The edge-quantised Dirichlet-to-Neumann operator

(z, ζ) :

K^s,γ(R+)

⊕ G_1,(z,ζ)

−→

K^{s−µ,γ−µ}(R+)

⊕ G_2,(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) i.e., N₊C₊∈ L¹_cl(int Y₊), and its homogeneous principal symbol is equal to c|η|.

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₊ : H^s−1/2(int Y₊) → H^s−1/2(int Y₊), composed with a specific extension operator E : H^s−1/2(int Y+) → H^s−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+)

⊕

H^s−1/2(Z, C^L(s−1/2))

→

Hs−3/2,s−3/2(Y+)

⊕

H^s−3/2(Z, C^L(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 H^s(int Y₊) → H^s−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.

Set

Γ_β := {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 R_ND as a continuous operator in such weighted spaces we write R_ND(γ). In particular, R_ND in (1.39) then has the meaning of R_ND(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

R_ND(γ) :

Hs−1/2,γ−1/2(Y₊)

⊕

H^s−1/2(Z, C^L(γ−1/2))

→

Hs−3/2,γ−3/2(Y₊)

⊕

H^s−3/2(Z, C^L(γ−3/2))

, (1.40) for every s, γ ∈ R with

γ > 2 and γ /∈ 1 + N. (1.41)

The operator R_ND(γ) = (R_ij(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) = P_β−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∧(R_ND(γ))(z, ζ) :

Ks−1/2,γ−1/2(R+)

⊕ C^L(γ−1/2)

→

Ks−3/2,γ−3/2(R+)

⊕ C^L(γ−3/2)

, (1.42)

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

R_ND(γ) :=RND K

T Q

 (γ) :

Hs−1/2,γ−1/2(Y+)

⊕

H^s−1/2(Z, C^L(γ−1/2))

⊕ H^s−1/2(Z, C^d1)

→

Hs−3/2,γ−3/2(Y+)

⊕

H^s−3/2(Z, C^L(γ−3/2))

⊕ H^s−3/2(Z, C^d2)

(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 R_ND(γ) belongs to the edge pseudo-differential calculus on Y+ which is a smooth manifold with boundary Z, and Z is interpreted as an edge. From Theorem 1.5 we know that the operator R_ND(s) belongs

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 R_ND(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 R_NDin 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 ⇒ R_ND 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:=R_ND K

T Q

 :

H^s−1/2(int Y+)

⊕ H^s−1/2(Z, C^d1)

→

H^s−3/2(int Y+)

⊕ H^s−3/2(Z, C^d2)

. (1.44)

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

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

id₁ := id_Hs−1/2(Z,C^d1), id₂:= id_Hs−3/2(Z,C^d2). (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 R_ND(γ) with an elliptic problem of the edge calculus of boundary value problems, applied to the Zaremba problem A_m 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.