op(·)e+, cf. the notation around (3.46) below. We employ the fact that

op+(`−) : S(R+) → S(R+) (2.31) and op+_(` +) γ0 : S(R+) → S(R+) ⊕ C (2.32)

are isomorphisms. The isomorphism (2.31) is a consequence of the Paley-Wiener Theorem, see, e.g., [10], while the isomorphism (2.32) holds since op+(`+) : S(R+) → S(R+) is

surjective and γ0 : ker op+(`+) → C is an isomorphism, using that ker op+(`+) = {ce−αxn :

c ∈ C}. By virtue of op+(`−)op+(`+) = op+(`−`+) it follows that also

op+_(` −) 0 0 1 op+_(` +) γ0 =α 2_{− ∂}2 xn γ0 : S(R+) → S(R+) ⊕ C (2.33)

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is an isomorphism. Let us compute the shape of the inverse. First we have

(op+(`−))−1= op+(`−1− ) (2.34)

since `−is a minus symbol, see also the terminology in Eskin’s book [10]. Moreover, observe

that op+_(` +) γ0 (op+(`−1₊ ) k0) = 1 0 0 1 (2.35) for the potential symbol k0: C → ce−αxn. Thus we proved that (2.30) is an isomorphism.

The relations (2.33), (2.34) and (2.35) allows us to express the inverse of (2.30), namely α2_{− ∂}2 xn γ0 −1 =op +_(` +) γ0 −1 op+_(` −)−1 0 0 1 = (op+(`−1₊ ) k0) op+_(`−1 − ) 0 0 1 = (p0 k0).

for p0 = (op+(`+)−1op+(`−)−1). In this case we have op+(`+)−1op+(`−)−1 =

op+(`−1₊ `−1− ) + g = op+(α2 + ξn2)−1 + g for a so-called Green operator g of type 0 in

the boundary symbolic calculus of Boutet de Monvel’s algebra, cf. [32, Theorem 4.3.17], characterised by the mapping properties g, g∗ : L2_(R

+) → S(R+). Let −α2 _{= −|ζ|}2_{− ξ}2 n−1, ζ ∈ Rn−2\ {0}, ξn−1 ∈ R. Then a0(ζ, ξn−1) := −|ζ|2_−ξ2 n−1+∂2xn γ0hξn−1i3/2 belongs to S_cl2Rξn−1; H s_(R +), Hs−2(R+) ⊕ C

for every ζ ∈ Rn−2 \ {0}, cf. (3.21)

be-low. Thus Op_x_n−1(a0)(ζ) : Hs(R+2) = Ws(R, Hs(R+)) → Ws−2 R, Hs−2(R+) ⊕ C = Hs−2(R2+) ⊕ Hs−2(R) is a family of isomorphisms with the inverses Opxn−1(a

−1

0 )(ζ). Now

(2.29) itself is obtained by diag 1, Op_x_n−1(hξn−1i−3/2)Opxn−1(a0)(ζ).

Let us now prove the assertion for j = 1. By virtue of Proposition 2.5 for j = 1 the operator

α2_−∂2 xn γ0 : S(R+) → S(R+) ⊕ C

is an isomorphism for every real α 6= 0, or, equivalently,

α2_−∂2 xn γ0 : H s_(R +) → Hs−2(R+) ⊕ C

, s > 3/2. Then similarly as (1.1) we have

α2_{− ∂}2 xn γ1 (p0 k0) = 1 0 γ1p0 γ1k0 = 1 0 γ1p0 −α where γ1k0 = γ0∂xn(e −αxn_{) = −αe}−αxn_| xn=0= −α. Thus α2_{− ∂}2 t γ1 −1 = (p0 k0) 1 0 −γ₁p0α−1 −α−1 = (p0− k0γ1p0α−1 − k0α−1) =: (p1 k1) for p1 = p0 − k0γ1p0α−1, k1 = −k0α−1. Let a1(ζ, ξn−1) := −|ζ|2_−ξ2 n−1+∂xn2 γ1hξn−1i1/2 which be-longs to S2 Rξn−1; H s_(R +), Hs−2_(R +) ⊕ C for ζ ∈ Rn−2 \ {0}. Thus Op_x n−1(a1)(ζ) :

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2 MELLIN-EDGE QUANTISATIONS 31 Hs(R2+) = Ws(R, Hs(R+)) → Ws−2 R, Hs−2(R+) ⊕ C = Hs−2(R2+) ⊕ Hs−2(R) are

iso-morphisms with the inverses Op_x_n−1(a−1₁ )(ζ). Then (2.29) for j = 1 itself is obtained by diag 1, Op_x_n−1(hξn−1i−1/2)Opxn−1(a1)(ζ) which completes the proof.

The next steps in the proof of Theorem 2.3 is that we construct analogues of the iso-morphisms of Ad, ˜A0, D0, AD and AD, cf. the formulas (1.20), (1.22), (1.23), (1.24) and

(2.3), now starting with σ(A0)(ζ) rather than A0. The resulting operator functions will

be denoted by

σ(Ad)(ζ), σ( ˜A0)(ζ), σ(D0)(ζ), σ(AD)(ζ), σ∧(AD)(ζ). (2.36)

We shall see that the operators (2.36) are also isomorphisms which finally shows the σ∧

-ellipticity of AD. We form σ(Ad)(ζ) : Hs(R2+) → Hs−2(R2+) ⊕ Hs−1/2(R−) ⊕ Hs−1/2_(R +) , R∓ = {xn−1 ≶ 0}, and σ( ˜A0)(ζ) := σ(A0)(ζ) 0 0 id : Hs_(R2 +) ⊕ CL(s−1/2) → Hs−2(R2+) ⊕ Hs−1/2(R) ⊕ CL(s−1/2) .

Using a corresponding analogue of Lemma 1.2 we have

σ(D0)(ζ) :=   id_Hs−2_(R2 +) 0 0 0 r− σ(d−) 0 r+ σ(d+)  (ζ) : Hs−2(R2+) ⊕ Hs−1/2(R) ⊕ Ck(s−1/2) → Hs−2(R2+) ⊕ Hs−1/2(R−) ⊕ Hs−1/2(R+) .

Concerning the decomposition R = R−∪ R+ we employ the existence of an isomorphisms

r− _σ(d −) r+ _σ(d +) (ζ) : Hs−1/2_(R) ⊕ Ck(s−1/2) → Hs−1/2_(R −) ⊕ Hs−1/2(R+) .

for s ∈ R, s > 1, s − 1 /∈ N. The isomorphisms σ(AD)(ζ) have the form

σ(AD)(ζ) =   σ(∆) 0 σ(D−) σ(d−) σ(D+) σ(d+)  (ζ) : Hs(R2+) ⊕ CL(s−1/2) → Hs−2(R2+) ⊕ Hs−1/2(R−) ⊕ Hs−1/2(R+) ,

The edge algebra structure of the Zaremba problem

The edge algebra structure of the Zaremba problem

Contents

Introduction

1

The Dirichlet-to-Neumann operator

2

Mellin-edge quantisations