We now continue studying edge boundary value problems on M when the edge Z is of codimension 1 in ∂M. In this case the base of the model cone is S+1, and we have a decomposition
∂M = Y−∪ Y+ for Z = Y−∩ Y+.
In this case the weighted spaces over ∂M occurring in (3.112) appear as direct sums. In fact, Hs,γ(∂M ) = Hs,γ(Y−) ⊕ Hs,γ(Y+) with Y∓ being interpreted as manifolds with edge Z. In other words we have
Hs−1/2−αm,γ−1/2−αm(∂M ) = Hs−1/2−αm,γ−1/2−αm(Y−) ⊕ Hs−1/2−αm,γ−1/2−αm(Y+) for αm ∈ R and the same for βl. In mixed problems with different orders of boundary conditions on the ∓ sides we need different αm, βl on both sides, namely, pairs (α−m, α+m) and (β−l , β+l ), and to replace the former spaces with αm and βl by
Hs−1/2−α−m,γ−1/2−α−m(Y−) ⊕ Hs−1/2−α+m,γ−1/2−α+m(Y+) =: Hs−1/2−αm,γ−1/2−αmY−
Y+
(3.120) for any pair of orders αm := (α−m, α+m), and similarly for βl := (βl−, βl+).
Definition 3.45. On a compact manifold M with boundary ∂M and edge Z the space
Bµ,d(M, g; w) (3.121)
for µ ∈ Z, d ∈ N, w := (j1, j2; d1, d2) is defined to be the set of all continuous block matrix operators
A:
Hs,γ(M )
⊕
⊕jm=11 Hs−1/2−αm,γ−1/2−αmY−
Y+
⊕
⊕di=11 Hs−1−πi(Z)
→
Hs−µ,γ−µ(M )
⊕
⊕jl=12 Hs−1/2−µ−βl,γ−1/2−µ−βlY−
Y+
⊕
⊕dk=12 Hs−1−µ−τk(Z)
(3.122)
for s > d − 1/2, and µ, g defined in an analogous manner as in Definition 3.34 as the schemes of orders and weight shifts (to be read off from the entries) such that
Anl:= diag (1, P−β, R−τ) A diag (1, Pα, Rπ) is again of normalised form, here
Pα:= diag
Pα−1 0 0 Pα+1
!
, . . . , Pα
−
j1 0
0 Pα
+ j1
!
,
and similarly P−β, are matrices of order reductions in the respective edge algebras over Y∓ separately, while Rπ, etc. are as in Definition 3.34.
By Bµ,dM +G(M, g; w) (Bµ,dG (M, g; w)) the subspaces of operators in (3.121) which are smoothing Mellin plus Green or Green analogously as in Definition 3.34.
Let us illustrate the shape of symbols which are involved in the assertion. The principal symbolic hierarchy in this case distinguishes the ∓ sides of the boundary. Similarly as (3.115) we have
σ(A) = (σψ(A), σ∂(A), σ∧(A))
but with a modified meaning of σ∂ and σ∧. Writing A = (Aij)i,j=1,...,4 the interior sym-bol σψ is coming from A11 ∈ Lµcl(M \ ∂M ), i.e., we obtain σψ(A) := σψ(A11)(x, ξ), (x, ξ) ∈ T∗(M \ ∂M ) \ 0.
Moreover, the configuration determines two non-compact manifolds with boundary, namely, M \ Y+ with ∂(M \ Y+) = int Y−, M \ Y− with ∂(M \ Y−) = int Y+ Thus the computation of the principal boundary symbols concerns A11 close to int Y∓together with all entries in the 3 × 3 upper left corner of A belonging to trace, potential, etc., operators referring to Y− or Y+ alone. This determines a pair
σ∂(A) := σ∂−(A), σ∂+(A)
of matrix-valued boundary symbols, depending on (y, η) ∈ T∗(int Y∓) \ 0. Clearly in mixed problems the twisted homogeneities of the components of σ∂∓(A) may be different. The orders are determined by the orders involved in (3.122), at the end of Subsection 3.2.
3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 69 The edge symbol σ∧(A)(z, ζ) contains contributions from all entries of A. The scheme of the family of mappings is formally similar to (3.122). In this case we have
σ∧(A)(ζ) : op-erators in (3.121) we have
Lµ0(∂M, g0; w) for w = (j1, j2; d1, d2), (3.124) the subspace of lower right 2 × 2 corners A0 of (3.122) (more precisely, the lower right (2j2+ d2) × (2j1+ d1) corners ) with orders µ0 and weight data g0 of a similar meaning as in (3.114). The operators coming from the M + G (G) classes form subspaces
LµM +G0 (∂M, g0; w) (LµG0(∂M, g0; w)). (3.125) The elements of (3.124) represent transmission problems on ∂M with respect to the em-bedded interface Z of codimension 1. Those are also a variant of edge operators with Z being the edge where R− and R+ form the model cones of associated local wedges.
At the same time they can be interpreted as boundary value problems on ∂M with respect to (disjoint) boundary components int Y∓, here without the transmission property at Z.
For A0∈ Lµ0(∂M, g0; w) we have matrix-valued principal symbols of the form σ0(A0) := (σψ0 (A0), σ∧0 (A0))
where the components of σ0ψ(A0) := (σψ,−0 (A0), σ0ψ,+(A0)) are furnished by the homogeneous principal components of operators separately in Lµ+β
−
Remark 3.46. The results on the calculus of operators in (3.111), including continuity in weighted Sobolev spaces, compositions, ellipticity and parametrices, have natural analogues for (3.45) as well as for Lµ0(∂M, g0; w).
Let us pass to another specialisation of the above calculus, namely, to a variant only referring to Y− or Y+ alone. The manifolds Y∓ with smooth boundary Z are interpreted as manifolds with edge Z, and the only change is now, for instance, when we talk about the plus part only consider operators A0+ consisting of (j2 + d2) × (j1+ d1) lower right corners of (3.122). Denoting by
Lµ0+(Y+, g0+; w) (3.126) the corresponding space of operators (a subspace of (3.124)) we have again all features of the calculus as mentioned in Remark 3.46 and also the subclasses
Lµ
respectively. In particular, we have matrices of principal symbols σ+0 (A0+) := (σψ+(A0+), σ0∧(A0+)) especially, continuity, compositions, ellipticity, and parametrices.
Let us finally observe that every mixed problem
A =
A T−
T+
: Hs(int M ) →
Hs−µ(int M )
⊕
⊕jl=12 Hs−νl−−1/2(int Y−)
⊕
⊕jl=12 Hs−νl+−1/2(int Y+)
, (3.128)
for a differential operator A of order µ ∈ N on our manifold M with boundary ∂M and edge Z with differential boundary conditions T∓ over Y∓ belongs to the calculus of operators (3.121). It suffices to observe that A itself belongs to the calculus as an upper left corner and that the boundary operators T∓ correspond to trace operators in (3.121). Locally near Z the operator A is given in the variables (z, xn−1, xn) ∈ Rn−2z × R2+. In cylindrical coordinates, i.e., polar corrdinates in R2+ with S+1 = S1∩ R2+ as the base of the cone and the axial variable r we obtain A in edge-degenerate form, namely,
A = r−µ X
j+|α|≤µ
ajα(r, z)(−r∂r)j(rDz)α
for coefficients ajα ∈ C∞(R+× Rdim Z). Such edge-degenerate differential operators are automatically elements in (3.121) as upper left corners. Writing A as a continuous operator A : Hs(int M ) → Hs−µ(int M ), then, by applying Proposition 3.9 (ii) we obtain
Γs−µAΓ−1s :
Hs,s(M )
⊕ Hs(Z, CN (s))
→
Hs−µ,s−µ(M )
⊕
Hs−µ(Z, CN (s−µ))
(3.129)
which corresponds to a realisation in weighted edge spaces plus spaces over the edge Z, occurring in (3.122).
Concerning the trace operators T∓we consider, for instance, the plus case. The components of T+ in the differential case have the form r0+B : Hs(int M ) → Hs−ν−1/2(int Y+) for a differential operator B of order ν which is given in a collar neighbourgood of ∂M, and r0+ is the restriction to int Y+. In the case of the Zaremba problem B is simply the derivative in normal direction. Using Lemma 1.1 (ii) for s − ν − 1/2 > 1/2, s − ν − 1 /∈ N, we can pass to the operator
Ps−ν−1/2+ Ts−ν−1/2+
(r0+B)Γ−1s :
Hs,s(M )
⊕ Hs(Z, CN (s))
→
Hs−ν−1/2,s−ν−1/2(M )
⊕
Hs−ν−1/2(Z, CN (s))
. (3.130)
The entries of (3.130) belong to (3.121). In fact, the lower right corner is an L(s − ν − 1/2) × N (s) matrix of classical pseudo-differential operators on Z of order ν + 1/2. The first row consists of a pair of trace operators of order ν + 1/2 mapping to Y+ or Z and the
3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 71 lower left corner is a potential operator of order ν + 1/2. In this characterisation it is not necessary to normalise the orders by applying order reducing isomorphisms occurring in Definition 3.45 since it suffices to separatly look at the entries. Clearly we can transform the full mixed problem to the normalised form. Recall that till now the base N of the local model cones of M near Z is an arbitrary smooth compact manifold with boundary ∂N . In mixed boundary value problems, our main application here, we have N = S+1, and ∂N consists of two isolated points. In the calculus of operators Bµ,d(M, g; w) of Definition 3.34 we have a characterisation of the operators ADin (2.3) and ANin (2.5). The manifold M with boundary and edge in this case is
M = G, ∂M = Y−∪ Y+, Z = Y−∩ Y+.
In particular, the weighted spaces over ∂M may be written as direct sums of spaces over Y−and Y+, respectively. We already know that entry-wise we are in the edge calculus, i.e., we may interpret s at the second place, say, in Hs,s(G), as a weight γ while s at the first place remains in the meaning of a smoothness. For the smoothness we only assume s to be so large that the actions in the sense of Boutet de Monvel’s calculus are well-defined, including for parametrices.
In addition without loss of generality we rearrange the spaces, i.e., we write them down according to the scheme in (3.112). In such a reinterpretation the operator AD in (2.3) represents an operator
AD(γ) :
Hs,γ(G)
⊕
⊕N (γ)i
1=1Hs(Z)
⊕
⊕L(γ−1/2)l=1 Hs−1/2(Z)
→
Hs−2,γ−2(G)
⊕
Hs−1/2,γ−1/2(Y−)
⊕
Hs−1/2,γ−1/2(Y+)
⊕
⊕N (γ−2)k
1=1 Hs−2(Z)
⊕
⊕L(γ−1/2)l=1 Hs−1/2(Z)
⊕
⊕L(γ−1/2)l=1 Hs−1/2(Z)
(3.131)
for
s > 3/2 γ > 1, γ − 1 /∈ N. (3.132) Here we tacitly replaced s in the the corresponding weighted spaces by γ, similarly as in Subsection 1.5. This change is indicated by (γ). In a similar manner we proceed with the
operator that comes from the Neumann problem. For AN in (2.5) we obtain an operator Let us point out that the notation AD(γ) and AN(γ) does not only indicate the presence of γ in the spaces for the continuous actions in (3.131) and (3.133), but also an explicit dependence on γ of the operators themselves. For instance, the operators d∓in (1.24) and n∓in (2.4) depend on γ and there are exceptional values of γ given by (3.132) and (3.134), respectively, together with exceptions from certain discrete sets D in C.
Remark 3.47. (i) The operator (3.131) defines an element AD(γ) ∈ BµD,1(G, gD; wD)
3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 73 We now reformulate the operator (2.11). Similarly as before we replace s by γ where it occurs in the meaning of a weight. This gives us
AM(γ) :
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