Parameter-dependent pseudo-differential boundary value problems

H^s(R^q, C^{N (s)})

, with the inverse (E_s K_s) where P_s= 1−K_sT_s: W^s(R^q, H^s(R²)) → W^s(R^q, H_θ^s(R²)), is a projection, cf. [14, Theorem 1.1.68],

(ii) moreover, we have isomorphisms ^P_T^s

s
: H^s(R^q+2+ ) →

W^s(R^q, H_θ^s(R²+))

⊕ H^s(R^q, C^{N (s)})

, with the inverse (Es Ks) and Ps being of analogous meaning as in (i).

A similar result holds for 2G and G, respectively, where 2G is regarded as a manifold with embedded edge Z of codimension 2 while G is interpreted as a manifold with boundary Y and embedded edge Z in Y . According to the general notation of the edge calculus we have the weighted edge spaces

H^s,γ(2G) and H^s,γ(G)

locally modelled on W^s(R^q, K^s,γ((S¹)^∧)) and W^s(R^q, K^s,γ((S¹₊)^∧)), respectively, q = dim Z.

The global analogues of the isomorphisms in Proposition 3.8 are denoted again by the same letters.

Proposition 3.9. Let s > 1, s − 1 /∈ N.

(i) We have isomorphisms ^P_T^s

s

: H^s(2G) →

H^s,s(2G)

⊕ H^s(Z, C^{N (s)})

, with the inverse (E_s K_s) and the projection P_s= 1 − K_sT_s: H^s(2G) → H^s,s(2G).

(ii) Analogously we have isomorphisms Γ_s:= ^P_T^s

s
: H^s(int G) →

H^s,s(G)

⊕ H^s(Z, C^{N (s)})

, with the inverse (Es Ks) and Ps of analogous meaning as in (i).

3.2 Parameter-dependent pseudo-differential boundary value problems In our calculus we need pseudo-differential boundary value problems depending on param-eters. The operators are defined on a smooth manifold N of dimension n with boundary

3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 45

∂N . The local model of a manifold with boundary is Ω × R+ 3 (y, t), Ω ⊆ Rⁿ⁻¹ open, and the boundary value problems represent families of continuous operators

A(λ) :

H_comp,y^s (Ω × R+)

⊕ Hcomp^s−1/2(Ω, C^j−)

→

H_loc,y^s−µ(Ω × R+)

⊕

H_loc^s−µ−1/2(Ω, C^j+)

, λ ∈ R^l.

Here H_comp,y^s (Ω × R+) := W_comp^s (Ω, H^s(R+)), H_loc,y^s (Ω × R+) := W_loc^s (Ω, H^s(R+)). Mod-ulo smoothing operators to be formulated below the operators A(λ) are pseudo-differential in the sense

A(λ)u(y) := Op_y(a)(λ)u(y) = Z Z

e^i(y−y0)ηa(y, η, λ)u(y⁰)dy⁰d¯η,

d¯η = (2π)⁻⁽ⁿ⁻¹⁾dη, and a(y, η, λ) is a 2 × 2 block matrix family of operator-valued symbols

a(y, η, λ) = a_ij(y, η, λ)

i,j=1,2∈ S^µ

Ω × R^n−1+l_η,λ ;

H^s(R+)

⊕ C^j−

,

H^s−µ(R+)

⊕ C^j+



κ,κ, (3.43) of a specific structure, namely, as (3.23) for  := 1/2, i.e.,

µ =

 µ µ − 1/2

µ + 1/2 µ



. (3.44)

The group action in the spaces (3.43) is defined as

κ = {κδ}_δ∈R+, κδ:= diag(κδ, id) for (κδu)(t) := δ^1/2u(δt), δ ∈ R+, (3.45) and the identity in the respective finite-dimensional spaces. Note that κ_δ is unitary in L²(R+), i.e., κ^∗_δ= κ⁻¹_δ , δ ∈ R+.

The symbols (3.18) in boundary value problems contain different ingredients. The main part is connected with scalar symbols

p(y, t, η, τ, λ) ∈ S_cl^µ(R+× Ω × R^n+l_η,τ,λ)

that have the transmission property at the boundary t = 0. This means for the components p_(µ−j)(y, t, η, τ, λ) of homogeneity µ − j in (η, τ, λ) 6= 0, j ∈ N, that the relations

D^k_tD^α_η,λ{p_(µ−j)(y, t, η, τ, λ) − (−1)^µ−jp_(µ−j)(y, t, −η, −τ, −λ)}|t=0,(η,λ)=0= 0 hold for all y ∈ Ω, τ ∈ R \ {0}, k ∈ N, α ∈ N^n−1+l. In this context we assume µ ∈ Z. In addition, since the behaviour for large t is not essential, for convenience we assume that p does not depend on t for large t. Then we have

op_t(p)(y, η, λ) ∈ S^µ(Ω × R^n−1+l; H^s(R), H^s−µ(R))

for every s ∈ R. The notation op or opt, i.e., op(p)u(t) =RR e^i(t−t0)τp(τ )u(t⁰)dt⁰d¯τ is chosen in order to distinguish the one-dimensional action in connection with boundary symbols from the higher-dimensional action. Setting

op⁺(p)(y, η, λ) := r⁺op(p)(y, η, λ)e⁺ (3.46)

where e⁺ is the operator of extension by zero from R+ to R and r⁺ the operator of restriction to R+, for s > −1/2 we have

op⁺(p)(y, η, λ) ∈ S^µ(Ω × R^n−1+l; H^s(R+), H^s−µ(R+)). (3.47) Parameter-dependent pseudo-differential operators with the transmission property at the boundary on the half-space are then given (modulo smoothing operators) by Op_y(op⁺_t (p))(λ).

Examples are differential operators as well as pseudo-differential operators where p is the Leibniz inverse of the symbol of an elliptic differential operator (with parameter; clearly we always admit l = 0). Another important category of operator-valued symbols are Green, trace and potential symbols. Green symbols appear in considerations of symbols of the kind (3.47) as remainders while trace and potential symbols play a role in formulating ellipticity of boundary value problems.

Certain details concerning different entries in block matrix operators can be voluminous, as we already saw in Section 2. Therefore, we manage both the pseudo-differential BVPs in Boutet de Monvel’s calculus, as well as the calculus of BVPs on manifolds with edge below in two steps. We first establish the structures for model operators of a normalised form with a minimal numbers of entries and specific orders. In a second step, in order to reach the operators in our applications, we pass to larger block matrices. We admit more gen-eral orders, and an arbitrary number of rows and columns. For the (parameter-dependent) BVPs-calculus this means normalised orders in the sense of (3.44) and j− = j₊ = 1, cf.

Definition 3.13 below. The general definition will be given afterwards, cf. Definition 3.16 below. Then we require every individual entry to belong to the normalised class after suit-able reductions of orders.

In the following definition we set ν :=

 ν ν − 1/2 cf. the notation in connection with (3.22).

Definition 3.10. (i) Let R^ν,0_G (Ω × R^n−1+l) for ν ∈ R be the space of all

referring to the group actions (3.45) such that g(y, η, λ) and its (y, η, λ)-wise adjoint g^∗(y, η, λ) belong to 0, and g₁₂(y, η, λ) a potential symbol. The right lower corner is a scalar symbol in S_cl^ν(Ω × R^n−1+l).

(ii) R^ν,d_G (Ω × R^n−1+l), ν ∈ R, d ∈ N, is defined as the set of all g(y, η, λ) such that g₁₂(y, η, λ) and g₂₂(y, η, λ) are as before, while g₁₁(y, η, λ) =

3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 47 Pd

i=0g_11,i(y, η, λ)∂_tⁱ, g₂₁(y, η, λ) = Pd

i=0g_21,i(y, η, λ)∂_tⁱ for any Green symbols g_11,i(y, η, λ) of order ν − i and trace symbols g_21,i(y, η, λ) in R^ν+1/2−i,0_G (Ω × R^n−1+l), i = 0, . . . , d. The symbols g11(y, η, λ) are called Green symbols of order ν and type d, and g₂₁(y, η, λ) a trace symbols of order ν + 1/2 and type d.

Remark 3.11. Every g(y, η, λ) ∈ R^ν,d_G (Ω × R^n−1+l) induces a symbol

g(y, η, λ) ∈ S^ν_cl Ω × R^n−1+l_η,λ ;

H^s(R+)

⊕ C

,

S(R+)

⊕ C

!

for every s > d − 1/2. Moreover, in the case d = 0 the formal adjoint induces a

g^∗(y, η, λ) ∈ S_cl^ν∗ Ω × R^n−1+l_η,λ ;

H^s(R+)

⊕ C

,

S(R+)

⊕ C

!

for every s > −1/2.

Definition 3.12. We define R^µ,d(Ω × R^n−1+l) for µ ∈ Z, d ∈ N, to be the space of all a(y, η, λ) = diag(op⁺_t (p)(y, η, λ), 0) + g(y, η, λ)

for g(y, η, λ) = gij(y, η, λ)

i,j=1,2 where p(y, t, η, τ, λ) ∈ S_cl^µ(Ω × R × R^n+l_η,τ,λ) has the trans-mission property at t = 0 and g(y, η, λ) ∈ R^µ,d_G (Ω × R^n−1+l).

For manifold N with boundary (not necessarily compact) we form the double 2N , and we then have the spaces H_comp^s (2N ), H_loc^s (2N ), s ∈ N. Now we set

H_[comp)^s (int N ) := {u|int N : u ∈ H_comp^s (2N )}, H_[loc)^s (int N ) := {u|int N : u ∈ H_loc^s (2N )}.

By B^−∞,0(N ) we denote the set of all C that induce continuous mappings

C, C^∗ :

H_[comp)^s (int N )

⊕ H_comp^s (∂N )

→

H_[loc)^∞ (int N )

⊕ H_loc^∞(∂N )

for all s > −1/2, where C^∗is the formal adjoint of C with respect to a local L²pairing on N . The entires may be identified with operators with kernels in C^∞(N × N ), C^∞(∂N × N ), C^∞(N × ∂N ), and C^∞(∂N × ∂N ), respectively, via the Riemannian metric on N and the induced one on ∂N .The space B^−∞,0(N ) is Fr´echet in a natural way, and we set B^−∞,0(N ; R^l) := S(R^l, B^−∞,0(N )). Moreover, for any d ∈ N by B^−∞,d(N ; R^l) we denote the set of all C(λ) =Pd

j=0C_j(λ)diag(D^j, 0) for arbitrary C_j(λ) ∈ B^−∞,0(N ; R^l) and some differential operator D^j on N which is close to ∂N of the form ∂_t^j where t is the normal variable to ∂N .

Definition 3.13. Let N be a (not necessarily compact ) smooth manifold with boundary.

By

B^µ,d(N ; R^l)

for µ =

 µ µ − 1/2

µ + 1/2 µ



, µ ∈ Z, d ∈ N, we denote the set of all families of operators A(λ) = (A_ij)i,j=1,2(λ) of the form A(λ) =P

j∈NΦjχ⁻¹_j,∗Op_y(aj)(λ)Φ⁰_j+ Aint(λ) + C(λ) for a_j ∈ R^µ,d(Ω × R^n−1+l_η,λ ), A_int(λ) := diag(1 − ω, 0)A_int(λ) 0

0 0



diag(1 − ω⁰⁰, 0), A_int(λ) ∈ L^µ_cl(int N ; R^l), C(λ) ∈ B^−∞,d(N ; R^l).

By B^µ,d_G (N ; R^l) we denote the subspace defined by Aint(λ) = 0 and aj ∈ R^µ,d_G (Ω × R^n−1+l_η,λ ).

By definition the space B^µ,d(N ; R^l) (B^µ,d_G (N ; R^l)) consists of 2 × 2 block matrices. Those induce families of continuous operators

A =A K

T Q

 :

H_[comp)^s (int N )

⊕ Hcomp^s−1/2(∂N )

→

H_[loc)^s−µ(int N )

⊕ H_loc^s−1/2−µ(∂N )

for every s ∈ R, s > d − 1/2. The operators A in the upper left corner constitute a space that we denote by

B^µ,d(N ; R^l) (B^µ,d_G (N ; R^l)) (3.49) while the entries T and K form spaces

B^µ+1/2,d_trace (N ; R^l) and B^µ−1/2,0_potential(N ; R^l).

They have the meaning of trace and potential operators, respectively, in the space B^µ,d(N ; R^l). The lower right corners Q belong to L^µ_cl(∂N ; R^l). Observe, cf. [16, Remark 3.2.13], that an A ∈ B^µ,d(N ; R^l) of the form of an upper left corner belongs to B_G^µ,d(N ; R^l) if and only if A|_{int N} ∈ L^−∞(int N ; R^l).

We now turn to block matrix operators

A = (A_lm)l=0,1,...,j2,m=0,1,...,j1

where A₀₀∈ B^µ,d(N ; R^l) is of the form of an upper left corner, A_l0 ∈ B^µ+1/2+βl,d(N ; R^l)_trace for l ≥ 1 is a trace operator of order µ + 1/2 + βl and type d, A0m ∈ B^{µ−1/2−αm,0}(N ; R^l)_potential for m ≥ 1 is a potential operator of order µ − 1/2 − αm. The entries of (Alm)l=1,...,j2,m=1,...,j1 belong to L^µ+β_cl ^l−αm(∂N ; R^l).

For every ν ∈ R the space L^ν_cl(∂N ; R^l) contains a properly supported parameter-dependent elliptic element P^ν of order ν, and there exists a properly supported parametrix (P^ν)⁽⁻¹⁾ of P^ν in such that (P^ν)⁽⁻¹⁾P^ν and P^ν(P^ν)⁽⁻¹⁾ are both the identity modulo a properly supported element of L^−∞(∂N ; R^l).

Lemma 3.14. (i) For every Gnl∈ B^µ,d_G (N ; R^l), and properly supported elements P^−α∈ L^−α_cl (∂N ; R^l), P^β ∈ L^β_cl(∂N ; R^l), α, β ∈ R, the composition

1 0 0 P^β



G_nl1 0 0 P^−α



=:G K

T Q



defines a potential operator K ∈ B^{µ−1/2−α,0}(N ; R^l)_potential, a trace operator T ∈ B^µ+1/2+β,d(N ; R^l)_trace of type d, and a Q ∈ L^µ+β−α_cl (∂N ; R^l).

3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 49 (ii) If P^β and P^−α are parameter-dependent elliptic of order −α and β, respectively,

then we have

G_nl=1 0 0 (P^β)⁽⁻¹⁾

 G K

T Q

 1 0

0 (P^−α)⁽⁻¹⁾



+ C (3.50)

for some C ∈ B^−∞,d(N ; R^l).

Remark 3.15. For ∂N compact we can (and will ) choose P^−α and P^β as order reducing operators in such a way that the λ-wise inverses belong to L^α_cl(∂N ; R^l) and L^−β_cl (∂N ; R^l), re-spectively, and then the remainder C in (3.50) vanishes when we set (P^−α)⁽⁻¹⁾= (P^−α)⁻¹ and (P^β)⁽⁻¹⁾= (P^β)⁻¹.

Let us set

µ := µ (µ − 1/2 − αm)m=1,...,j1

t(µ + 1/2 + β_l)_l=1,...,j2 (µ + β_l− α_m)m=1,...,j1

l=1,...,j2

!

(3.51) for tuples of reals α = (α₁, . . . , α_j1), β = (β₁, . . . , β_j2). Here upper t means the transpose.

Definition 3.16. Let N be a smooth manifold with boundary. Then

B^µ,d(N, v; R^l) (3.52)

for µ ∈ Z, d ∈ N, and dimension data v := (j1, j2), is defined to be the space of all block matrix families of continuous operators

A :

H_[comp)^s (int N )

⊕

⊕^j_m=1¹ H_comp^{s−1/2−αm}(∂N )

→

H_[loc)^s−µ(int N )

⊕

⊕^j_l=1² H_loc^{s−1/2−µ−βl}(∂N )

(3.53)

for s > d − 1/2, such that the upper left corner A belongs to B^µ,d(N ; R^l), cf. the notation (3.49), while the other entries of

A_nl= diag (1, (P^β1)⁽⁻¹⁾, . . . , (P^βj2)⁽⁻¹⁾)A diag (1, P^α1, . . . , P^αj1) (3.54) are (according to the position in the block matrix ) potential operators of order µ − 1/2, trace operators of order µ + 1/2 of type d or pseudo-differential operators on ∂N of order µ, cf. Definition 3.13. By B_G^µ,d(N, v; R^l) we denote the space of all elements of (3.52) such that the upper left corner belongs to B^µ,d_G (N ; R^l), cf. the formula (3.49).

Remark 3.17. (i) The type d ∈ N in notation (3.52) indicates that all entries of the first column are of type ≤ d.

(ii) Note that the entries in the lower right j₂× j₁ corner of A have a scheme of orders as in Douglis-Nirenberg systems, see, in particular, [2].

(iii) By notation we have the normalised operator spaces

B^µ,d(N ; R^l) = B^µ,d(N, v_nl; R^l) for v_nl:= ((1, 1));

subscript “nl” indicates normalisations expressed in Definition 3.16.

Remark 3.18. For elements in (3.16) the notion of being properly supported also makes sense, and again every A ∈ B^µ,d(N ; v; R^l) admits a decomposition A = A0+ C where A0

is properly supported and C ∈ B^−∞,d(N ; v; R^l). A properly supported A induces continuous operators analogously as (3.53), now between spaces with [comp)/comp or [loc)/loc on both sides.

In our applications the boundary ∂N can be represented as

∂N = Y−∪ Y₊∪ Z

where Y∓ are embedded smooth manifolds with common boundary Z which is a smooth interface of codimension 1 in ∂N , and Y− ∩ Y₊. Then N \ Z is also a (non-compact) manifold with smooth boundary, namely, ∂(N \ Z) = int Y−∪ int Y₊. A simple example is N := {|z| ≤ 1} in the complex z-plane where ∂N = S¹ is the unit circle. Taking Z = {−1} ∪ {+1} and setting

Y∓:= S_∓¹ = S¹∩ {Im z T} int Y∓= S¹∩ {Im z ≷ 0}.

The calculus of operators of Definition 3.13 has a natural generalisation to trace and potential conditions separately over the disjoint components int Y∓of the boundary where we also may admit independent orders. Since this generalisation is straightforward we content ourselves with the remark, that, in order to make the calculus work, we replace αm and β_l by pairs of reals (α⁻_m, α⁺_m) and (β_l⁻, β_l⁺) , respectively, and the Sobolev spaces over ∂(N \ Z) by direct sums of the ones over int Y∓ with the respective smoothness. For any A ∈ B^µ,d(N, v; R^l) the upper left corner A can be written as

A = r⁺Ae˜ ⁺

for an ˜A ∈ L^µ_cl(2N ) where e⁺ extends distributions from int N by zero to the opposite side of the double 2N and r⁺restricts from 2N to int N . There is then a parameter-dependent homogeneous principal symbol σ_ψ( ˜A) of order µ as a function in C^∞(T^∗(2N ) × R^l\ 0), where 0 indicates (ξ, λ) = 0, and we set

σψ(A)(x, ξ, λ) := σψ( ˜A)|_T^∗_{N ×R}^l_\0(x, ξ, λ) =: σψ(A)(x, ξ, λ). (3.55) The boundary symbol of A will be expressed near a point of ∂N in local coordinates in Rⁿ+, depending on (y, η, λ) ∈ Rⁿ⁻¹× (R^n−1+l_η,λ \ {0}). By definition of A we have

A = diag (A, 0, . . . , 0) + G (3.56) where G ∈ B^µ,d_G (N, v; R^l), and we set

σ_∂(A)(y, η, λ) := diag (σ_∂(A)(y, η, λ), 0, . . . , 0) + σ_∂(G)(y, η, λ). (3.57) Here

σ_∂(A)(y, η, λ) := r⁺op(σ_ψ(A)|_t=0)(y, η, λ))e⁺ (3.58) where op is the pseudo-differential action in direction of the variable t normal to the boundary, referring to the local splitting of variables x = (y, t) and the covariables ξ = (η, τ ). The boundary symbol σ∂(G) of G is defined as follows. Restricting (3.54) to the subspace of Green operators G, analogously as (3.50) every individual entry of

G_nl= diag (1, (P^β1)⁽⁻¹⁾, . . . , (P^βj2)⁽⁻¹⁾)G diag (1, P^α1, . . . , P^αj1) (3.59)

3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 51 may be regarded locally as a family of operators Op_y(g)(λ) for a symbol g(y, η, λ) = (gij(y, η, λ))i,j=1,2 as in Definition 3.10, now for µ rather than ν. The entries are classical symbols of orders from the scheme

 µ µ − 1/2

µ + 1/2 µ



, and the matrix of the respective twisted homogeneous principal symbols just furnish

σ∂(Op_y(g))(y, η, λ). (3.60)

In other words we know the boundary symbol of G in (3.59). Then we set σ∂(G)(y, η, λ)

= σ∂(diag (1, P^β1, . . . , P^βj2))(y, η, λ)σ∂(Gnl)(y, η, λ)σ∂((diag (1, P^α1, . . . , P^αj1)⁻¹)(y, η, λ) where σ_∂(1) = id, and σ_∂(P^β)(y, η, λ) is the homogeneous principal symbol in (η, λ) 6= 0 of order β which is non-zero, because of the order reducing property of these factors. Because of (3.56) we obtain altogether the boundary symbol (3.57) as a (y, η, λ)-dependent family of operators

σ∂(A)(y, η, λ) :

H^s(R+)

⊕ C^j1

→

H^s−µ(R+)

⊕ C^j2

(3.61) for s > d − 1/2. The entries are twisted homogeneous of orders from σ_∂(A)(y, δη, δλ) = δ^µκ_δσ_∂(A)(y, η, λ)κ⁻¹_δ for the upper left corner, together with the orders coming from (3.53) specialised to G. Moreover, writing G for the upper left corner of G, because of (3.56) the upper left corner of A is equal to A + G, and for G we have again the twisted homogeneity σ_∂(G)(y, δη, δλ) = δ^µκ_δσ_∂(G)(y, η, λ)κ⁻¹_δ . The remaining entries of A are determined by G. We have

ord A_l0= µ + 1/2 + β_l, ord A0m= µ − 1/2 − αm, ord A_lm = µ + β_l− α_m, l = 1, . . . , j2, m = 1, . . . , j₁, l = 1, . . . , j₂.

The associated boundary symbols

σ_∂(A_l0)(y, η, λ) : H^s(R+) → C, σ∂(A_0m)(y, η, λ) : C → H^s−µ(R+), σ_∂(A_lm)(y, η, λ) : C → C have the twisted homogeneities

σ∂(Al0)(y, δη, δλ) = δ^µ+1/2+βlid_Cσ∂(Al0)(y, η, λ)κ⁻¹_δ , l = 1, . . . , j2, σ_∂(A0m)(y, δη, δλ) = δ^{µ−1/2−αm}κ_δσ_∂(A0m)(y, η, λ)id_C, m = 1, . . . , j1, and σ∂(Alm)(y, δη, δλ) = δ^µ+βl−βmid_Cσ∂(Alm)(y, η, λ)id_C, δ ∈ R+.

We set

σ(A) := (σ_ψ(A), σ_∂(A)).

In the following composition theorem we employ a notation µ◦ν between matrices of orders of the kind (3.51). The meaning is evident when we observe the orders of compositions of matrices of operators. Therefore, we postpone the formal definition to Subsection 3.4, formula (3.118).

Theorem 3.19. Let A ∈ B^µ,d(M, v; R^l), B ∈ B^ν,e(M, w; R^l) for v = (j0, j2), w = (j1, j0).

Then we have AB ∈ B^µ◦ν,c(M, v ◦ w; R^l) for c = max{ν + d, e}, v ◦ w = (j₁, j₂), and σ(AB) = σ(A)σ(B) with componentwise composition, i.e., σψ(AB) = σψ(A)σψ(B), etc.

If A or B belongs to the subclass with subscript G then the same is true of the composition.

在文檔中 The edge algebra structure of the Zaremba problem (頁 44-52)