Limiting absorption principle and perfectly matched layer method for Dirichlet Laplacians in quasi-cylindrical domains
aa r X i v : . [ m a t h . A P ] O c t LIMITING ABSORPTION PRINCIPLE AND PERFECTLYMATCHED LAYER METHOD FOR DIRICHLET LAPLACIANS INQUASI-CYLINDRICAL DOMAINS. ∗ VICTOR KALVIN † Abstract.
We establish a limiting absorption principle for Dirichlet Laplacians in quasi-cylindrical domains. Outside a bounded set these domains can be transformed onto a semi-cylinderby suitable diffeomorphisms. Dirichlet Laplacians model quantum or acoustically-soft waveguidesassociated with quasi-cylindrical domains. We construct a uniquely solvable problem with perfectlymatched layers of finite length. We prove that solutions of the latter problem approximate outgoingor incoming solutions with an error that exponentially tends to zero as the length of layers tendsto infinity. Outgoing and incoming solutions are characterized by means of the limiting absorptionprinciple.
Key words.
Perfectly Matched Layers, PML, quasi-cylindrical domains, Dirichlet Laplacian,limiting absorption principle, resonances, compound expansions
AMS subject classifications.
1. Introduction.
The perfectly matched layer (PML) method, originally intro-duced in [1], is in common use for the numerical analysis of a wide class of problems.For some of them stability and convergence of the method have been proved math-ematically; see, e.g., [2, 3, 4, 5, 6, 7, 20]. In the present paper we develop the PMLmethod for Dirichlet Laplacians in quasi-cylindrical domains
G ⊂ R n +1 , see Fig. 1.1.These are unbounded domains that outside a bounded set can be transformed ontoa semi-cylinder by suitable diffeomorphisms. Intuitively, one can understand G as / / ζ O O η G _______ _______ / / ζ O O η G _______ _______ Fig. 1.1 . Examples of quasi-cylindrical domains in R . a domain whose boundary asymptotically approaches at infinity the boundary of asemi-cylinder (0 , ∞ ) × Ω, where the cross-section Ω of G at infinity is a bounded do-main in R n . Dirichlet Laplacians ∆ model quantum or acoustically-soft waveguidesassociated with quasi-cylindrical domains. In order to characterize outgoing and in-coming solutions of the Helmholtz equation (∆ − µ ) u = f we establish a limitingabsorption principle. Then we construct a uniquely solvable problem with PMLs offinite length. This is a Dirichlet problem in the domain G truncated at a finite dis-tance. We prove that solutions of the latter problem locally approximate outgoing orincoming solutions of the Helmholtz equation with an error that exponentially tendsto zero as the length of PMLs tends to infinity. In other words, we prove stability andexponential convergence of the PML method. We find that the rate of exponential ∗ This work was supported by grant number 108898 awarded by the Academy of Finland. † Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve West,Montreal, H3G 1M8 Quebec, Canada ( [email protected]) VICTOR KALVIN convergence depends only on the spectral parameter µ and on the infinitely distantcross-section Ω. Thus the rate is the same as in the particular case of a domain G that coincides with the semi-cylinder (0 , ∞ ) × Ω outside a bounded set.As is known, construction of PMLs is closely related to the complex scaling.The complex scaling involves complex dilation of variables and has a long tradi-tion in mathematical physics and numerical analysis; for a historical account seee.g. [16, 10, 34]. Although there are several papers utilizing different approaches tothe complex scaling in waveguide-type of geometry, e.g. [8, 11, 12, 20], the complexscaling has not been used in quasi-cylindrical domains before. Our approach to thecomplex scaling originates from the one developed in [18] for a Schr¨odinger operatorin R (see also [16]). Deformations of the Dirichlet Laplacian by means of the com-plex scaling give rise to an analytic family of non-selfadjoint operators in G . Theseoperators correspond to a Dirichlet problem with infinite PMLs. Localization of theessential spectra of these operators together with certain relations between their re-solvents justifies a limiting absorption principle. For locating the essential spectra weemploy methods of the theory of elliptic boundary value problems [24, 25, 28]. Rela-tions between the resolvents are obtained with the help of Hardy spaces of analyticfunctions. Note that Hardy spaces in context of the complex scaling were originallyused in [37, 38]. As we mention in Remark 6.2, our methods also make it possi-ble to develop an analog of the celebrated Aguilar-Balslev-Combes-Simon theory ofresonances.As is typically the case, solutions to the Helmholtz equation satisfying the limitingabsorption principle locally coincide with solutions to the problem with infinite PMLs.Moreover, under certain assumptions on the right hand side solutions to the latterproblem are of some exponential decay at infinity. This allows us to prove uniquesolvability of the problem with finite PMLs and establish exponential convergence ofthe PML method by using the compound expansion technique [27, 25]. In [20] weused a similar approach to study the PML method for inhomogeneous media. Herewe study the PML method for a wide class of quasi-cylindrical domains.This paper is organized as follows. Section 2 consists of preliminaries, wherewe introduce notations, formulate our assumptions on the quasi-cylindrical domains,and give a formal definition of operators corresponding to the problem with infinitePMLs. In Section 3 we demonstrate that the operators are well-defined and derivesome estimates on their coefficients. As shown in Section 4, these operators give riseto an analytic family of m-sectorial operators. In Section 5 we introduce and studyHardy spaces of analytic functions. In Section 6 we formulate and prove a limitingabsorption principle. In Section 7 we show that under certain assumptions on theright hand side solutions to the problem with infinite PMLs are of some exponentialdecay at infinity. Finally, in Section 9 we formulate and study the problem with finitePMLs and prove exponential convergence of the PML method.
2. Preliminaries.
In this section we introduce basic notations that are in usethroughout the paper. We formulate our assumptions on the quasi-cylindrical do-mains and introduce differential operators corresponding to the problem with infinitePMLs. Recall that PMLs are artificial strongly absorbing layers designed so thatwaves coming from a non-PML medium to PMLs do not reflect at the interface.Let ( x, y ) and ( ζ, η ) be two systems of the Cartesian coordinates in R n +1 , n ≥ x, ζ ∈ R , while y = ( y , . . . , y n ) and η = ( η , . . . , η n ) are in R n . Let ∂ x = ddx , ∂ y m = ddy m , and ∂ ζ = ddζ , ∂ η m = ddη m .Consider a bounded domain Ω ⊂ R n , and the semi-cylinder R + × Ω, where R + = IRICHLET LAPLACIANS IN QUASI-CYLINDRICAL DOMAINS { x ∈ R : x > } . We say that C ⊂ R n +1 is a quasi-cylinder, if there exists adiffeomorphism R + × Ω ∋ ( x, y ) κ ( x, y ) = ( ζ, η ) ∈ C , (2.1)such that the elements κ ′ ℓm ( x, · ) of its Jacobian matrix κ ′ tend to the Kronecker delta δ ℓm in the space C ∞ (Ω) as x → + ∞ .Let G be a domain in R n +1 with smooth boundary ∂ G . We suppose that the set { ( ζ, η ) ∈ G : ζ ≤ } is bounded, and the set { ( ζ, η ) ∈ G : ζ > } coincides with aquasi-cylinder C . (Extension of our results to the case of a domain G that coincidesoutside a bounded set with several quasi-cylinders is straightforward.) Following [25],we say that G is a quasi-cylindrical domain.Introduce the notation ∇ ζη = ( ∂ ζ , ∂ η , . . . , ∂ η n ) ⊤ . In the domain G we considerthe Dirichlet Laplacian ∆ = −∇ ζη · ∇ ζη , which is initially defined on the set C ∞ ( G )of all smooth compactly supported functions u in G satisfying the Dirichlet boundarycondition u ↾ ∂ G = 0.Consider the complex scaling x x + λs ( x − r ) with parameters r > λ ∈ C .Here s ( x ) is a smooth scaling function possessing the properties: s ( x ) = 0 for x ≤ , (2.2)0 ≤ s ′ ( x ) ≤ x ∈ R , (2.3) s ′ ( x ) = 1 for x ≥ C > , (2.4)where s ′ ( x ) = ∂ x s ( x ), and C is arbitrary. For all real λ ∈ ( − ,
1) the function R + ∋ x x + λs ( x − r ) is invertible, and R + × Ω ∋ ( x, y ) κ λ,r ( x, y ) = ( x + λs ( x − r ) , y ) ∈ R + × Ωis a selfdiffeomorphism of the semi-cylinder. Therefore ϑ λ,r ( ζ, η ) = (cid:26) κ ◦ κ λ,r ◦ κ − ( ζ, η ) for ( ζ, η ) ∈ C , ( ζ, η ) for ( ζ, η ) ∈ G \ C , (2.5)is a selfdiffeomorphism of G . In other words, ϑ λ,r with λ ∈ ( − ,
1) and r > C along the curvilinear coordinate x . Let ( ϑ ′ λ,r ) ⊤ bethe transpose of the Jacobian matrix ϑ ′ λ,r . Then e λ,r = ( ϑ ′ λ,r ) ⊤ ϑ ′ λ,r is the matrixcoordinate representation of a metric e λ,r on G , and∆ λ,r = − (cid:0) det e λ,r (cid:1) − / ∇ ζη · (cid:0) det e λ,r (cid:1) / e − λ,r ∇ ζη (2.6)is the Laplace-Beltrami operator on the Riemannian manifold ( G , e λ,r ). As the pa-rameter r increases, the equalities ϑ λ,r ( ζ, η ) = ( ζ, η ), e λ,r ( ζ, η ) = Id, where Id is the( n + 1) × ( n + 1) identity matrix, and the equality ∆ λ,r = ∆ become valid on a largerand larger subset of G . In the case λ = 0 the scaling is not applied. Therefore e ,r ≡ e is the Euclidean metric and ∆ ,r ≡ ∆.In order to consider complex values of the scaling parameter λ , we impose addi-tional assumptions on the diffeomorphism (2.1): i. the function R + ∋ x κ ( x, · ) ∈ C ∞ (Ω) has an analytic continuation from R + to some sector S α = { z ∈ C : | arg z | < α < π/ } ; (2.7) VICTOR KALVIN ii. the elements κ ′ ℓm ( z, · ) of the Jacobian matrix κ ′ tend to the Kronecker delta δ ℓm in the space C ∞ (Ω) uniformly in z ∈ S α as z → ∞ .For instance, the assumptions i,ii are satisfied for the following quasi-cylinders: C = { ( ζ, η ) ∈ R : ( ζ, η ) = ( x, y + log( x + 2)) , x ∈ R + , y ∈ [0 , } , C = n ( ζ, η ) ∈ R : ζ = Z x ϕ ( t ) dt, η = yψ ( x ) , x ∈ R + , y ∈ [0 , o , where as ϕ ( x ) and ψ ( x ) we can take the functions 1, 1 + e − x , 1 + ( x + 1) − s with s > / log( x + 2), 1 + 1 / log(1 + log( x + 2)), and so on. These examples show thatquasi-cylinders can have very different shapes comparing with the semi-cylinder.In the next section we will show that for all sufficiently large r > i, ii on κ together with (2.2) and (2.3) lead to the analyticity of the coefficients ofthe differential operator (2.6) with respect to the scaling parameter λ in the disk D α = { λ ∈ C : | λ | < sin α < / √ } . (2.8)Thus the equality (2.6) defines ∆ λ,r for all λ ∈ D α . Clearly, ∆ λ,r coincides with theDirichlet Laplacian ∆ on the set G r = ( G \ C ) ∪ { ( ζ, η ) ∈ C : ( ζ, η ) = κ ( x, y ) , x < r, y ∈ Ω } . (2.9)We will show that on the set G \ G r the operator ∆ λ,r with λ ∈ D α \ R describes aninfinite PML for ∆. In the case ℑ λ > ℑ λ <
0) this PML is an artificialnonreflective strongly absorbing layer for the outgoing (resp. incoming) solutions.
Remark 2.1.
For simplicity we consider in this paper only Dirichlet Laplacians.However, similar methods can be used to develop and study PML method for theSchr¨odinger operator ∆ + V in G , where ∆ is the Dirichlet Laplacian and V ∈ C ∞ ( G ) is a real-valued potential with the following properties: for some r > and α > the function x V ◦ κ ( x, · ) ∈ L (Ω) extends by analyticity to the sector { z ∈ C : | arg( z − r ) | < α } , where for all y ∈ Ω we have | V ◦ κ ( z, y ) | ≤ C ( | z | ) → as z → ∞ .One can also include into consideration potentials with moderate local singularitiesand relatively bounded operator-valued potentials.
3. Construction of infinite PMLs.
In this section we show that for all suf-ficiently large r > λ in the disk (2.8). We also obtain some estimates on thematrix e λ,r .Consider the quasi-cylinder C as a manifold endowed with the Euclidean metric e . We will use the coordinates ( ζ, η ) in G and ( x, y ) in R + × Ω, and identify theRiemannian metrics on G and R + × Ω with their matrix coordinate representations.Let g = κ ∗ e be the pullback of the metric e by the diffeomorphism κ in (2.1). Thenthe matrix g = [ g ℓm ] n +1 ℓ,m =1 is given by the equality g = ( κ ′ ) ⊤ κ ′ , where ( κ ′ ) ⊤ is thetranspose of the Jacobian κ ′ . Since the diffeomorphism κ satisfies the assumptions i,ii of Section 2, we conclude that the metric matrix elements S α ∋ z g ℓm ( z, · ) ∈ C ∞ (Ω) (3.1)are analytic functions. Moreover, g ℓm ( z, · ) tends to the Kronecker delta δ ℓm in thespace C ∞ (Ω) uniformly in z ∈ S α as z → ∞ or, equivalently, we have (cid:12)(cid:12) ∂ qy ( g ℓm ( z, y ) − δ ℓm ) (cid:12)(cid:12) ≤ C q ( | z | ) → z → ∞ , z ∈ S α , y ∈ Ω , | q | ≥
0; (3.2)
IRICHLET LAPLACIANS IN QUASI-CYLINDRICAL DOMAINS / / ℜ z uuuuuuuuuu IIII α O O ℑ z ppppppp • r arg(1+ λ ) L λ,r Fig. 3.1 . The curve L λ,r for complex values of λ . here ∂ qy = ∂ q y ∂ q y . . . ∂ q n y n with a multiindex q = ( q , . . . , q n ), and | q | = P q j .Consider the selfdiffeomorphism κ λ,r , λ ∈ ( − , R + × Ω.We define the metric g λ,r = κ λ,r ∗ g on R + × Ω as the pullback of the metric g by κ λ,r . As a result we get the manifold ( R + × Ω , g λ,r ) parameterized by λ ∈ ( − ,
1) and r >
0. We deduce g λ,r ( x, y ) = diag { λs ′ ( x − r ) , Id } g ( x + λs ( x − r ) , y ) diag { λs ′ ( x − r ) , Id } , (3.3)where Id stands for the n × n -identity matrix, and diag { λs ′ ( x − r ) , Id } is theJacobian of κ λ,r .Let us consider complex values of the scaling parameter λ . We suppose that λ is inthe complex disk D α , where α is the same as in our assumptions on the partial analyticregularity of the diffeomorphism κ ; cf. (2.7), (2.8). The curve L λ,r = { z ∈ C : z = x + λs ( x − r ) , x > } lies in the sector S α , see Fig. 3.1. We define the matrix g λ,r forall non-real λ in the disk by the equality (3.3), where g ( x + λs ( x − r ) , y ) stands for thevalue of the analytic in z ∈ S α function g ( z, y ) at z = x + λs ( x − r ). By analyticityin λ we conclude that g λ,r is a complex symmetric matrix, the Schwarz reflectionprinciple gives g λ,r = g λ,r , where the overline stands for the complex conjugation. If λ ∈ D α is non-real, then the matrix g λ,r does not correspond to a Riemannian metric.However, g λ,r is invertible for all λ ∈ D α provided r > s ( x − r ) = 0 and g − λ,r ( x, y ) = g − ( x, y ) for all x < r . On the other hand, the matrix g ( x + λs ( x − r ) , y ) from (3.3) is invertible for x ≥ r with large r > | x + λs ( x − r ) | ≥ r .The derivatives ∂ px ∂ qy g − λ,r are analytic functions of λ ∈ D α . From the condi-tions (3.1) and (3.2) together with (2.4) and (3.3) it follows that (cid:13)(cid:13)(cid:13) ∂ px ∂ qy (cid:0) g − λ,r ( x, y ) − diag (cid:8) (1 + λ ) − , Id (cid:9)(cid:1)(cid:13)(cid:13)(cid:13) → x → + ∞ , y ∈ Ω , λ ∈ D α , (3.4)and the estimate (cid:13)(cid:13)(cid:13) ∂ px ∂ qy (cid:0) g − λ,r ( x, y ) − diag (cid:8) (1 + λs ′ ( x − r )) − , Id (cid:9)(cid:1)(cid:13)(cid:13)(cid:13) ≤ C pq ( r ) (3.5)holds uniformly in ( x, y ) ∈ [ r, ∞ ) × Ω and λ ∈ D α , where k · k is the matrix norm k A k = max ℓm | a ℓm | , and p + | q | ≥
0. The constants C pq ( r ) in (3.5) tend to zero as r → + ∞ .We define the complex scaling ϑ λ,r for all λ in the disk D α by the equality (2.5),where κ ◦ κ λ,r ( x, y ) is the value of the analytic in z ∈ S α function κ ( z, y ) at the point VICTOR KALVIN z = x + λs ( x − r ). Consider the matrix e λ,r = ( ϑ ′ λ,r ) ⊤ ϑ ′ λ,r . It is clear that for all( ζ, η ) ∈ G \ C the matrix e λ,r ( ζ, η ) coincides with the ( n + 1) × ( n + 1)-identity. Forall real λ ∈ D α we have g λ,r ( x, y ) = (cid:0) κ ′ ( x, y ) (cid:1) ⊤ (cid:0) e λ,r ◦ κ ( x, y ) (cid:1) κ ′ ( x, y ) , ( x, y ) ∈ R + × Ω , (3.6)where g λ,r = κ ∗ e λ,r is the pullback of the corresponding metric e λ,r on C by the dif-feomorphism κ . Therefore e λ,r is analytic in λ ∈ D α and invertible for all sufficientlylarge r >
0. By analyticity in λ we conclude that e λ,r ( ζ, η ) is a complex symmetricmatrix, the Schwarz reflection principle gives e λ,r = e λ,r .Differentiating the equality (3.6), we see that the matrices ∂ pζ ∂ qη e λ,r and ∂ pζ ∂ qη e − λ,r are analytic in λ ∈ D α . Moreover, from (3.4), (3.5), and our assumptions on κ weobtain (cid:13)(cid:13)(cid:13) ∂ pζ ∂ qη (cid:0) e − λ,r ( ζ, η ) − diag (cid:8)(cid:0) λ (cid:1) − , Id (cid:9)(cid:1)(cid:13)(cid:13)(cid:13) → ζ → + ∞ , (cid:13)(cid:13)(cid:13) ∂ pζ ∂ qη (cid:0) e − λ,r ( ζ, η ) − diag (cid:8)(cid:0) λ s ′ r ( ζ, η ) (cid:1) − , Id (cid:9)(cid:1)(cid:13)(cid:13)(cid:13) ≤ c ( r ) , where p + | q | ≤ , and c ( r ) → r → + ∞ . (3.7)Here ( ζ, η ) ∈ C and s ′ r ( ζ, η ) stands for the function s ′ ( x − r ) written in the coordinates( ζ, η ). We extend s ′ r from C to G by zero. Note that the estimate (3.7) remains validfor all ( ζ, η ) ∈ G and the constant c ( r ) is independent of λ ∈ D α and ( ζ, η ) ∈ G . Nowwe see that for all sufficiently large r > λ in the disk D α , and its coefficients are subjected to the estimate (3.7).
4. Analytic families of operators.
In this section we study the unboundedoperator ∆ λ,r in the Hilbert space L ( G ) with the usual norm k u ; L ( G ) k = (cid:18)Z G | u ( ζ, η ) | dζ dη (cid:19) / . The operator ∆ λ,r is initially defined on the set C ∞ ( G ). We show that the operatoris closable, and the closure defines an analytic family D α ∋ λ ∆ λ,r of type (B) [23].In a standard way this implies that the resolvent (∆ λ,r − µ ) − is an analytic functionof λ and µ on some open subset of D α × C . The latter fact will be used for justificationof a limiting absorption principle in Section 6.We intend to show that the operator ∆ λ,r in L ( G ) with the domain C ∞ ( G ) issectorial and to study its Friedrichs extension. With the operator ∆ λ,r we associatethe quadratic form q λ,r [ u, u ] = Z G (cid:10)(cid:0) det e λ,r (cid:1) / e − λ,r ∇ ζη u, ∇ ζη (cid:0) det e λ,r (cid:1) − / u (cid:11) dζ dη, (4.1)where h· , ·i is the Hermitian inner product in C n +1 and u ∈ C ∞ ( G ). Let ( · , · ) standfor the inner product in L ( G ). We represent the quadratic form as follows: q λ,r [ u, u ] = ( −∇ ζη · e − λ,r ∇ ζη u, u )+ Z G (cid:10)(cid:0) det e λ,r (cid:1) / e − λ,r ∇ ζη u, u ∇ ζη (cid:0) det e λ,r (cid:1) − / (cid:11) dζ dη. (4.2) IRICHLET LAPLACIANS IN QUASI-CYLINDRICAL DOMAINS
Lemma 4.1.
Assume that r > is sufficiently large. Then there exist ϕ < π/ and δ > such that for all λ ∈ D α and u ∈ C ∞ ( G ) we have | arg( −∇ ζη · e − λ,r ∇ ζη u, u ) | ≤ ϕ, δ (∆ u, u ) ≤ ℜ ( −∇ ζη · e − λ,r ∇ ζη u, u ) ≤ δ − (∆ u, u ) . In other words, the form ( −∇ ζη · e − λ,r ∇ ζη u, u ) is sectorial.Proof . It is clear that( −∇ ζη · e − λ,r ∇ ζη u, u ) = Z G (cid:10) e − λ,r ∇ ζη u, ∇ ζη u (cid:11) dζ dη. (4.3)Let us estimate the numerical range of the matrix e − λ,r ( ζ, η ). We shall rely on theestimate (3.7). Let ξ = ∇ ζη u ( ζ, η ) ∈ C n +1 . Observe that by virtue of 0 ≤ s ′ r ( ζ, η ) ≤ | λ | < sin α < − / we have | ξ | / ≤ (cid:12)(cid:12)(cid:12) ξ · diag (cid:8)(cid:0) λ s ′ r ( ζ, η ) (cid:1) − , Id (cid:9) ξ (cid:12)(cid:12)(cid:12) ≤ | ξ | , (cid:12)(cid:12)(cid:12) arg (cid:16) ξ · diag (cid:8)(cid:0) λ s ′ r ( ζ, η ) (cid:1) − , Id (cid:9) ξ (cid:17)(cid:12)(cid:12)(cid:12) < α. (4.4)Since r is sufficiently large, the constant c ( r ) in (3.7) is small. In particular c ( r )meets the estimate 4( n + 1) c ( r ) ≤ sin( σ/
2) with some σ ∈ (0 , π/ − α ). Then (3.7)together with (4.4) gives (cid:12)(cid:12)(cid:12) arg (cid:16) ξ · e − λ,r ( ζ, η ) ξ (cid:17)(cid:12)(cid:12)(cid:12) ≤ ϕ < π/ , δ | ξ | ≤ (cid:12)(cid:12)(cid:12) ξ · e − λ,r ( ζ, η ) ξ (cid:12)(cid:12)(cid:12) ≤ δ − | ξ | , (4.5)uniformly in λ ∈ D α and ( ζ, η ) ∈ G , where ϕ = 2 α + σ and δ = min n / − ( n + 1) c ( r ) , (cid:0)
12 + ( n + 1) c ( r ) (cid:1) − o . Taking into account (4.3) we complete the proof.
Remark 4.2.
Throughout the paper we say that r > is sufficiently large if thematrix e λ,r ( ζ, η ) is invertible and its inverse meets the estimates (4.5) uniformly in ( ζ, η ) ∈ G and λ ∈ D α . In the next lemma we show that in the right hand side of (4.2) the second termhas an arbitrarily small relative bound with respect to the first term uniformly in λ ∈ D α . Lemma 4.3.
For any ǫ > and u ∈ C ∞ ( G ) the estimate (cid:12)(cid:12)(cid:12)Z G (cid:10)(cid:0) det e λ,r (cid:1) / e − λ,r ∇ ζη u,u ∇ ζη (cid:0) det e λ,r (cid:1) − / (cid:11) dζ dη (cid:12)(cid:12)(cid:12) ≤ ǫ | ( −∇ ζη · e − λ,r ∇ ζη u, u ) | + Cǫ − k u ; L ( G ) k holds, where the constant C is independent of ǫ , u , and λ ∈ D α .Proof . We have (cid:12)(cid:12)(cid:12) Z G (cid:10)(cid:0) det e λ,r (cid:1) / e − λ,r ∇ ζη u, u ∇ ζη (cid:0) det e λ,r (cid:1) − / (cid:11) dζ dη (cid:12)(cid:12)(cid:12) ≤ C (cid:16)Z G | (cid:10) e − λ,r ∇ ζη u, ∇ ζη (cid:0) det e λ,r (cid:1) − / (cid:11) | dζ dη (cid:17) / (cid:16)Z G | u | dζ dη (cid:17) / ≤ C (cid:16) ˜ ǫ Z G | (cid:10) e − λ,r ∇ ζη u, ∇ ζη (cid:0) det e λ,r (cid:1) − / (cid:11) | dζ dη + ˜ ǫ − k u ; L ( G ) k (cid:17) (4.6) VICTOR KALVIN with arbitrarily small ˜ ǫ > C = sup ( ζ,η ) (cid:0) det e λ,r ( ζ, η ) (cid:1) / < ∞ , cf. (3.7).From (4.5) it follows that (cid:12)(cid:12) ℑ{ ξ · e − λ,r ( ζ, η ) ξ (cid:9)(cid:12)(cid:12) ≤ (tan ϕ ) ℜ{ ξ · e − λ,r ( ζ, η ) ξ } , where the form ℜ{ ξ · e − λ,r ( ζ, η ) ξ (cid:9) defines an inner product in C n +1 . This and theCauchy-Schwarz inequality give (cid:12)(cid:12) τ · e − λ,r ( ζ, η ) ξ (cid:12)(cid:12) ≤ (1 + tan ϕ ) ℜ{ ξ · e − λ,r ( ζ, η ) ξ (cid:9) ℜ{ τ · e − λ,r ( ζ, η ) τ (cid:9) , (4.7)cf. [23, Chapter VI.2]. We substitute ξ = ∇ ζη u ( ζ, η ) and τ = ∇ ζη (cid:0) det e λ,r ( ζ, η ) (cid:1) − / .Thanks to (3.7) we have the uniform bound |∇ ζη (cid:0) det e λ,r ( ζ, η ) (cid:1) − / | ≤ c, λ ∈ D α , ( ζ, η ) ∈ G . This together with (4.7) and (4.5) implies | (cid:10) e − λ,r ∇ ζη u, ∇ ζη (cid:0) det e λ,r (cid:1) − / (cid:11) | ≤ cδ − (1 + tan ϕ ) ℜ (cid:10) e − λ,r ∇ ζη u, ∇ ζη u (cid:11) . Now we make use of (4.6) and establish the assertion for C = C cδ − (1 + tan ϕ ) andan arbitrarily small ǫ = C cδ − (1 + tan ϕ ) ˜ ǫ .As a consequence of the equality (4.2) and Lemmas 4.1, 4.3 for all sufficientlylarge r > | arg (cid:0) q λ,r [ u, u ] + γ k u ; L ( G ) k (cid:1) | ≤ ϕ < π/ , (4.8) δq ,r [ u, u ] − γ k u ; L ( G ) k ≤ ℜ q λ,r [ u, u ] ≤ δ − ( q ,r [ u, u ] + k u ; L ( G ) k ) (4.9)with some angle ϕ and some positive constants δ and γ , which are independent of u ∈ C ∞ ( G ) and λ ∈ D α . The symmetric form q ,r [ u, u ] = R G h∇ ζη u, ∇ ζη u (cid:11) dζ dη is independent of r , it corresponds to the Dirichlet Laplacian ∆ ≡ ∆ ,r . Clearly, q λ,r [ u, u ] = (∆ λ,r u, u ). Estimate (4.8) implies that the numerical range { µ ∈ C : µ = (∆ λ,r u, u ) , u ∈ C ∞ ( G ) , k u ; L ( G ) k = 1 } is a subset of the sector { µ ∈ C : | arg( µ + γ ) | ≤ ϕ < π/ } . By definition this meansthat the operator ∆ λ,r with the domain C ∞ ( G ) is sectorial.We introduce the Sobolev space ◦ H ( G ) as the completion of the set C ∞ ( G ) withrespect to the norm k u ; ◦ H ( G ) k = q q ,r [ u, u ] + k u ; L ( G ) k . Recall that i ) a sequence { u j } is said to be q λ,r − convergent, if u j is in the domainof q λ,r , k u j − u ; L ( G ) k → q λ,r [ u j − u m , u j − u m ] → j, m → ∞ ; ii ) theform q λ,r is closed, if every q λ,r -convergent sequence { u j } has a limit u in the domainof q λ,r , and q λ,r [ u − u j , u − u j ] →
0. From (4.8), (4.9) it immediately follows that q λ,r with the domain ◦ H ( G ) is a closed densely defined sectorial form [23, 34], andits sector { µ ∈ C : | arg( µ + γ ) | ≤ ϕ } is independent of λ ∈ D α . As known [23,Chapter VI.2.1], to every closed densely defined sectorial form there corresponds aunique m-sectorial operator. Namely, to the form q λ,r there corresponds a unique IRICHLET LAPLACIANS IN QUASI-CYLINDRICAL DOMAINS λ,r in L ( M ) such that its sector is the sector of q λ,r , thedomain D(∆ λ,r ) is dense in ◦ H ( G ), and q λ,r [ u, v ] = (∆ λ,r u, v ) for all u ∈ D(∆ λ,r )and v ∈ ◦ H ( G ). (Here and elsewhere m-sectorial means that the numerical range { µ = ( Au, u ) H ∈ C : u ∈ D( A ) , ( u, u ) H = 1 } and the spectrum σ ( A ) of a closedunbounded operator A in a Hilbert space H with the inner product ( · , · ) H both lie insome sector { µ ∈ C : arg( µ − γ ) ≤ ϕ } with γ ∈ R and ϕ < π/ q ,r there corresponds the selfadjoint Dirichlet Laplacian∆ ≡ ∆ ,r . The m-sectorial operator ∆ λ,r in L ( M ) is the Friedrichs extension of thesectorial operator ∆ λ,r defined on C ∞ ( G ), see [23, Chapter VI.2.3]. As we show inassertion 1 of the next proposition the set C ∞ ( G ) is a core of the Friedrichs extension. Proposition 4.4.
Assume that r > is sufficiently large. Then the followingassertions are valid.1. For λ ∈ D α the m-sectorial operator ∆ λ,r in L ( G ) with the domain D(∆ λ,r ) is the closure of the operator ∆ λ,r defined on the set C ∞ ( G ) .2. The family of m-sectorial operators D α ∋ λ ∆ λ,r in L ( G ) is an analyticfamily of type (B) .3. The resolvent Γ ∋ ( λ, µ ) (∆ λ,r − µ ) − : L ( G ) → L ( G ) is an analyticfunction of two variables on the set Γ = (cid:8) ( λ, µ ) : λ ∈ D α , µ ∈ C \ σ (∆ λ,r ) (cid:9) ,where σ (∆ λ,r ) is the spectrum of ∆ λ,r .Proof . Consider the domain D(∆ λ,r ) as a Hilbert space endowed with the graphnorm k u ; D(∆ λ,r ) k = k u ; L ( G ) k + k ∆ λ,r u ; L ( G ) k . Let µ be a point outside of thesector of the m-sectorial operator ∆ λ,r . Then the setC(∆ λ,r ) = { u : u = (∆ λ,r − µ ) − f, f ∈ C ∞ ( G ) } is dense in D(∆ λ,r ) because the resolvent (∆ λ,r − µ ) − : L ( G ) → D(∆ λ,r ) is boundedand the set C ∞ ( G ) is dense in L ( G ). From (4.5) it follows the estimate ℜ ( ξ · e − λ,r ( ζ, η ) ξ ) ≥ c | ξ | , ξ ∈ R n +1 , λ ∈ D α , ( ζ, η ) ∈ G , (4.10)on the principal symbol of ∆ λ,r , where c >
0. Hence ∆ λ,r is a strongly ellipticoperator. As is well-known, a strongly elliptic operator and the Dirichlet boundarycondition set up an elliptic boundary value problem, e.g. [26]. The usual argumenton the regularity of solutions to the elliptic boundary value problems [26, 25] impliesthat the set C(∆ λ,r ) consists of smooth in G functions u with u ↾ ∂ G = 0. Multiplying u ∈ C(∆ λ,r ) by appropriate cutoff functions χ j with expanding compact supportssupp χ j ⊂ supp χ j +1 , it is easy to see that for any u ∈ C(∆ λ,r ) there is a sequence { χ j u } ∞ j =1 such that χ j u ∈ C ∞ ( G ) tends to u in D(∆ λ,r ) as j → + ∞ . Assertion 1 isproven.The family D α ∋ λ q λ,r is analytic in the sense of Kato (i.e. q λ,r is a closeddensely defined sectorial form, its domain ◦ H ( G ) is independent of λ , and the function D α ∋ λ q λ,r [ u, u ] is analytic for any u ∈ ◦ H ( G )). By definition [23, 34] this meansthat the family of m-sectorial operators D α ∋ λ ∆ λ,r is an analytic family of type(B). This proves assertion 2.As is well-known [23, 34], any analytic family of type (B) is also an analytic familyof operators in the sense of Kato. Now a standard argument justifies assertion 3; see,e.g., [34, Theorem XII.7].
5. Spaces of analytic functions.
We have shown that for all µ outside thesector of the family D α ∋ λ ∆ λ,r of m-sectorial operators the resolvent (∆ λ,r − µ ) − VICTOR KALVIN is an analytic function of λ , see Proposition 4.4.3. In order to get some relationsbetween the resolvents (∆ − µ ) − and (∆ λ,r − µ ) − we will use a sufficiently largeHilbert space H α ( G ) of analytic functions D α ∋ λ f ◦ ϑ λ,r ∈ L ( G ); (5.1)here ϑ λ,r is the complex scaling (2.5). The goal of this section is two-fold:1. To introduce a Hilbert space H α ( G ), which is sufficiently large in the sensethat for any λ ∈ D α and r > { f ◦ ϑ λ,r ∈ L ( G ) : f ∈ H α ( G ) } isdense in L ( G );2. To derive the uniform in λ ∈ D α and f ∈ H α ( G ) estimate k f ◦ ϑ λ,r ; L ( G ) k ≤ C r k f ; H α ( G ) k , r > . (5.2)Introduce the Hardy class H ( S α ) of all analytic functions S α ∋ z F ( z ) ∈ L (Ω)satisfying the uniform in ψ estimate Z ∞ k F ( e iψ x ); L (Ω) k dx ≤ C F , ψ ∈ ( − α, α ) . (5.3)Below we cite some facts from the theory of Hardy classes. Lemma 5.1.
1. Every F ∈ H ( S α ) has boundary limits F ± ∈ L ( R + × Ω) such that Z ∞ k F ( e iψ x ) − F ± ( x ); L (Ω) k dx → as ψ → ± α.
2. The Hardy class H ( S α ) endowed with the norm k F ; H ( S α ) k = (cid:0) k F − ; L ( R + × Ω) k + k F + ; L ( R + × Ω) k (cid:1) / is a Hilbert space.3. For every compact set K ⊂ S α there is an independent of F ∈ H ( S α ) constant C ( K ) such that for all z ∈ K we have k F ( z ); L (Ω) k ≤ C ( K ) k F ; H ( S α ) k .
4. For any F ∈ H ( S α ) , z ∈ S α , and ψ ∈ ( − α, α ) we have Z ∞ k F ( z + e iψ x ); L (Ω) k dx ≤ k F ; H ( S α ) k . Proof . Assertions 1–3 are direct consequences of well-known facts from the theoryof Hardy spaces of functions analytic in strips, e.g. [35]. In fact, the proof reduces tothe conformal mapping of the sector S α to the strip { z ∈ C : − α < ℑ z < α } , we omitthe details.Let us prove assertion 4. A standard argument, see e.g. [36], shows that any F ∈ H ( S α ) can be represented by the Cauchy integral F ( z ) = 12 πi + ∞ Z −∞ e iα F + ( x ) z − e iα x dx − πi + ∞ Z −∞ e − iα F − ( x ) z − e − iα x dx, z ∈ S α , (5.4) IRICHLET LAPLACIANS IN QUASI-CYLINDRICAL DOMAINS F ± ( x ) are extended to x < L (Ω). As is well-known [35],the first integral in (5.4) defines an element f + of the Hardy space H ( C − α ) of L (Ω)-valued functions analytic in the half-plane C − α = { z ∈ C : ℑ ( e − iα z ) < } . As shownin [37, 38], the norm in H ( C − α ) can be defined by the equality k f + ; H ( C − α ) k = sup ψ ∈ ( α − π,α ) Z ∞ k f + ( z + + e iψ x ); L (Ω) k dx ! / , e − iα z + ∈ R . Then k f + ; H ( C − α ) k = k F + ; L ( R + × Ω) k . Similarly, the second integral in (5.4) definesa function f − from the Hardy space H ( C + α ) in C + α = { z ∈ C : ℑ ( e iα z ) > } , and k f − ; H ( C + α ) k = k F − ; L ( R + × Ω) k , where k f − ; H ( C + α ) k = sup ψ ∈ ( − α,π − α ) Z ∞ k f − ( z − + e iψ x ); L (Ω) k dx ! / , e iα z − ∈ R . As a consequence, for all z ∈ S α and ψ ∈ ( − α, α ) we have Z ∞ k F ( z + e iψ x ); L (Ω) k dx ≤ k f + ; H ( C − α ) k + k f − ; H ( C + α ) k = k F + ; L ( R + × Ω) k + k F − ; L ( R + × Ω) k = k F ; H ( S α ) k . Consider the algebra E of all entire functions C ∋ z F ( z ) ∈ C ∞ (Ω) with thefollowing property: in any sector |ℑ z | ≤ (1 − ǫ ) ℜ z with ǫ > k F ( z ); L (Ω) k decays faster than any inverse power of ℜ z as ℜ z → + ∞ . Examples of functions F ∈ E are F ( z ) = e − γz P ( z ), where γ > P ( z ) is an arbitrary polynomial in z with coefficients in C ∞ (Ω). Clearly, E ⊂ H ( S α ). The next lemma is an adaptationof [18, Theorem 3], we omit the proof. Lemma 5.2.
The set of functions { R + × Ω ∋ ( x, y ) F ◦ κ λ,r ( x, y ) : F ∈ E } isdense in the space L ( R + × Ω) for any λ ∈ D α and r > . Here F ◦ κ λ,r ( x, · ) standsfor the value of the entire function z F ( z ) at the point z = x + λs ( x − r ) . Now we are in position to prove the following proposition.
Proposition 5.3.
1. The estimate Z ∞ k F ◦ κ λ,r ; L (Ω) k dx ≤ C r k F ; H ( S α ) k , r > , (5.5) holds uniformly in λ ∈ D α and F ∈ H ( S α ) .2. The space H ( S α ) is sufficiently large in the sense that for any λ ∈ D α and r > the set { R + × Ω ∋ ( x, y ) F ◦ κ λ,r ( x, y ) : F ∈ H ( S α ) } is dense in L ( R + × Ω) .3. For any F ∈ H ( S α ) and r > the function D α ∈ λ F ◦ κ λ,r ∈ L ( R + × Ω) is analytic. The proof is preceded by a discussion. By Proposition 5.3.1 for any r > κ λ,r induces the uniformly bounded injection H ( S α ) ∋ F F ◦ κ λ,r ∈ L ( R + × Ω) , λ ∈ D α . (5.6)2 VICTOR KALVIN
Any function F ∈ H ( S α ) can be reconstructed from its trace x F ◦ κ λ,r ( x, · ) ∈ L (Ω)by analytic continuation from the curve { x + λs ( x − r ) ∈ C : x > } to the sector S α .Therefore we can always identify the space H ( S α ) with the range of injection (5.6).By Proposition 5.3.2 the range is dense in L ( R + × Ω).
Proof . 1. For all λ ∈ D α the curve { x + λs ( x − r ) ∈ C : x > } lies in the sector S α . Observe that this curve is differ from a ray only inside an independent of λ ∈ D α compact subset K r ⊂ S α . Now the uniform in λ ∈ D α and F ∈ H ( S α ) estimate (5.5)follows from assertions 3 and 4 of Lemma 5.1.2. The assertion is an immediate consequence of the embedding E ⊂ H ( S α ),Lemma 5.2, and the estimate (5.5).3. It is easy to see that the function D α ∋ λ F ◦ κ λ,r ∈ L ( R + × Ω) is weakly(and therefore strongly) analytic.Introduce the Hilbert space H α ( G ) with the norm k f ; H α ( G ) k = k f ; L ( G ) k + k f ◦ κ − ; H ( S α ) k . The space H α ( G ) consists of all functions f ∈ L ( G ) such that f ◦ κ − is an elementof the Hardy space H ( S α ); here κ is the diffeomorphism (2.1). As a consequence ofProposition 5.3 and definition (2.5) of the complex scaling ϑ λ,r we immediately getthe following assertions. Corollary 5.4.
1. For any r > the estimate (5.2) holds with an independent of f ∈ H α ( G ) and λ ∈ D α constant C r .2. For any λ ∈ D α and r > the set { f ◦ ϑ λ,r ∈ L ( G ) : f ∈ H α ( G ) } is densein the space L ( G ) .3. For any f ∈ H α ( G ) and r > the function (5.1) is analytic.
6. Limiting absorption principle.
Introduce the Sobolev space H ( G ) of func-tions satisfying the homogeneous Dirichlet boundary condition on ∂ G as the comple-tion of the core C ∞ ( G ) with respect to the graph norm k u ; H ( G ) k = k u ; L ( G ) k + k ∆ u ; L ( G ) k . (6.1)(Integrating by parts and using the Cauchy-Schwarz inequality one can easily see thatthe norm (6.1) is equivalent to the traditional norm ( P ℓ + | m |≤ k ∂ ℓζ ∂ mη u ; L ( G ) k ) / .)By Proposition 4.4.1 the space H ( G ) is the domain D(∆) of the selfadjoint DirichletLaplacian ∆. For the points µ ∈ σ (∆) the resolvent (∆ − µ − iǫ ) − does not havelimits in the space of bounded operators B ( L ( G ) , H ( G )) as ǫ tends to zero frombelow ( ǫ ↑
0) or from above ( ǫ ↓ H α ( G ) constructed in the previous section.As a target space we take the reflexive Fr´echet space H , loc ( G ). The space H , loc ( G )consists of all distributions u such that ̺u ∈ H ( G ) with any ̺ ∈ C ∞ c ( G ), the topologyin H , loc ( G ) is induced by the family of seminorms u
7→ k ̺u ; H ( G ) k ; here C ∞ c ( G ) isthe set of all smooth functions with compact supports in G . Theorem 6.1.
Let σ (∆ Ω ) stand for the spectrum of the selfadjoint DirichletLaplacian ∆ Ω in L (Ω) . Assume that µ ∈ R \ σ (∆ Ω ) is not an eigenvalue of theselfadjoint Dirichlet Laplacian ∆ in L ( G ) . Then the following assertions hold.1. For all sufficiently large r > and λ ∈ D α \ R the resolvent (∆ λ,r − µ ) − : L ( G ) → H ( G ) IRICHLET LAPLACIANS IN QUASI-CYLINDRICAL DOMAINS is a bounded operator.2. The resolvent (∆ − µ − iǫ ) − , ǫ ≷ , viewed as a bounded operator actingfrom H α ( G ) to H , loc ( G ) , has limits as ǫ ↓ and ǫ ↑ .3. Suppose that r > is sufficiently large, λ ∈ D α \ R , and f ∈ H α ( G ) . Let u λ,r ∈ H ( G ) be given by the equality u λ,r = (∆ λ,r − µ ) − ( f ◦ ϑ λ,r ) . Then theoutgoing u − ∈ H , loc ( G ) and the incoming u + ∈ H , loc ( G ) solutions definedby the limiting absorption principle u + = lim ǫ ↑ (∆ − µ − iǫ ) − f, u − = lim ǫ ↓ (∆ − µ − iǫ ) − f, (6.2) meet the relation u λ,r ↾ G r = (cid:26) u + ↾ G r , ℑ λ < u − ↾ G r , ℑ λ > . Here the bounded domain G r is the same as in (2.9) . The proof is preceded by a discussion. From Theorem 6.1.3 we see that theequation (∆ λ,r − µ ) u λ,r = f ◦ ϑ λ,r with non-real parameter λ ∈ D α describes infinitePMLs on G \ G r for all µ satisfying the assumptions of the theorem. The layers areperfectly matched in the sense that for ℑ λ > ℑ λ < u λ,r coincides in G r with the outgoing solution u − (resp. with the incoming solution u + ). The PMLs areabsorbing because in contrast to u ± the function u λ,r decays at infinity in the meanas an element of H ( G ). In the next section we will refine results of Theorem 6.1by showing that under an additional assumption on f ◦ ϑ λ,r the solution u λ,r is ofsome exponential decay at infinity. For instance, this assumption is a priori met for f ∈ L ( G ) supported in G r . Then f ◦ ϑ λ,r ≡ f and the operator ∆ λ,r completelydescribes infinite PMLs on G \ G r . Proof . The proof consists of two steps.
Step 1.
In Section 8 below we will show that the graph norm of ∆ λ,r is anequivalent norm in H ( G ). This immediately implies that D(∆ λ,r ) = H ( G ) as the set C ∞ ( G ) is dense in both spaces, cf. Proposition 4.4.1. We will also localize the essentialspectrum σ ess (∆ λ,r ) of the unbounded m-sectorial operator ∆ λ,r in L ( G ). As is well-known, the spectrum σ (∆ Ω ) consists of infinitely many positive isolated eigenvalues.It turns out that σ ess (∆ λ,r ) consists of an infinite number of rays emanating fromevery point ν ∈ σ (∆ Ω ), cf. Figure 6.1. By definition σ (∆ Ω ) is the set of thresholds of ℜ µ / / O O ℑ µ • • • − λ ) • thresholds ν ∈ σ (∆ Ω ); — rays of σ ess (∆ λ,r ). Fig. 6.1 . Essential spectrum of the m-sectorial operator ∆ λ,r for ℑ λ > . the Dirichlet Laplacian ∆. As λ varies, the ray { µ ∈ C : arg( µ − ν ) = − λ ) } ofthe essential spectrum σ ess (∆ λ,r ) rotates about the threshold ν ∈ σ (∆ Ω ) and sweeps4 VICTOR KALVIN the sector { µ ∈ C : | arg( µ − ν ) | < α } . In order to avoid several repetitions weorganized the paper so that the proofs of these results in some greater generality arepostponed to Section 8; here we take these results for granted.Recall that µ is said to be a point of the essential spectrum σ ess ( A ) of a closedunbounded operator A in the space L ( G ) with the domain H ( G ), if the boundedoperator A − µ : H ( G ) → L ( G ) is not Fredholm (a linear bounded operator betweentwo Banach spaces is Fredholm, if its kernel and cokernel are finite dimensional, andits range is closed). Since the operator ∆ λ,r is m-sectorial, there exists a regular pointof ∆ λ,r in the simply connected set C \ σ ess (∆ λ,r ). Therefore C \ σ ess (∆ λ,r ) ∋ µ ∆ λ,r − µ : H ( G ) → L ( G )is a Fredholm holomorphic operator function, e.g. [24, Appendix]. Recall that thespectrum of a Fredholm holomorphic operator function consists of isolated eigenvaluesof finite algebraic multiplicity, e.g. [24, Proposition A.8.4]. As a consequence, theresolvent C \ σ ess (∆ λ,r ) ∋ µ (∆ λ,r − µ ) − : L ( G ) → H ( G ) (6.3)is a meromorphic operator function.For f, g ∈ L ( G ) and a sufficiently large r > f, g ) λ,r = Z G f g p det e λ,r dζ dη, λ ∈ D α . This form is bounded in L ( G ). Indeed, thanks to the estimate (3.7) on e − λ,r we have0 < c ≤ | det e λ,r ( ζ, η ) | ≤ c uniformly in λ ∈ D α and ( ζ, η ) ∈ G .Assume that λ ∈ D α is real. Then the form ( · , · ) λ,r is the inner product inducedon G by the metric e λ,r , the norm p ( f, f ) λ,r is equivalent to the norm k f ; L ( G ) k ,and ∆ λ,r is the Laplace-Beltrami operator on ( G , e λ,r ). We have(∆ − µ ) u = (cid:0) (∆ λ,r − µ )( u ◦ ϑ λ,r ) (cid:1) ◦ ϑ − λ,r ∀ u ∈ C ∞ ( G ) . (6.4)Assume that µ is not in the sector of ∆ λ,r . Then (∆ λ,r − µ ) − is a bounded operatorand we can rewrite (6.4) in the form(∆ − µ ) − f = (cid:0) (∆ λ,r − µ ) − ( f ◦ ϑ λ,r ) (cid:1) ◦ ϑ − λ,r , (6.5)where f is in the set { f = (∆ − µ ) u : u ∈ C ∞ ( G ) } . This set is dense in L ( G ),because C ∞ ( G ) is dense in H ( G ), and the operator ∆ − µ : H ( G ) → L ( G ) yieldsan isomorphism. It is clear that ( f ◦ ϑ λ,r , f ◦ ϑ λ,r ) λ,r = ( f, f ). As a consequence, the(real) scaling f f ◦ ϑ λ,r realizes an isomorphism in L ( G ), and the equality (6.5)extends by continuity to all f ∈ L ( G ). Taking the inner product of the equality (6.5)with g ∈ L ( G ), and passing to the variables (˜ ζ, ˜ η ) = ϑ λ ( ζ, η ) in the right hand side,we obtain (cid:0) (∆ − µ ) − f, g (cid:1) = (cid:0) (∆ λ,r − µ ) − ( f ◦ ϑ λ,r ) , g ◦ ϑ λ,r (cid:1) λ,r . (6.6)Now we assume that f, g ∈ H α ( G ). Then f ◦ ϑ λ,r and g ◦ ϑ λ,r are L ( G )-valued analyticfunctions of λ in the disk D α , see Corollary 5.4.3. This together with Proposition 4.4.4implies that the right hand side of (6.6) extends by analyticity from λ ∈ D α ∩ R to all IRICHLET LAPLACIANS IN QUASI-CYLINDRICAL DOMAINS λ ∈ D α . The right hand side of (6.6) extends from all µ outside of the sector of ∆ λ,r to a meromorphic function of µ ∈ C \ σ ess (∆ λ,r ). In particular, for all λ ∈ D α \ R wehave µ ∈ C \ σ ess (∆ λ,r ), cf. Figure 6.1. Here µ is the same as in the formulation ofthe theorem.Now we are in position to prove assertion 1. Consider the projection P = s-lim ǫ ↓ iǫ (∆ − µ − iǫ ) − onto the eigenspace of the selfadjoint operator ∆. Suppose, by contradiction, that theresolvent (6.3) has a pole at the point µ ∈ C \ σ ess (∆ λ,r ). By Corollary 5.4.2 thereexist f and g in the space H α ( G ) such that µ is a pole of the right hand side of (6.6).The equality (6.6) implies that ( P f, g ) = 0, and thus ker(∆ − µ ) = { } . This is acontradiction. Assertion 1 is proven. Step 2.
We need to show that for any ̺ ∈ C ∞ c ( G ) the operator ̺ (∆ − µ − iǫ ) − tends to some limits in the space of bounded operators B ( H α ( G ) , H ( G )) as ǫ ↓ ǫ ↑
0. We take a sufficiently large r = r ( ̺ ) > ̺ ⊂ G r . Then ̺ ◦ ϑ λ,r = ̺ for all λ ∈ D α . Now we can pass from (6.5) to the equality ̺ (∆ − µ ) − f = ̺ (∆ λ,r − µ ) − ( f ◦ ϑ λ,r ) . (6.7)For f ∈ H α ( G ) the equality (6.7) extends by analyticity to all λ ∈ D α . Consider, forinstance, the case ℑ λ > ℑ λ < C + = { µ ∈ C : ℑ µ > } and a complex neighborhood of the point µ do not containpoints of σ ess (∆ λ,r ), cf. Figure 6.1. Therefore the right hand side of (6.7) has ameromorphic continuation in µ to the union of C + and a complex neighborhood of µ . Hence the left hand side of (6.7) has the same meromorphic continuation. Clearly,a pole at µ may only appear due to a pole of the resolvent (6.3) at µ , but it is aregular point by assertion 1. Since ̺ is an arbitrary smooth function supported in G r this proves assertion 3. In order to prove assertion 2 it remains to note that k lim ǫ ↓ ̺ (∆ − µ − iǫ ) − f ; H ( G ) k = k ̺ (∆ λ,r − µ ) − ( f ◦ ϑ λ,r ); H ( G ) k≤ C ( ̺ ) k (∆ λ,r − µ ) − ; L ( G ) → H ( G ) kk f ◦ ϑ λ,r ; L ( G ) k≤ C ( ̺, r, µ ) k f ; H α ( G ) k . In the last inequality we used Corollary 5.4.1 and assertion 1.In the following two remarks we collect some results that can be obtained bymethods developed in the proof of Theorem 6.1. Although these results are not usedin this paper, they provide additional insights of the problem.
Remark 6.2.
On the basis of the equality (6.6) , Corollary 5.4, and the descriptionof σ ess (∆ λ,r ) for λ ∈ D α , one can develop an analog of the celebrated Aguilar-Balslev-Combes-Simon theory of resonances [10, 16, 18, 34]. We announce some results below,for the proof we refer to [21].1. The selfadjoint Dirichlet Laplacian ∆ in G has no singular continuous spec-trum, its eigenvalues can accumulate only at thresholds ν ∈ σ (∆ Ω ) . (Exam-ples of accumulating eigenvalues can be found e.g. in [13].)2. The spectrum σ (∆ λ,r ) of the m-sectorial operator ∆ λ,r does not depend on thechoice of the scaling function satisfying (2.2) – (2.4) . Moreover, the spectrum σ (∆ λ,r ) lies in the half-plain C + in the case ℑ λ ≥ and σ (∆ λ,r ) ⊂ C − inthe case ℑ λ ≤ , where C ± = { µ ∈ C : ℑ µ ≷ } and λ ∈ D α . VICTOR KALVIN
3. A point µ ∈ R \ σ (∆ Ω ) is an eigenvalue of ∆ if and only if it is an isolatedeigenvalue of ∆ λ,r , where λ ∈ D α \ R and r > is sufficiently large.4. The resolvent matrix elements ((∆ − µ ) − f, g ) , where f, g ∈ H α ( G ) , havemeromorphic continuations from the physical sheet C \ σ ess (∆) across σ ess (∆) to the set C \ σ ess (∆ λ,r ) . Moreover, µ is a pole of the continuation for some f, g ∈ H α ( G ) if and only if it is an isolated eigenvalue of ∆ λ,r . The non-realisolated eigenvalues of ∆ λ,r are naturally identified with resonances of ∆ .5. As λ changes continuously in the disk D α , an isolated eigenvalue of ∆ λ,r survive while it is not covered by one of the rotating rays of σ ess (∆ λ,r ) . Remark 6.3.
The argument of the second step in the proof of Theorem 6.1 allowsalso to see that the resolvent (∆ − µ ) − : H α ( G ) → H , loc ( G ) has a meromorphiccontinuation from the physical sheet C \ σ ess (∆) across the intervals ( ν − , ν + ) betweenthe neighboring thresholds ν ± ∈ σ (∆ Ω ) to a Riemann surface. The surface consistsof the physical sheet C \ σ ess (∆) of the Dirichlet Laplacian and an infinite numberof the sectors { µ ∈ C : 0 > arg( µ − ν − ) > − α } attached to C + ⊂ C \ σ ess (∆) andof the sectors { µ ∈ C : 0 < arg( µ − ν − ) < α } attached to C − ⊂ C \ σ ess (∆) alongthe intervals ( ν − , ν + ) . Indeed, for any ̺ ∈ C ∞ c ( G ) we can take a sufficiently large r = r ( ̺ ) > such that supp ̺ ⊂ G r . As λ varies in D α ∩ C + (resp. in D α ∩ C − )the strip between the neighboring rays { µ ∈ C : arg( z − ν ± ) = − λ ) } of σ ess (∆ λ,r ) sweeps the sector { µ ∈ C : 0 > arg( µ − ν − ) > − α } (resp. the sector { µ ∈ C : 0 < arg( µ − ν − ) < α } ) and the right hand side of (6.7) provides the lefthand side with a meromorphic continuation to the strip. The poles of the continuationare resonances of the Dirichlet Laplacian ∆ [39]. The results listed in Remarks 6.2 and 6.3 are new and might be of their owninterest. Traditionally, when studying Laplacians in the waveguide-type of geometry,one imposes more or less restrictive assumptions on the rate of convergence of themetric g on (0 , ∞ ) × Ω to its limit at infinity; see, e.g., [9, 11, 12, 13, 14, 15, 19, 29, 30,31]. Contrastingly, our assumptions on the diffeomorphism κ allow for arbitrarily slowconvergence of the metric g = κ ∗ e to the Euclidean metric e at infinity, see Section 3.As a substitution for the assumptions on the rate of convergence of the metric g atinfinity we use assumptions on the analytic regularity of the diffeomorphism κ .Let us also mention here the paper [33], where general elliptic problems whosecoefficients slowly converge to their limits at infinity are considered. It is shown thatany finite accumulation point of eigenvalues corresponding to exponentially decayingeigenfunctions is a threshold, these accumulations are characterized in terms of somenon-classical “augmented scattering matrices.” An additional investigation on decayof eigenfunctions at infinity is required in order to say whether these results describeaccumulations of eigenvalues of the Dirichlet Laplacian ∆ or not. This goes beyondthe scope of the present paper, we refer to [22].
7. Exponential decay of solutions in infinite PMLs.
In this section weprove the following theorem.
Theorem 7.1.
Assume that µ ∈ R \ σ (∆ Ω ) is not an eigenvalue of the selfadjointDirichlet Laplacian ∆ in L ( G ) , the scaling parameter λ ∈ D α is not real, and ≤ β < min ν ∈ σ (∆ Ω ) |ℑ{ (1 + λ ) √ µ − ν }| . (7.1) Let s ( ζ, η ) stand for the scaling function R + × Ω ∋ ( x, y ) s ( x ) written in the coor-dinates ( ζ, η ) ∈ C and extended to G by zero; see (2.2) – (2.4) . Then for all sufficientlylarge r > and all F ∈ L ( G ) satisfying e β s F ∈ L ( G ) the estimate k e β s (∆ λ,r − µ ) − F ; H ( G ) k ≤ C k e β s F ; L ( G ) k (7.2) IRICHLET LAPLACIANS IN QUASI-CYLINDRICAL DOMAINS is valid with a constant C independent of F . Theorem 7.1 together with Theorem 6.1 shows that under the additional assump-tion e β s ( f ◦ ϑ λ,r ) ∈ L ( G ) on f ∈ H α ( G ) infinite PMLs absorb outgoing or incomingsolutions (depending on the sign of ℑ λ ) so effectively that the function u λ,r in Theo-rem 7.1.3 exponentially decays at infinity in the mean. Proof . Consider the conjugated operator e β s ∆ λ,r e − β s as an unbounded operatorin L ( G ) with the domain C ∞ ( G ). With this operator we associate the quadratic form q βλ,r [ u, u ] = ( e β s ∆ λ,r e − β s u, u ) λ,r . Observe that q βλ,r [ u, u ] − q λ,r [ u, u ] = − β Z G (cid:10) e − λ,r u ∇ ζη s , u ∇ ζη s (cid:11) dζ dη − β Z G (cid:10)(cid:0) det e λ,r (cid:1) / e − λ,r u ∇ ζη s , ∇ ζη (cid:0) det e λ,r (cid:1) − / u (cid:11) dζ dη + β Z G (cid:10) e − λ,r ∇ ζη u, u ∇ ζη s (cid:11) dζ dη, where q λ,r is the same as in (4.1). Since the right hand side depends linearly on ∇ ζη u ,similarly to the the proof of Lemma 4.3 one can deduce (cid:12)(cid:12) q βλ,r [ u, u ] − q λ,r [ u, u ] (cid:12)(cid:12) ≤ ǫ | ( −∇ ζη · e − λ,r ∇ ζη u, u ) | + Cǫ − k u ; L ( G ) k with an arbitrary small ǫ > C independent of u ∈ C ∞ ( G ). Thisestimate together with (4.8), (4.9), and Lemma 4.1 implies that for all sufficientlylarge r > | arg (cid:0) q βλ,r [ u, u ] + γ k u ; L ( G ) k (cid:1) | ≤ ϕ (7.3)with some angle ϕ < π/ γ >
0, which are independent of u ∈ C ∞ ( G ). Therefore e β s ∆ λ,r e − β s with the domain C ∞ ( G ) is a densely defined sectorial operator in L ( G ).Let D( e β s ∆ λ,r e − β s ) be the domain of its m-sectorial Friedrichs extension [23, ChapterVI.2]. The Friedrichs extension will also be denoted by e β s ∆ λ,r e − β s .As in the proof of Proposition 4.4.1 we conclude that C ∞ ( G ) is a core of them-sectorial operator e β s ∆ λ,r e − β s . In Section 8 we will show that the graph norm of e β s ∆ λ,r e − β s is equivalent to the norm in H ( G ); hence D( e β s ∆ λ,r e − β s ) = H ( G ). Fur-thermore, we will localize the essential spectrum σ ess ( e β s ∆ λ,r e − β s ) of the m-sectorialoperator e β s ∆ λ,r e − β s , see Proposition 8.1. It turns out that the essential spectrumconsists of an infinite number of parabolas, see Figure 7.1. In the case β = 0 theparabolas collapse to the dashed rays originating from the thresholds ν ∈ σ (∆ Ω ) andwe obtain the essential spectrum σ ess (∆ λ,r ).The m-sectorial operator e β s ∆ λ,r e − β s defines the Fredholm holomorphic operatorfunction µ e β s ∆ λ,r e − β s − µ : H ( G ) → L ( G )on the simply connected subset of C \ σ ess ( e β s ∆ λ,r e − β s ) containing an infinite partof the real negative semiaxis (regular points of e β s ∆ λ,r e − β s ). Condition (7.1) on β guarantees that the point µ is in this simply connected subset. As the spectrumof a Fredholm holomorphic operator function consists of isolated eigenvalues of finitemultiplicity, µ is a regular point or an eigenvalue of e β s ∆ λ,r e − β s . The inclusionΨ ∈ ker( e β s ∆ λ,r e − β s − µ ) implies e − β s Ψ ∈ ker(∆ λ,r − µ ), and hence Ψ ≡ VICTOR KALVIN ℜ µ / / O O ℑ µ • • • − λ ) • thresholds ν ∈ σ (∆ Ω ) of ∆; — parabolas of σ ess ( e β s ∆ λ,r e − β s ). Fig. 7.1 . Essential spectrum of the conjugated operator e β s ∆ λ,r e − β s for ℑ λ > and β ≷ . Theorem 6.1.1. Thus the operator e β s ∆ λ,r e − β s − µ yields an isomorphism betweenthe spaces H ( G ) and L ( G ). This together with the equality( e β s ∆ λ,r e − β s − µ ) − e β s F = e β s (∆ λ,r − µ ) − F , e β s F ∈ L ( G ) , justifies the estimate (7.2).
8. Localization of the essential spectrum.
In this section we localize theessential spectrum σ ess ( e β s ∆ λ,r e − β s ) of the m-sectorial operator e β s ∆ λ,r e − β s in L ( G )with parameters λ ∈ D α and β ∈ R ; here s is the same as in Theorem 7.1. Inparticular, in the case β = 0 we find σ ess (∆ λ,r ). We also prove that D( e β s ∆ λ,r e − β s ) = H ( G ). In other words, we show that the information on the essential spectrum andthe domain of e β s ∆ λ,r e − β s we used in the proofs of Theorem 6.1 and Theorem 7.1 iscorrect.Let us note that for fixed λ and β the spectrum σ ess ( e β s ∆ λ,r e − β s ) depends onlyon the behavior of the scaling function s and the matrix e λ,r outside any compactregion of G . In order to control σ ess ( e β s ∆ λ,r e − β s ) we imposed the condition (2.4). Proposition 8.1.
Assume that λ ∈ D α , β ∈ R , and r > is sufficiently large.Then the following assertions hold.1. The Hilbert spaces D( e β s ∆ λ,r e − β s ) and H ( G ) are coincident and their normsare equivalent.2. The bounded operator e β s ∆ λ,r e − β s − µ : H ( G ) → L ( G ) (8.1) is not Fredholm (or, equivalently, µ ∈ σ ess ( e β s ∆ λ,r e − β s ) ) if and only if theparameters µ , λ , and β meet the condition ν − µ = (1 + λ ) − ( β + iξ ) for some ν ∈ σ (∆ Ω ) and ξ ∈ R . (8.2) Proof . The proof is essentially based on methods of the theory of elliptic non-homogeneous boundary value problems [24, 25, 28, 26]. We will rely on the followinglemma due to Peetre, see e.g. [26, Lemma 5.1], [25, Lemma 3.4.1] or [32]:
Let X , Y and Z be Banach spaces, where X is compactly embedded into Z .Furthemore, let L be a linear bounded operator from X to Y . Then the nexttwo assertions are equivalent: (i) the range of L is closed in Y and dim ker L < ∞ , (ii) there exists a constant C , such that k u ; X k ≤ C ( kL u ; Yk + k u ; Zk ) ∀ u ∈ X . (8.3) IRICHLET LAPLACIANS IN QUASI-CYLINDRICAL DOMAINS µ , λ , and β does not meet the condition (8.2) and establishthe coercive estimate k u ; H ( G ) k ≤ C ( k ( e β s ∆ λ,r e − β s − µ ) u ; L ( G ) k + k w u ; L ( G ) k ) ∀ u ∈ H ( G ) . (8.4)Here w ∈ C ∞ ( G ) is a positive rapidly decreasing at infinity weight, such that theembedding of H ( G ) into the weighted space L ( G ; w ) with the norm k w · ; L ( G ) k iscompact. Note that (8.4) is an estimate of type (8.3) for the operator (8.1).The strongly elliptic differential operator e β s ∆ λ,r e − β s endowed with the Dirichletboundary condition set up a regular elliptic boundary value problem. Solutions ofa regular elliptic boundary value problem satisfy local coercive estimates, e.g. [26]or [25]. Thus we have the local coercive estimate k ρ T u ; H ( G ) k ≤ C ( k ̺ T ( e β s ∆ λ,r e − β s − µ ) u ; L ( G ) k + k ̺ T u ; L ( G ) k ) . (8.5)Here ρ T and ̺ T are smooth compactly supported cutoff functions in G such that ρ T ( ζ, η ) = 1 for | ζ | < T + 1 and ̺ T ρ T = ρ T , where T is a large fixed number.Let χ T ∈ C ∞ ( G ) be another cutoff function such that χ T ( ζ, η ) = 1 for | ζ | > T and χ T ( ζ, η ) = 0 for | ζ | < T −
1. On the next step we establish the estimate (8.4)with u replaced by χ T u . We will do it in the coordinates ( x, y ) ∈ R + × Ω.Let L ( R × Ω) be the space of functions in the infinite cylinder R × Ω with thenorm (cid:0)R R k u ( x ); L (Ω) k dx (cid:1) / . Introduce the Sobolev space H ( R × Ω) of functionswith zero Dirichlet data on R × ∂ Ω as the completion of the set C ∞ ( R × Ω) withrespect to the norm k u ; H ( R × Ω) k = (cid:16) X ℓ + | m |≤ k ∂ ℓx ∂ my u ; L ( R × Ω) k (cid:17) / . Denote u = ( χ T u ) ◦ κ , where κ is the diffeomorphism (2.1). Let △ λ,r = − (cid:0) det g λ,r (cid:1) − / ∇ xy · (cid:0) det g λ,r (cid:1) / g − λ,r ∇ xy , λ ∈ D α , (8.6)be the operator ∆ λ,r written in the coordinates ( x, y ). Here g λ,r is the matrix (3.3)and ∇ xy = ( ∂ x , ∂ y . . . ∂ y n ) ⊤ . Due to our assumptions on κ the estimates 0 < ǫ ≤ det κ ′ ( x, y ) ≤ /ǫ hold uniformly in ( x, y ) ∈ R + × Ω. Hence for some independent of u ∈ C ∞ ( G ) constants c , c , and c we have k χ T u ; H ( G ) k = k ∆( χ T u ); L ( G ) k + k χ T u ; L ( G ) k≤ c ( k△ ,r u ; L ( R × Ω) k + k u ; L ( R × Ω) k ) ≤ c k u ; H ( R × Ω) k , k ( e βs △ λ,r e − βs − µ ) u ; L ( R × Ω) k ≤ c k ( e β s ∆ λ,r e − β s − µ ) χ T u ; L ( G ) k . (8.7)Here the functions u , s , and △ λ,r u ≡ (∆ λ,r ( χ T u )) ◦ κ are extended from R + × Ωto the infinite cylinder R × Ω by zero, and k△ ,r u ; L ( R × Ω) k ≤ C k u ; H ( R × Ω) k because the coefficients of the Laplacian △ ,r are bounded, cf. (8.6) and (3.5). As T is large, the function u is supported in a small neighborhood of infinity. Due to thestabilization condition (3.4) on g − λ and the condition (2.4) on the scaling function s the coefficients of the differential operator e β s △ λ,r e − β s − ∆ Ω + (1 + λ ) − ( ∂ x + β ) VICTOR KALVIN are small on the support of u . As a result we get the estimate (cid:13)(cid:13)(cid:0) e β s △ λ,r e − β s − ∆ Ω + (1 + λ ) − ( ∂ x + β ) (cid:1) u ; L ( R × Ω) (cid:13)(cid:13) ≤ ǫ k u ; H ( R × Ω) k , (8.8)where ǫ is small and independent of u ∈ C ∞ ( G ); moreover, ǫ → T → + ∞ .Consider the bounded operator∆ Ω − (1 + λ ) − ( ∂ x + β ) − µ : H ( R × Ω) → L ( R × Ω) . (8.9)Applying the Fourier transform F x ξ we pass from the operator (8.9) to the selfad-joint Dirichlet Laplacian ∆ Ω + (1 + λ ) − ( β + iξ ) − µ in L (Ω). Since µ , λ , and β donot meet the condition (8.2), the spectral parameter µ − (1 + λ ) − ( β + iξ ) is outsideof the spectrum of ∆ Ω for all ξ ∈ R . Then a known argument [25, Theorem 5.2.2], [24,Theorem 2.4.1], which is also used as a part of the proof of Lemma 9.2 below, impliesthat the operator (8.9) realizes an isomorphism. In particular, the estimate k u ; H ( R × Ω) k ≤ c (cid:13)(cid:13)(cid:0) ∆ Ω − (1 + λ ) − ( ∂ x + β ) − µ (cid:1) u ; L ( R × Ω) (cid:13)(cid:13) is valid with an independent of u ∈ H ( R × Ω) constant c. As a consequence of thisestimate and (8.8) we obtain(1 − ǫ c) k u ; H ( R × Ω) k ≤ c (cid:13)(cid:13)(cid:0) ∆ Ω − (1 + λ ) − ( ∂ x + β ) − µ (cid:1) u ; L ( R × Ω) (cid:13)(cid:13) − c (cid:13)(cid:13)(cid:0) e βs △ λ,r e − βs − ∆ Ω + (1 + λ ) − ( ∂ x + β ) (cid:1) u ; L ( R × Ω) (cid:13)(cid:13) ≤ c k ( e βs △ λ,r e − βs − µ ) u ; L ( R × Ω) k . If T is sufficiently large, then ǫ c <
1. This together with (8.7) gives k χ T u ; H ( G ) k ≤ C k ( e β s ∆ λ,r e − β s − µ ) χ T u ; L ( G ) k , (8.10)where the constant C = c(1 − ǫ c) − c c is independent of u ∈ C ∞ ( G ). By continuitythe estimate (8.10) extends to all u ∈ H ( G ).Now we combine (8.10) with (8.5), and arrive at the estimates k u ; H ( G ) k ≤ k χ T u ; H ( G ) k + k ρ T u ; H ( G ) k≤ C ( k χ T ( e β s ∆ λ,r e − β s − µ ) u ; L ( G ) k + k [ e β s ∆ λ,r e − β s , χ T ] u ; L ( G ) k + k ̺ T ( e β s ∆ λ,r e − β s − µ ) u ; L ( G ) k + k ̺ T u ; L ( G ) k ) ≤ C ( k ( e β s ∆ λ,r e − β s − µ ) u ; L ( G ) k + k ̺ T u ; L ( G ) k ) . (8.11)Here we used that ρ T = 1 on the support of the commutator [ e β s ∆ λ,r e − β s , χ T ], andhence k [ e β s ∆ λ,r e − β s , χ T ] u ; L ( G ) k ≤ C k ρ T u ; H ( G ) k . For an arbitrary positive weight w we have k ̺ T u ; L ( G ) k ≤ C k w u ; L ( G ) k with anindependent of u ∈ H ( G ) constant C . Thus the estimate (8.3) is a direct consequenceof (8.11). By the Peetre’s lemma we conclude that the range of the operator (8.1) isclosed and the kernel is finite-dimensional.Clearly, the graph norm k u ; L ( G ) k + k e β s ∆ λ,r e − β s u k of u ∈ C ∞ ( G ) is majorizedby k u ; H ( G ) k . The estimate (8.3) with w ≡ k u ; H ( G ) k ismajorized by the graph norm of u . Since the set C ∞ ( G ) is dense in D( e β s ∆ λ,r e − β s )and in H ( G ), this proves assertion . IRICHLET LAPLACIANS IN QUASI-CYLINDRICAL DOMAINS e β s ∆ λ,r e − β s − µ ) = ker (cid:0) ( e β s ∆ λ,r e − β s ) ∗ − µ (cid:1) of the operator (8.1) is finite-dimensional (if µ , λ , and β does not meet the condi-tion (8.2)) we derive the coercive estimate k u ; H ( G ) k ≤ C ( k (cid:0) ( e β s ∆ λ,r e − β s ) ∗ − µ (cid:1) u ; L ( G ) k + k w u ; L ( G ) k ) (8.12)for the adjoint ( e β s ∆ λ,r e − β s ) ∗ of the m-sectorial operator e β s ∆ λ,r e − β s and apply thePeetre’s lemma. The m-sectorial operator ( e β s ∆ λ,r e − β s ) ∗ corresponds to the closeddensely defined sectorial form q λ,r [ u, u ] with the domain ◦ H ( G ). The proof of theestimate (8.12) is similar to the proof of (8.4), we omit it.We have proved that the operator (8.1) is Fredholm provided the condition (8.2)is not satisfied. Now we assume that the condition (8.2) is met, and show that theoperator (8.1) is not Fredholm.Let χ be a smooth cutoff function on the real line, such that χ ( x ) = 1 for | x − | ≤ χ ( x ) = 0 for | x − | ≥
2. Consider the functions u ℓ ( x, y) = χ ( x/ℓ ) exp (cid:0) i (1 + λ ) √ µ − νx − βx (cid:1) Φ(y) , ( x, y) ∈ R × Ω , (8.13)where Φ is an eigenfunction of ∆ Ω , corresponding to the eigenvalue ν ∈ σ (∆ Ω ). Theexponent in (8.13) is an oscillating function of x . Straightforward calculation showsthat (cid:13)(cid:13)(cid:0) ∆ Ω − (1 + λ ) − ( ∂ x + β ) − µ (cid:1) u ℓ ; L ( R × Ω) (cid:13)(cid:13) ≤ C, k u ℓ ; H ( R × Ω) k → ∞ (8.14)as ℓ → + ∞ . Similarly to (8.8) we conclude that (cid:13)(cid:13)(cid:0) e βs △ λ,r e − βs − ∆ Ω +(1+ λ ) − ( ∂ x + β ) (cid:1) u ℓ ; L ( R × Ω) (cid:13)(cid:13) ≤ ǫ ℓ k u ℓ ; H ( R × Ω) k , (8.15)where ǫ ℓ → ℓ → + ∞ . Let the functions u ℓ = u ℓ ◦ κ − be extended from C to G byzero. If, on the contrary, the operator (8.1) is Fredholm, then by the Peetre’s lemmathe estimate (8.4) holds with any weight w , such that H ( G ) ֒ → L ( G ; w ) is a compactembedding. Without loss of generality we can assume that k w u ℓ ; L ( G ) k ≤ C for all ℓ ≥
1. After the change of variables ( ζ, η ) ( x, y ) the estimate (8.4) implies k u ℓ ; H ( R × Ω) k ≤ C ( k ( e βs △ λ,r e − βs − µ ) u ℓ ; L ( R × Ω) k + 1) , where the function e βs △ λ,r e − βs u ℓ = ( e β s ∆ λ,r e − β s u ℓ ) ◦ κ is extended from R + × Ω to R × Ω by zero. This together with (8.15) justifies the estimate k u ℓ ; H ( R × Ω) k ≤ C (cid:0)(cid:13)(cid:13)(cid:0) ∆ Ω − (1 + λ ) − ( ∂ x + β ) − µ (cid:1) u ℓ ; L ( R × Ω) (cid:13)(cid:13) + 1 (cid:1) , which contradicts (8.14).
9. Problem with finite PMLs.
Consider the truncated domain G R with piece-wise smooth boundary, see (2.9). Introduce the Sobolev space H ( G R ) as the comple-tion of the set C ∞ ( G R ) in the norm k v ; H ( G R ) k = ( P ℓ + | m |≤ k ∂ ℓζ ∂ mη v ; L ( G R ) k ) / .In this section we study the problem with finite PMLs: Given g ∈ L ( G R ) find asolution v ∈ H ( G R ) of the equation (∆ λ,r − µ ) v = g in G R , (9.1) where R > r . The next theorem presents a stability result for this problem.2
VICTOR KALVIN
Theorem 9.1.
Assume that µ ∈ R \ σ (∆ Ω ) is not an eigenvalue of the selfadjointDirichlet Laplacian ∆ in L ( G ) . Take a sufficiently large r > and λ ∈ D α \ R . Thenthere exists R > r such that for all R > R and g ∈ L ( G R ) the equation (9.1) has aunique solution v ∈ H ( G R ) . Moreover, the estimate k v ; H ( G R ) k ≤ C k g ; L ( G R ) k (9.2) holds with an independent of R > R and g constant C . The proof of Theorem 9.1 will be carried out by the compound expansion tech-nique. This requires construction of an approximate solution to the equation (9.1)compounded of solutions to limit problems. As the first limit problem we take theproblem with infinite PMLs. As the second limit problem we take a Dirichlet problemin the semi-cylinder ( −∞ , R ) × Ω studied in the next lemma.
Lemma 9.2.
Introduce the weighted Sobolev space H ,β (cid:0) ( −∞ , R ) × Ω (cid:1) of functionssatisfying the homogeneous Dirichlet boundary condition as the completion of the set C ∞ (cid:0) ( −∞ , R ] × Ω (cid:1) with respect to the norm (cid:13)(cid:13) u ; H ,β (cid:0) ( −∞ , R ) × Ω (cid:1)(cid:13)(cid:13) = X ℓ + | m |≤ Z R −∞ k e − βx ∂ ℓx ∂ my u ( x ); L (Ω) k dx / . Let L β (cid:0) ( −∞ , R ) × Ω (cid:1) be the weighted L -space with the norm (cid:13)(cid:13) f ; L β (cid:0) ( −∞ , R ) × Ω (cid:1)(cid:13)(cid:13) = Z R −∞ k e − βx f ( x ); L (Ω) k dx ! / . Assume that λ ∈ D α \ R , µ ∈ R \ σ (∆ Ω ) , and β is in the interval (7.1) . Then forany f ∈ L β (cid:0) ( −∞ , R ) × Ω (cid:1) there exists a unique solution u ∈ H , (cid:0) ( −∞ , R ) × Ω (cid:1) tothe equation (∆ Ω − (1 + λ ) − ∂ x − µ ) u = f . (9.3) Moreover, the estimate (cid:13)(cid:13) u ; H ,β (cid:0) ( −∞ , R ) × Ω (cid:1)(cid:13)(cid:13) ≤ C (cid:13)(cid:13) f ; L β (cid:0) ( −∞ , R ) × Ω (cid:1)(cid:13)(cid:13) (9.4) holds, where the constant C is independent of f and R .Proof . It suffices to prove the assertion for R = 0. Then the general case can beobtained by the change of variables x x − R .The set C ∞ c (cid:0) ( −∞ , × Ω (cid:1) of smooth functions with compact supports is densein L β (cid:0) ( −∞ , × Ω (cid:1) . We first assume that f ∈ C ∞ c (cid:0) ( −∞ , × Ω (cid:1) and extend f to a function in C ∞ c ( R × Ω) by setting f ( − x ) = − f ( x ) for x <
0. Consider theequation (9.3) in the infinite cylinder R × Ω. As is known [36], the Fourier transformˆ f ( ξ ) = R R e ixξ f ( x ) dx is an entire function of ξ with values in L (Ω); it is rapidlydecaying at infinity in any strip { ξ ∈ C : |ℑ ξ | < β } in the sense that the estimates k ˆ f ( ξ ); L (Ω) k ≤ C β,k (1+ | ξ | ) − k hold for k = 0 , , , . . . and some constants C β,k . Since β is in the interval (7.1), the distance d between the set { µ − (1+ λ ) − ξ : 0 ≤ ℑ ξ < β } and the spectrum σ (∆ Ω ) of the selfadjoint operator ∆ Ω in L (Ω) with domain H (Ω)is positive. Hence forΨ( ξ ) = (cid:0) ∆ Ω + (1 + λ ) − ξ − µ (cid:1) − ˆ f ( ξ ) , ≤ ℑ ξ < β, IRICHLET LAPLACIANS IN QUASI-CYLINDRICAL DOMAINS k Ψ( ξ ); L (Ω) k ≤ d − k ˆ f ( ξ ); L (Ω) k . This together with theelliptic coercive estimate k Ψ( ξ ); H (Ω) k ≤ c ( k ˆ f ( ξ ); L (Ω) (cid:13)(cid:13) + k Ψ( ξ ); L (Ω) k )for the Dirichlet Laplacian ∆ Ω gives k Ψ( ξ ); H (Ω) k ≤ ( c + d − ) k ˆ f ( ξ ); L (Ω) k , ≤ ℑ ξ < β. (9.5)The differential operator ∆ Ω − (1 + λ ) − ∂ x is strongly elliptic. Therefore the localcoercive estimate k ̺ u ; H ( R × Ω) k ≤ c (cid:16)(cid:13)(cid:13) ς f ; L ( R × Ω) (cid:13)(cid:13) + k ς u ; L ( R × Ω) k (cid:17) (9.6)is valid, where ̺ and ς are smooth functions of the variable x with compact supportssuch that ̺ ̺ς = ̺ . We substitute u ( x, y ) = e iξx Ψ( ξ, y ) into (9.6). Aftersimple manipulations we arrive at the estimate X ℓ + | m |≤ | ξ | ℓ k ∂ my Ψ( ξ ); L (Ω) k ≤ C (cid:16)(cid:13)(cid:13) ˆ f ( ξ ); L (Ω) (cid:13)(cid:13) + k Ψ( ξ ); L (Ω) k (cid:17) , (9.7)where the constant C depends on ̺ and ς , but not on ξ or f . If | ξ | > C with sufficientlylarge C >
0, then the last term in (9.7) can be neglected. This together with (9.5)justifies the estimate X ℓ + | m |≤ | ξ | ℓ k ∂ my Ψ( ξ ); L (Ω) k ≤ C k ˆ f ( ξ ); L (Ω) k , (9.8)where 0 ≤ ℑ ξ < β and the constant C is independent of ξ and Ψ( ξ ). Therefore theanalytic in strip 0 ≤ ℑ ξ < β function ξ Ψ( ξ ) ∈ H (Ω) is rapidly decaying at infinity.This together with the Cauchy integral theorem allows us to replace the contour ofintegration in the inverse Fourier transformation u ( x ) = (2 π ) − R R e − iξx Ψ( ξ ) dξ . Weobtain u ( x ) =(2 π ) − Z R e − iξx (cid:0) ∆ Ω + (1 + λ ) − ξ − µ (cid:1) − ˆ f ( ξ ) dξ =(2 π ) − Z ξ − iβ ∈ R e − iξx (cid:0) ∆ Ω + (1 + λ ) − ξ − µ (cid:1) − ˆ f ( ξ ) dξ. The Parseval equality gives2 π Z R k e − βx ∂ ℓx ∂ my u ( x ); L (Ω) k dx = Z ξ − iβ ∈ R | ξ | ℓ k ∂ my Ψ( ξ ); L (Ω) k dξ, π Z R k e − βx f ( x ); L (Ω) k dx = Z ξ − iβ ∈ R k ˆ f ( ξ ); L (Ω) k dξ. Integrating (9.8) with respect to ξ , ξ − iβ ∈ R , we deduce the estimate X ℓ + | m |≤ Z ∞−∞ k e − βx ∂ ℓx ∂ my u ( x ); L (Ω) k dx ≤ C Z ∞−∞ k e − βx f ( x ); L (Ω) k dx ; (9.9)4 VICTOR KALVIN in the case β = 0 this estimate takes the form k u ; H ( R × Ω) k ≤ C k f ; L ( R × Ω) k . Thusfor any f ∈ C ∞ c ( R × Ω) there exists a solution u ∈ H ( R × Ω) to the equation (9.3)and the estimate (9.9) holds with any β in the interval (7.1). Usual argument onsmoothness of solutions to elliptic problems gives u ∈ C ∞ ( R × Ω). From the equality f ( − x ) = − f ( x ) it follows that ˆ f ( − ξ ) = − ˆ f ( ξ ) and therefore Ψ( ξ ) = − Ψ( − ξ ). Hence u ( x ) = − u ( − x ) and u (0) = 0. As in the proof of Proposition 4.4 we conclude that inthe norm of H ,β (cid:0) (0 , ∞ ) × Ω (cid:1) one can approximate u by functions in C ∞ (cid:0) (0 , ∞ ) × Ω (cid:1) .Hence u ∈ H ,β (cid:0) ( −∞ , × Ω (cid:1) . By continuity our construction extends to all f ∈ L β (cid:0) ( −∞ , × Ω (cid:1) . In particular, for any f ∈ L (cid:0) ( −∞ , × Ω (cid:1) we can find a solution u ∈ H , (cid:0) ( −∞ , × Ω (cid:1) to the equation (9.3). If f ∈ L β (cid:0) ( −∞ , × Ω (cid:1) with some β in the interval (7.1), then the estimate (9.4) with R = 0 is a direct consequenceof (9.9). It remains to note that u ∈ H , (cid:0) ( −∞ , × Ω (cid:1) is a unique solution asour argument also shows that for any f ∈ L (cid:0) ( −∞ , × Ω (cid:1) the adjoint equation(∆ Ω − (1 + λ ) − ∂ x − µ ) u = f is solvable in the space H , (cid:0) ( −∞ , × Ω (cid:1) .Now we are in position to prove Theorem 9.1. Proof . The scheme of the proof is similar to the one we used in the proof of [20,Theorem 4.1]. We rely on a modification of the compound expansion method [27].We say that w = w ( R ) ∈ H ( G R ) is an approximate solution of the problem withfinite PML if the following conditions are satisfied:i. The estimate k w ; H ( G R ) k ≤ c k g ; L ( G R ) k holds with an independent of g and R constant c ;ii. The estimate k (∆ λ,r − µ ) w − g ; L ( G R ) k ≤ C R k g ; L ( G R ) k is valid, wherethe constant C R is independent of g and C R → R → + ∞ .Due to condition i w continuously depends on g . Condition ii implies that the discrep-ancy, left by w in the equation (9.1), tends to zero as R → + ∞ . Once an approximatesolution w is found, it is not hard to verify the assertion of the theorem.Let ρ ∈ C ∞ ( R ) be a cutoff function such that ρ ( x ) = 1 for x ≤ χ ( x ) = 0for x ≥ /
2. We set ρ R = ρ ( x − R ), ̺ R ( ζ, η ) = ρ R ◦ κ − ( ζ, η ) for ( ζ, η ) ∈ C , and ̺ R ( ζ, η ) = 1 for ( ζ, η ) ∈ G \ C . Let F = ̺ R/ g and f = ( g − f ) ↾ G R ◦ κ . Weextend F from G R to G and f from (0 , R ) × Ω to ( −∞ , R ) × Ω by zero. We findan approximate solution w compounded of u λ,r = (∆ λ,r − µ ) − F and a solution u ∈ H , (cid:0) ( −∞ , R ) × Ω (cid:1) to the equation (9.3) in the form w = ̺ R u λ,r + (1 − ̺ R/ )( u ◦ κ − );here the second term in the right hand side is extended from G R ∩ C to G R by zero.Let us show that w is an approximate solution. Observe that on the support of f we have e β s ≤ Ce βR/ and on the support of f we have e − βx ≤ Ce − βR/ uniformlyin R . Hence k e β s F ; L ( G ) k + e βR (cid:13)(cid:13) f ; L β (cid:0) ( −∞ , R ) × Ω (cid:1)(cid:13)(cid:13) ≤ Ce βR/ k g ; L ( G ) k (9.10)with an independent of R and g constant C . Similarly to (8.7) we conclude that k w ; H ( G R ) k ≤ k ̺ R u λ,r ; H ( G ) k + c (cid:16)(cid:13)(cid:13) △ λ,r (1 − ρ R/ ) u ; L (cid:0) ( −∞ , R ) × Ω (cid:1)(cid:13)(cid:13) + (cid:13)(cid:13) u ; L (cid:0) ( −∞ , R ) × Ω (cid:1)(cid:13)(cid:13)(cid:17) ≤ C k u λ,r ; H ( G ) k + C (cid:13)(cid:13) u ; H , (cid:0) ( −∞ , × Ω (cid:1)(cid:13)(cid:13) , where C is independent of R and △ λ,r is the operator (8.6). This together with theestimates (9.4), (9.10) for β = 0 and Theorem 6.1.1 implies that the condition i issatisfied. IRICHLET LAPLACIANS IN QUASI-CYLINDRICAL DOMAINS λ,r − µ ) w − g = [∆ λ,r , ̺ R ] u λ,r + (1 + λ ) − (cid:0) [ ∂ x , ρ R/ ] u (cid:1) ◦ κ − + (cid:16)(cid:0) △ λ,r − ∆ Ω + (1 + λ ) − ∂ x (cid:1) (1 − ρ R/ ) u (cid:17) ◦ κ − . (9.11)The support of the term [∆ λ,r , ̺ R ] u λ,r is a subset of the image of ( R, R + 1 / × Ωunder the diffeomorphism κ . On this support the weight e β s is bounded from belowby ce βR uniformly in R >
0. As a consequence we get the uniform in R estimates k [∆ λ,r , ̺ R ] u λ,r ; L ( G R ) k ≤ C e − βR k e β s u λ,r ; H ( G ) k ≤ C e − βR k e β s F ; L ( G ) k , (9.12)where we used Theorem 7.1. Now we estimate the second term in the right hand sideof (9.11). On the support of [ ∂ x , ρ R/ ] u we have e − βx ≥ Ce − βR/ . Relying on (9.4)we obtain (cid:13)(cid:13) (1 + λ ) − (cid:0) [ ∂ x , ρ R/ ] u (cid:1) ◦ κ − ; L ( G R ) (cid:13)(cid:13) ≤ c (cid:13)(cid:13) [ ∂ x , ρ R/ ] u ; L (cid:0) ( −∞ , R ) × Ω (cid:1)(cid:13)(cid:13) ≤ C e βR/ (cid:13)(cid:13) u ; H ,β (cid:0) ( −∞ , R ) × Ω (cid:1)(cid:13)(cid:13) ≤ C e βR/ (cid:13)(cid:13) f ; L (cid:0) ( −∞ , R ) × Ω (cid:1)(cid:13)(cid:13) . (9.13)Finally, consider the last term in the right hand side of (9.11). On the support of(1 − ρ R/ ) u the coefficients of the operator △ λ,r − ∆ Ω + (1 + λ ) − ∂ x tend to zero as R → + ∞ , cf. (3.4) and (8.6). This together with the estimate (9.4) for β = 0 gives (cid:13)(cid:13)(cid:13)(cid:16)(cid:0) △ λ,r − ∆ Ω + (1 + λ ) − ∂ x (cid:1) (1 − ρ R/ ) u (cid:17) ◦ κ − ; L ( G R ) (cid:13)(cid:13)(cid:13) ≤ c R (cid:13)(cid:13) f ; L (cid:0) ( −∞ , R ) × Ω (cid:1)(cid:13)(cid:13) , (9.14)where c R → R → + ∞ . From (9.10)–(9.14) it follows that w meets the conditionii. Thus w = w ( R ) is indeed an approximate solution to the problem with finite PML.Now we are in position to prove the assertion of the theorem. Observe that(∆ λ,r − µ ) w − g = O ( R ) g with some operator O ( R ) in L ( G R ), whose norm ||| O ( R ) ||| tends to zero as R → + ∞ because of the condition ii on w . For all R > R witha sufficiently large R we have ||| O ( R ) ||| ≤ ||| O ( R ) ||| <
1. Hence there exists theinverse ( I + O ( R )) − : L ( G R ) → L ( G R ) and its norm is bounded by the constant1 / (1 − ||| O ( R ) ||| ) uniformly in R > R . We set ˜ g = ( I + O ( R )) − g . In the same wayas before we construct the approximate solution w for the problem (9.1), where g isreplaced by ˜ g . Then for v = w we have (∆ λ,r − µ ) v = ˜ g + O ( R )˜ g = g and k v ; H ( G R ) k ≤ c k ˜ g ; L ( G R ) k ≤ c/ (1 − ||| O ( R ) ||| ) k g ; L ( G R ) k , where C is independent of R > R . Thus for R > R and g ∈ L ( G R ) there existsa solution v ∈ H ( G R ) to the equation (9.1) satisfying the estimate (9.2), where theconstant C is independent of R and g . In the remaining part of the proof we showthat this solution is unique.Let ∆ Rλ,r be the unbounded operator in L ( G R ) such that for any v in its domain H ( G R ) we have ∆ Rλ,r v = ∆ λ,r v . Note that ∆ Rλ,r is the operator of the problem withfinite PML. To the operator ∆
Rλ,r there corresponds the quadratic form q Rλ,r [ v, v ] = Z G R (cid:10)(cid:0) det e λ,r (cid:1) / e − λ,r ∇ ζη v, ∇ ζη (cid:0) det e λ,r (cid:1) − / v (cid:11) dζ dη in L ( G R ) with the domain H ( G R ). In the same way as in Section 4 we con-clude that the form q Rλ,r admits a densely defined sectorial closure with the do-main ◦ H ( G R ), where ◦ H ( G R ) is the completion of H ( G R ) with respect to the norm6 VICTOR KALVIN k v ; ◦ H ( G R ) k = ( P ℓ + | m |≤ k ∂ ℓζ ∂ mη v ; L ( G R ) k ) / . This together with the argumentabove implies that all µ < Rλ,r . Hence the sectorial operator ∆
Rλ,r coincides with its m-sectorialFriedrichs extension. Moreover, thanks to the argument above we know that underthe assumptions of theorem for any g ∈ L ( G R ) there exists v ∈ H ( G R ) such that(∆ Rλ,r − µ ) v = g . Similarly, one can study the adjoint m-sectorial operator (∆ Rλ,r ) ∗ .It turns out that under the assumptions of theorem for any g ∈ L ( G R ) there exists v in the domain H ( G R ) of (∆ Rλ,r ) ∗ such that (cid:0) (∆ Rλ,r ) ∗ − µ (cid:1) v = g . Therefore theequation (9.1) is uniquely solvable in H ( G R ).In the next theorem we show that under some natural assumptions solutions v = v ( R ) of the problem with finite PMLs converge in the domain G r to outgoingor incoming solutions u ± with an exponential rate as R → + ∞ . In other words, weestimate the error produced by truncation of infinite PMLs. Solutions to the problemwith finite PMLs can be found numerically with the help of finite element solvers;certainly, discretization produces yet another error that we do not estimate here. Theorem 9.3.
Assume that µ ∈ R \ σ (∆ Ω ) is not an eigenvalue of the selfadjointDirichlet Laplacian ∆ in L ( G ) , the parameter r > is sufficiently large in the senseof Remark 4.2, λ ∈ D α \ R , and β is in the interval (7.1) . Let f ∈ H α ( G ) satisfythe inclusion e β s ( f ◦ ϑ λ,r ) ∈ L ( G ) , where s is the same function as in Theorem 7.1.In (9.1) we set g = f ◦ ϑ λ,r . Then there exists R > r such that for R > R a uniquesolution v = v ( R ) ∈ H ( G R ) of the problem with finite PMLs converges in G r
1. to the outgoing solution u − ∈ H , loc ( G ) of the equation (∆ − µ ) u = f in thecase ℑ λ >
2. to the incoming solution u + ∈ H , loc ( G ) of the equation (∆ − µ ) u = f in thecase ℑ λ < in the sense that as R → + ∞ the estimate X ℓ + | m |≤ k ∂ ℓζ ∂ mη ( u ± − v R ); L ( G r ) k ≤ Ce − βR k e β s ( f ◦ ϑ λ,r ); L ( G ) k (9.15) holds with a constant C independent of R > R and f . Let us remark here that the assumptions of Theorem 9.3 on the right hand side f are a priori met for all f ∈ L ( G ) such that f ↾ C = F ◦ κ with some F ∈ E ; here E isthe algebra defined in Section 5 and κ is the diffeomorphism (2.1). From Lemma 5.2it follows that the set of functions f satisfying the assumptions of Theorem 9.3 isdense in L ( G ). In particular, we can take any f ∈ L ( G ) supported in G r . Proof . Let ρ ∈ C ∞ ( R ) be a cutoff function such that ρ ( x ) = 1 for x ≤ χ ( x ) = 0 for x ≥ /
2. We set ρ R = ρ ( x − R ), ̺ R ( ζ, η ) = ρ R ◦ κ − ( ζ, η ) for ( ζ, η ) ∈ C ,and ̺ R ( ζ, η ) = 1 for ( ζ, η ) ∈ G \ C . Thanks to Theorem 6.1.3 it suffices to prove theestimate (9.15) with u ± replaced by ̺ R u λ,r . The difference ̺ R u λ,r − v R ∈ H ( G R )satisfies the problem (9.1) with g = ( ̺ R − f ◦ ϑ λ,r ) + [∆ λ,r , ̺ R ] u λ,r . Observe that k ( ̺ R − f ◦ ϑ λ,r ); L ( G R ) k ≤ Ce − βR k e β s ( f ◦ ϑ λ,r ); L ( G ) k , k [∆ λ,r , ̺ R ] u λ,r ; L ( G R ) k ≤ Ce − βR k e β s u λ,r ; H ( G ) k , because the functions ( ̺ R − f ◦ ϑ λ,r ) and [∆ λ,r , ̺ R ] u λ,r , being written in the co-ordinates ( x, y ), are equal to zero for x < R , while e β s ( ζ,η ) = ce βx for x ≥ R > C ,cf. (2.4). This together with Theorem 7.1 gives k g ; L ( G R ) k ≤ ce − βR k e β s ( f ◦ ϑ λ,r ); L ( G ) k . (9.16) IRICHLET LAPLACIANS IN QUASI-CYLINDRICAL DOMAINS k ̺ R u λ,r − v R ; H ( G R ) k ≤ C k g ; L ( G R ) k , R > R . (9.17)It remains to note that X ℓ + | m |≤ k ∂ ℓζ ∂ mη ( u ± − v R ); L ( G r ) k ≤ k ̺ R u λ,r − v R ; H ( G R ) k , R > R > r. This together with (9.17) and (9.16) completes the proof of the estimate (9.15).
Remark 9.4.
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