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Dirichlet spectrum of the Fichera layer
Monique Dauge, Yvon Lafranche, Thomas Ourmières-Bonafos
To cite this version:
Monique Dauge, Yvon Lafranche, Thomas Ourmières-Bonafos. Dirichlet spectrum of the Fichera layer.
Integral Equations and Operator Theory, Springer Verlag, 2018, 90 (5, article 60), �10.1007/s00020- 018-2486-y�. �hal-01645083v2�
MONIQUE DAUGE, YVON LAFRANCHE, AND THOMAS OURMI `ERES-BONAFOS
ABSTRACT. We investigate the spectrum of the three-dimensional Dirichlet Laplacian in a prototypal infinite polyhedral layer, that is formed by three perpendicular quarter-plane walls of constant width joining each other. Alternatively, this domain can be viewed as an octant from which another “parallel”
octant is removed. It contains six edges (three convex and three non-convex) and two corners (one convex and one non-convex). It is a canonical example of non-smooth conical layer. We name it after Fichera because near its non-convex corner, it coincides with the famous Fichera cube that illustrates the interaction between edge and corner singularities. This domain could also be called an octant layer.
We show that the essential spectrum of the Laplacian on such a domain is a half-line and we char- acterize its minimum as the first eigenvalue of the two-dimensional Laplacian on a broken guide. By a Born-Oppenheimer type strategy, we also prove that its discrete spectrum is finite and that a lower bound is given by the ground state of a special Sturm-Liouville operator. By finite element computa- tions taking singularities into account, we exhibit exactly one eigenvalue under the essential spectrum threshold leaving a relative gap of 3%. We extend these results to a variant of the Fichera layer with rounded edges (for which we find a very small relative gap of 0.5%), and to a three-dimensional cross where the three walls are full thickened planes.
1. INTRODUCTION AND MAIN RESULTS
1.1. Motivations. In the usual three-dimensional Euclidean space, the termlayer commonly de- notes a tubular neighborhood of a reference surface. Such geometries are physically relevant because for example, in mesoscopic physics, properties of thin-films (also known as quantum layers) can be deduced by a spectral study of the Laplacian in hard-wall layer domains.
From a mathematical point of view, such operators have been considered in [18,10] and the main results can be roughly summed up as follows: under adequate geometric conditions on the reference surface the essential spectrum is a half-line of the form[a,+∞)(a∈ R+) whereas the existence of bound states depends on curvature properties (such a behavior is reminiscent of the study of quantum waveguides, see for instance the pioneering works [22,17]).
In fact, as mentioned in [18,10] and specifically studied in [20], reference surfaces being circular conical layers exhibit an interesting behavior: they have infinitely many bound states and thus, they accumulate to the threshold of the essential spectrum. Moreover, we know that the accumulation rate is logarithmic as shown in [16].
Recently, in [29], it has been proved that the infiniteness of bound states and the logarithmic ac- cumulation to the threshold of the essential spectrum still hold for conical layers constructed around any smooth reference conical surface (by smooth reference conical surface, we mean that the surface is smooth except in its vertex).
Then, a natural question is to know whether this structure of the spectrum is preserved if this smoothness hypothesis is violated. This is the very question we tackle in this paper for a reference surface that is a specific polyhedral cone and thus has edges. We show that the structure of the spectrum drastically differs from the smooth case: the threshold of the essential spectrum is lower than for smooth conical surfaces and there is only a finite number of bound states.
2010Mathematics Subject Classification. 35J05, 35P15, 35Q40, 81Q10, 65M60.
Key words and phrases. Laplace operator on infinite layer domains, Bound states, Essential spectrum, Quantum layers, Conical layers, Octant layer, Fichera corner.
1
Before going any further, let us mention similar works about various realizations of the Laplacian interplaying with conical geometries. Schr¨odinger operators with singular potential, modeled byδ- interactions supported on smooth cones, are investigated in [4,25] whereas the case of the Laplacian with Robin boundary conditions in smooth conical domains is dealt with in [8, 30]. Finally, for problems related with polyhedral geometries, let us mention [7] where the magnetic Laplacian in three-dimensional corner domains is studied as well as [9] where the bottom of the essential spectrum of the Robin Laplacian is characterized for polyhedral cones.
1.2. Main results. As a prototype for infinite polyhedral layers including edges, we investigate the layerΛobtained by removing the first octant(R+)3from the translated octant(R+)3−(1,1,1)(see Figure1, right), namely:
Λ =
(R+)3−(1,1,1) \(R+)3. (1.1) (HereR+denotes(0,∞)). This domain can be called anoctant layer, but we choose to name it after Fichera since its non-convex polyhedral corner sitting at the origin(0,0,0)is a celebrated example of interaction between edge and corner singularities: this interaction is described in [13, §17] and related numerical issues are addressed in [2, 1, 12] for instance. We are interested in the positive Dirichlet LaplacianLΛposed onΛ. We will show that its spectral properties heavily depend on its two-dimensional analogue posed on the broken guideΓof width1and angle π2 (see Figure1, left).
Γ (−1,−1)
(0,0)
Λ
•(0,0,0)
FIGURE 1. On the left: the two-dimensional broken guide Γwith angle π2. On the right: a representation of the three-dimensionalFichera layerΛ.
Theorem 1.1([15, 28]). With the broken guideΓ =
(R+)2−(1,1) \(R+)2 andLΓ the positive Dirichlet Laplacian onΓ, there holds:
i) The essential spectrum ofLΓcoincides with[π2,+∞);
ii) The operatorLΓhas exactly one eigenvalue below its essential spectrum.
It is proved in [15] thatLΓ has a finite number of eigenvalues below its essential spectrum, and in [28] that this number is1, as expected after semi-analytical calculations [19], or finite element approximation [15]. We are going to revisit this result later on, see Remark 2.2. Let λ1(Γ)be the first eigenvalue ofLΓ. An approximate numerical value given in [15] isλ1(Γ)'0.929π2. Our main result concerning the Fichera layerΛis
Theorem 1.2. With the Fichera layerΛ =
(R+)3−(1,1,1) \(R+)3andLΛthe positive Dirichlet Laplacian onΛ, there holds:
i) The essential spectrum ofLΛcoincides with[λ1(Γ),+∞);
ii) LΛhas at most a finite number of eigenvalues below its essential spectrum.
Theorem 1.2 exhibits a significant difference between non-smooth and smooth conical layers:
First, the bottom of the essential spectrum in the non-smooth case is determined by the edge profile (the broken guide) and is lower than in the smooth case where it is given by the first eigenvalueπ2 of the one-dimensional Laplacian on the intervalI = (−1,0)across the layer. Second, in contrast with the present statement, for smooth conical layers the discrete spectrum is infinite as it was first observed for circular cones [20, Thm. 3.1], [16, Thm. 1.4], and next generalized to smooth conical layers [29, Thm. 2].
Before going further, let us comment on relations between smooth layers and our Fichera layer.
A smooth layer is defined like a shell in elasticity: from a reference unbounded smooth surfaceS[ without boundary (said midsurface) and a positive thickness parameter ε, we may define the layer Λ[[ε] as the set of points at distance strictly smaller than ε/2to S[. This same definition is also adopted in [29] whenS[ is a smooth conical surface. Trying this for our Fichera layer, we choose S[as the union of three quarter planes
S[ ={x∈R3 : min{x1, x2, x3}=−12}
and takeε = 1. However the layerΛ[ := Λ[[1]is distinct from Λoutside any compact set: We can see that the section ofΛby any planex1 = R,R > 0, is isometric to the broken guideΓ, whereas a similar section of Λ[ is isometric to the broken guide Γ[ with rounded exterior corner drawn in Figure2, left.
Γ[ S[
(0,0) (−12,−12)
Γ] S]
(0,0) (−1,−1)
FIGURE 2. Plane sectionsΓ[ andΓ]of the variantsΛ[andΛ]of the Fichera layer.
An alternative in the same spirit would be to set S0 = {x ∈ R3 : min{x1, x2, x3} = 0}and define the layerΛ]as the set of points at signed distance less than1fromS0where the sign is given by the outward normal to the octant(R+)3. We observe that the section ofΛ]by any planex1 =R, R > 0, is isometric to the broken guide Γ] with rounded exterior corner drawn in Figure 2, right.
Note that Λ] can also be viewed as a layer of thicknessε = 1associated with the midsurface S] drawn in the same figure.
In fact, the Dirichlet Laplacian onΓ[orΓ] has exactly one eigenvalue under the threshold of the essential spectrum [22, 31] and Theorem1.2 generalizes to the layers Λ[ and Λ] if we replace the guideΓbyΓ[andΓ], respectively, see Section5and AppendixB.
Another interesting related geometry is given by the cross-like domains. Let X and Y be the two- and three-dimensional “crosses”,cf.Figure3(hereIis the bounded interval(−1,0))
X = (R× I)∪(I ×R) and Y = (R2× I)∪(R× I ×R)∪(I ×R2). (1.2) Here, once more, the Dirichlet Laplacian on X has exactly one eigenvalue under the essential spectrum [31, Prop. 13] and Theorem1.2generalizes to the three-dimensional crossY if we replace the guideΓby the two-dimensional crossX, see Section5.
X
Y
FIGURE 3. The 2- and 3-dimensional crosses.
1.3. Notations. Letd∈ {1,2,3}and letx= (x1,· · · , xd)denote the Cartesian coordinates ofRd. ForL >0, we will make use ofL, the box domains ofRddefined as
L:={x∈Rd : maxd
j=1 |xj + 1|< L}. (1.3) 1.3.1. Three-dimensional domains. InR3, the Fichera layer (1.1) is the unbounded layer domainΛ that can alternatively be written as
Λ ={x∈R3 : −1<min{x1, x2, x3}<0}. (1.4) Bounded versions of Λ are obtained as the intersection ofΛ with the three-dimensional boxesL
centered at the external vertex(−1,−1,−1)ofΛ. Set forR >−1:
ΛR= Λ∩R+1. (1.5)
The residual parts “at infinity” ofΛare denoted byΩR
ΩR = Λ∩(R3\R+1) = Λ\ΛR. (1.6) The layerΛis invariant by the symmetries with respect to the three diagonal planesxj =xk(j < k).
It is natural to splitΛinto the three isometric partsΛ1,Λ2andΛ3, withΛ3defined as
Λ3 ={x∈Λ : x1 < x3 and x2 < x3} (1.7) and the other two by permutation of indices. We also introduce the finite and residual parts
ΛjR= Λj∩ΛR and ΩjR = Λj ∩ΩR. (1.8) 1.3.2. Two-dimensional domains. The two-dimensional domain corresponding to the Fichera layer is the broken guideΓof opening π2 and width1
Γ ={x∈R2 : −1<min{x1, x2}<0} (1.9) and the finite broken guideΓRis defined accordingly
ΓR= Γ∩R+1 for R >−1, (1.10) whereR+1is the two-dimensional box as defined in (1.3).
These domains are symmetric with respect to the diagonal linex1 =x2, which makes natural the definition of the subdomainsΓ1 ={x ∈Γ : x2 < x1}andΓ2 ={x∈ Γ : x1 < x2}together with their bounded analoguesΓjR = Γj∩R+1forj = 1,2. We particularize the parts of their boundaries at “distance”R, i.e. ΣjR =∂ΓjR∩∂R+1. Note that
Σ1R ={R} × I and Σ2R=I × {R} with I = (−1,0). (1.11) Finally
ΣR= Σ1R∪Σ2R =∂ΓR∩∂R+1. (1.12) We will also make use of the rectanglesTRj and half-stripsSRj
TR1 = (0, R)× I, TR2 =I ×(0, R) and SR1 = (R,∞)× I, SR2 =I ×(R,∞).
These domains are represented in Figure4and5.
Γ1 Γ2
(−1,−1)
(0,0)
Γ1R SR1 Σ1R Γ2R
SR2 Σ2R
(−1,−1)
(0,0)
(R,−1) (−1, R)
FIGURE 4. The guideΓand its associated subdomains.
In this paper, we will use in two occurrences the following simple uniform trace estimate:
Lemma 1.3. There exists a constantCsuch that for allR≥1and allu∈H1(ΓR)there holds kukL2(ΣR) ≤CkukH1(ΓR). (1.13) Proof. We bound theL2-norm of the trace ofuonΣRby theH1-norm onΓR\ΓR−1. 1.3.3. Operators. The Laplace operator∆inRdis the partial differential operator
∆ =
d
X
j=1
∂j2.
On a generic domainO ⊂Rd, the positive Laplacian−∆is associated with the quadratic form Q(u) =
Z
O
|∇u(x)|2dx,
defined for ubelonging to the Sobolev space H1(O). The boundary conditions considered in this paper are either Dirichlet or Neumann on parts of the boundary of O. Let us assume that O is a Lipschitz domain and that ∂DO and ∂NO are two Lipschitz subdomains of the boundary ∂O such that∂DO ∪∂NO =∂O. Then we can introduce the variational spaceDom(Q)(form domain forQ) with Dirichlet conditions on∂DO:
Dom(Q) =
u∈H1(O) : u ∂
DO = 0 .
DomainO Γ, cf.(1.9) ΓR, cf.(1.10) Λ, cf.(1.4) Neumann part∂NO ∂NΓ =∅ ∂NΓR = ΣR, cf.(1.12) ∂NΛ =∅
Laplace operator LΓ LΓR LΛ
Rayleigh quotients λ`(Γ) λ`(ΓR) λ`(Λ)
Condensed notations λ1(Γ) :=λ∞ λ1(ΓR) :=λR –
– – forR=x3: λ1(Γx3) := λ(x3) –
Main results Th.1.1 Cor.2.4 and Prop.2.6 Th.1.2 and Prop.3.8 TABLE 1. Notations for the main Laplace operators studied in this paper.
The associated self-adjoint operator isL :=−∆with domain Dom(L) =
u∈Dom(Q) : −∆u∈L2(Ω), ∂nu
∂NO = 0 .
The spectrum of L is denoted by σ(L), its discrete and essential parts byσdis(L) and σess(L), re- spectively. Likewise, we might also writeσ(Q),σess(Q)andσdis(Q), respectively. We particularize the notation ofL(respectivelyQ) on the domainOasLDirO (respectivelyQDirO ) if∂NO has measure zero andLMixO (respectivelyQMixO ) otherwise.
The`-th Rayleigh quotient ofQDirO (respectivelyQMixO ) on its form domain is denoted byλDir` (O) (respectivelyλMix` (O)). Except possibly in Section4, there is no risk of confusion, thus, for the sake of readability, we omit the mention of the superscriptsDirandMixusingLO,QO andλ`(O)instead.
In particularλ1(O)is the bottom of the spectrum ofLO: λ1(O) = min
u∈Dom(QO), u6=0
QO(u) kuk2 .
In Table 1, we list the main geometrical domains we are interested in as well as the associated Laplace operators, their Rayleigh quotients and the main related results.
1.3.4. Domain partition. We will often use a comparison principle of eigenvalues based on a domain partition. Let us introduce a finite partition(Oj)j∈J of the Lipschitz domainOin the sense that each Oj is Lipschitz, that they are pairwise disjoint, and
∪j∈JOj =O.
Then we define the broken quadratic formQBroO QBroO (u) =X
j∈J
Z
Oj
|∇u(x)|2dx, defined foruin the domain
Dom(QBroO ) =
u∈L2(O) : u O
j ∈Dom(QOj), j ∈ J , whereDom(QOj) ={u∈H1(Oj), u
∂DO∩∂Oj= 0}. LetλBro` (O)denote the`-th Rayleigh quotient ofQBroO . Since we have the obvious embedding between form domainsDom(QO) ⊂ Dom(QBroO ), the following inequalities between Rayleigh quotients hold
λBro` (O)≤λ`(O), ∀`≥1, (1.14a)
while
λBro` (O) is the`-th smallest term in the set [
j∈J
`
[
k=1
{λk(Oj)} (1.14b) with multiplicities taken into account.
1.4. Structure of the paper. In Section2we study the auxiliary question of the two-dimensional broken guides of finite length ΓR. We collect results regarding the first eigenvalue of the Dirichlet Laplacian in such domains, in particular its exponential convergence to the first eigenvalue of the infinite broken guide as the lengthR goes to infinity (see Corollary2.4) and a Dauge-Helffer type formula about its derivative with respect to the length of the finite guide (see Proposition2.6).
Section3is devoted to the proof of Theorem1.2. First, we investigate the structure of the essential spectrum using the form decomposition method as well as constructing adapted Weyl sequences for the operator. Second, we prove finiteness of the number of bound states by a Born-Oppenheimer strategy: we compare the number of eigenvalues below the threshold of the essential spectrum to that of a one-dimensional operator obtained after projection on the lowest eigenfunction of a transverse operator. Third, we conclude this section by giving a lower bound on the spectrum that turns out to be numerically close to the threshold of the essential spectrum: there is very little room left for bound states to exist (see Proposition3.8and Figure7).
In Section 4, we illustrate some of our results thanks to computations performed with the finite element library XLiFE++ [33] and we exhibit the existence of exactly one isolated eigenvalue for the Dirichlet problem on the Fichera layer. We address in Section5the extensions mentioned above about the Fichera layer with exterior rounded edgesΛ]and the crossY. We draw finally some con- clusions about other possible generalizations of our results in Section6, developing the discussion on the distinct spectral behaviors in the family of smooth conical layers and in a family of generalized Fichera layers. The two appendicesAandBclose the paper.
2. BROKEN GUIDES OF FINITE LENGTH
In this section, we address the finite plane broken guidesΓR(1.10) forR ≥0: we investigate the first eigenpair of the LaplacianLΓR with Dirichlet conditions on∂ΓR∩∂Γand Neumann conditions on the remaining partΣR of the boundary of∂ΓR. We start with rough estimates on the first two eigenvalues ofLΓR.
Lemma 2.1. With the notations summarized in Table1, there holds:
i) For allR ≥0,λ1(ΓR)≤λ1(Γ).
ii) For allR ≥0,λ2(ΓR)≥π2.
Proof. We use the domain partition as stated in§1.3.4.
i)For a chosenR ≥0, we split the broken guideΓinto three pieces: The finite guideΓR, and the two half-stripsSR1 = (R,∞)× I and SR2 = I ×(R,∞). On the half-stripsSR1 andSR2, Dirichlet conditions are applied on the unbounded sides. By (1.14a)-(1.14b) we find
λ1(Γ)≥min
λ1(ΓR), λ1(SR1), λ1(SR2) . Sinceλ1(SR1) =λ1(SR2) =π2, we deduce point i) thanks to Theorem1.1.
ii)Now we split the finite guideΓRinto the squareΓ0and the two finite rectanglesTR1 = (0, R)×I andTR2 =I ×(0, R)(see Figure5). On the rectanglesTR1 andTR2, Dirichlet conditions are applied on the sides(0, R)× {−1,0}and{−1,0} ×(0, R), respectively. By (1.14a)-(1.14b) we find
λ2(ΓR)≥ 2d smallest element of
λ1(Γ0), λ2(Γ0), λ1(TRj), λ2(TRj), j= 1,2 . (2.1)
TR1 TR2
Γ0 (−1,−1)
(0,0) (−1, R)
(R,−1)
FIGURE 5. The finite guideΓR, the squareΓ0 and the rectanglesTRj,j = 1,2.
Recall that the boundary conditions on the squareΓ0are Dirichlet on the sidesx1 =−1,x2 =−1, and Neumann on the sidesx1 = 0,x2 = 0. Therefore we have
λ1(Γ0) =1 4+ 1
4
π2 = π2
2 and λ2(Γ0) = 1 4 +9
4
π2 = 5π2 2 . Moreover, we get
λ1(TR1) =λ1(TR2) =π2 and λ2(TR1) = λ2(TR2)> π2.
Thus the second smallest element of the set in (2.1) isπ2 and pointii)of the lemma is proved.
Remark2.2. A similar proof as that of pointii)above yields that the second Rayleigh quotient ofLΓ is greater thanπ2. This is in the spirit of [31] and provides a more direct proof of the result [28] that LΓ has at most one eigenvalue under its essential spectrum.
2.1. Exponential decay of eigenvectors. In this section, we prove exponential decay estimates of the first eigenvector ofLΓR, uniformly asR → ∞. For convenience, in the rest of Section2we use the following condensed notation for the first eigenvalue on ΓR and Γ, cf. Table 1, as well as the L2-norm of their difference:
λR:=λ1(ΓR), λ∞:=λ1(Γ), and ω=p
π2−λ∞. (2.2)
Lemma 2.3. For R ≥ 0, let vR be the positive normalized eigenvector associated with the first eigenvalueλRofLΓR, i.e.
LΓRvR=λRvR, vR>0 on ΓR, and kvRkL2(ΓR) = 1. (2.3) Recall thatΣρ = ∂Γρ∩∂ρ+1, cf. (1.12). Then, withωgiven in (2.2), for all integers `, m ≥ 0, there exists a constantC`,msuch that
∀R ≥1, ∀ρ∈[1, R], k∂n`∂τmvRkL2(Σρ) ≤C`,me−ρω, where∂nand∂τ are the normal and tangential derivatives onΣρ, respectively.
Proof. We prove exponential decay for vR by a representation formula on the rectangle TR1 = (0, R)× I, the rectangle TR2 = I ×(0, R) being handled in a similar way. We note that vR is
solution of the mixed problem inTR1 = (0, R)× I
−∆vR = λRvR in(0, R)× I, vR(x1,−1) =vR(x1,0) = 0 ∀x1 ∈(0, R),
∂1vR(R, x2) = 0 ∀x2 ∈ I, vR(0, x2) = gR ∀x2 ∈ I,
(2.4)
wheregRis the trace ofvRon the segmentΣ10 ={0} × I. SincevRbelongs in particular toH1(Γ0), its trace belongs toH1/2(Σ10)with the estimates
kgRkH1/2(Σ10) ≤C0kvRkH1(Γ0) whereC0does not depend onR. Now, there holds
kvRkH1(Γ0) ≤ kvRkH1(ΓR) =p
λR+ 1kvRkL2(ΓR) =p
λR+ 1 ≤√
π2+ 1. Thus we have obtained in particular
kgRkL2(Σ1
0)≤C0
√
π2+ 1, ∀R≥0. (2.5)
We expand vR along the eigenvectors of the operator −∂y2, self-adjoint on (H2 ∩H01)(I). Its nor- malized eigenvectors are
sk(x2) =√
2 sin(kπx2) and we write
vR(x1, x2) =X
k≥1
uk(x1)sk(x2), where uk(x1) = Z
I
vR(x1, x2)sk(x2) dx2. Hence, (2.4) yields that for allk ≥1we have
−u00k+ (k2π2−λR)uk = 0 in(0, R), u0k(R) = 0,
uk(0) = gR,k
where gR,k = Z
I
gR(x2)sk(x2) dx2. (2.6)
The solution of (2.6) is given by
uk(x1) = gR,k cosh R√
k2π2−λR cosh (R−x1)p
k2π2−λR
which yields
vR(x1, x2) = X
k≥1
gR,k cosh RωR,k
cosh (R−x1)ωR,k
sk(x2), (2.7) where we have set ωR,k = √
k2π2−λR. Thanks to the uniform convergence of this series and its derivatives on any subdomain of the form [a, R]× I with a positivea, we deduce the formulas for
`, m∈N
∂1`∂2mvR(x1, x2) =P
k≥1(−1)m/2 cosh(RωωR,k` (kπ)m
R,k) gR,k cosh (R−x1)ωR,k
sk(x2) m even
∂1`∂2mvR(x1, x2) =P
k≥1(−1)(m−1)/2 ω
` R,k(kπ)m
cosh(RωR,k) gR,k cosh (R−x1)ωR,k
ck(x2) m odd if`is even and
∂1`∂2mvR(x1, x2) = P
k≥1(−1)m/2+1 ω
` R,k(kπ)m
cosh(RωR,k) gR,k sinh (R−x1)ωR,k
sk(x2) m even
∂1`∂2mvR(x1, x2) = P
k≥1(−1)(m+1)/2 cosh(RωωR,k` (kπ)m
R,k) gR,k sinh (R−x1)ωR,k
ck(x2) m odd
if` is odd, where we setck =√
2 cos(kπx2)the cosine basis. Thus we can calculate theL2-norm of their trace onΣ1ρ,i.e. atx1 =ρ, for anyρ∈[1, R]:
k∂1`∂2mvRk2L2(Σ1ρ) =
P
k≥1
ω`R,k(kπ)mcosh((R−ρ)ωR,k)
cosh(RωR,k) gR,k2
` even P
k≥1
ω`R,k(kπ)msinh((R−ρ)ωR,k)
cosh(RωR,k) gR,k2
` odd.
Since λR ≤ λ1(Γ)by point i) of Lemma 2.1, we notice that ωR,k is larger than p
k2π2−λ1(Γ), itself larger thankω. Thus we deduce
sinh (R−ρ)ωR,k
cosh RωR,k ≤ cosh (R−ρ)ωR,k
cosh RωR,k ≤2e−ρωR,k ≤2e−ρkω, hence
k∂1`∂2mvRk2L2(Σ1ρ)≤4X
k≥1
ωR,k` (kπ)m2
g2R,ke−2ρkω.
Using the upper boundωR,k ≤kπ, we find k∂1`∂m2 vRk2L2(Σ1ρ)≤4X
k≥1
(kπ)2(`+m)g2R,ke−2ρkω
≤4e−2ρω X
k≥1
gR,k2 maxk≥1
n
(kπ)2(`+m)e−2ρ(k−1)ωo
≤4e−2ρωkgRk2L2(Σ10)max
k≥1
(kπ)2(`+m)e−2(k−1)ω for ρ≥1.
Combining with (2.5), we find for any`, m ≥ 0the estimates k∂1`∂2mvRkL2(Σ1ρ) ≤ Cl,me−ρω with
constantsCl,mindependent ofR. The lemma is thus proved.
Corollary 2.4. With the positive numberω =√
π2−λ∞,cf. (2.2), there exists a constantC1 such that
∀R ≥0, 0≤λ∞−λR≤C1e−2Rω.
Proof. We know already thatλ∞ ≥λR. To prove the right estimate, we use the eigenvectorsvRas quasi-modes forLΓafter cutting them off on the rectangular regions(R−1, R)×IandI×(R−1, R):
assumeR ≥2for simplicity and introduce a smooth cut-offχsuch that χ(t) = 1 for t≤ −1 and χ(t) = 0 for t≥0.
We defineevR(x)byχ(x1−R)vR(x)ifx∈Γ1R, byχ(x2−R)vR(x)ifx∈Γ2R, and by0ifx∈Γ\ΓR. ThenevRbelongs to the domain ofLΓand we can evaluate its Rayleigh quotient. Lemma2.3implies the following estimates with a constantC independent ofR:
kevRk2L2(Γ)−1
≤C e−2Rω and
k∇evRk2L2(Γ)− k∇vRk2L2(ΓR)
≤C e−2Rω. (2.8) Hence, the min-max principle and (2.8) give
λ∞(1−Ce−2Rω)≤λ∞kevRk2L2(Γ) ≤ k∇evRk2L2(Γ) ≤ k∇vRk2L2(ΓR)+Ce−2Rω =λR+Ce−2Rω. Consequently, we get
λ∞−λR ≤C1e−2Rω,
withC1 =C(λ∞+ 1)and the corollary is proved.
2.2. Variation of eigenpairs. Before proving differential formulas for the first eigenpair of the operatorLΓR, let us give an argument stating the regularity of this first eigenpair with respect toR.
Lemma 2.5. Recall the abbreviated notationλR for the first eigenvalue ofLΓR and let vR be the associated normalized and positive eigenvector,cf. (2.3). Then the function R 7→λRis analytic on (0,∞)and the derivativewR :=∂RvRmakes sense inDom(QΓR).
Proof. For anyR >0, let us introduce the following change of variables that transformsΓRintoΓ1:
(x, y) =
(x1, x2) if (x1, x2)∈Γ0, (R−1x1, x2) if (x1, x2)∈ TR1, (x1, R−1x2) if (x1, x2)∈ TR2.
(2.9)
Foru ∈Dom(QΓR), settingu(x, y) =b u(x1, x2)we getQΓR(u) =QbR(bu), whereQbRis the param- eter dependentH1 semi-norm onΓ1defined as
QbR(bu) =k∇buk2L2(Γ0)+R−1
k∂xbuk2L2(T11)+k∂ybuk2L2(T12)
+R
k∂xukb 2L2(T12)+k∂yukb 2L2(T11)
, Dom(QbR) = {bu∈H1(Γ1) :u|b∂Γ1\Σ1 = 0}.
Remark that theL2-norm becomes
kuk2L2(ΓR)=NbR(bu)2 :=kukb 2L2(Γ0)+R
kbuk2L2(T11)+kukb 2L2(T12)
,
where the normNbRis equivalent to the usual normk · kL2(Γ1).
Now, asDom(QbR)does not depend onRand that for allub∈Dom(QbR)the application R∈(0,+∞)7→ QbR(u)b
NbR(bu)2
is analytic, using thatQbRis bounded from below (non-negative), the strategy of Kato [23, Chapt. 7
§4.2] can be adapted to obtain that the self-adjoint operators associated with the family(QbR)R>0via the first representation theorem is an analytic family of operators for R ∈ (0,∞). In particular, as for anyR >0the first eigenvalueλRis simple, we obtain the lemma.
Now we can prove a formula for the derivative with respect toRof the first eigenvalue ofLΓR in the spirit of [23, VII, sect 6(5)] and [14, Theorem (1.4)].
Proposition 2.6. With notations as in Lemma2.5, the derivative∂RλRsatisfies the formula (hereτ is a tangential variable in the Neumann partΣRof∂ΓR)
∂RλR= Z
ΣR
|∂τvR|2−λR|vR|2
dτ, R >0. (2.10)
Proof. SinceλRis a simple eigenvalue, and sinceLΓR commutes with the symmetry with respect to the diagonalD ={x: x1 =x2}, the eigenvectorvRis also symmetric with respect toD. Indeed, it satisfies Neumann conditions onD ∩Γ0, see [15, Prop.2.2]. Thus, we can reduce our analysis to the lower halfΓ1RofΓR. There holdskvRk2L2(Γ1R) = 12 and we are going to prove
∂RλR= 2 Z
Σ1R
|∂2vR|2−λR|vR|2
dx2, R >0. (2.11)
We follow the steps of the proof of [14, Theorem (1.4)]. Integrating by parts and using that∂nvRis zero onΣ1RandD ∩Γ0, we find for any chosenh >0
Z
Γ1R
(−∆−λR)vRvR+hdx1dx2 = Z
Γ1R
vR(−∆−λR)vR+hdx1dx2+ Z
Σ1R
vR∂1vR+hdx2. Hence, using the eigen-equations forvRandvR+h:
(λR+h−λR) Z
Γ1R
vRvR+hdx1dx2+ Z
Σ1R
vR∂1vR+hdx2 = 0.
Taking advantage of the condition∂1vR+h
Σ1R+h = 0we can write Z
Σ1R
vR∂1vR+hdx2 = Z
I
vR(R, x2)
∂1vR+h(R, x2)−∂1vR+h(R+h, x2) dx2.
Putting together the last two identities, dividing byhand lettinghtend to0, we obtain the relation
∂RλR
Z
Γ1R
|vR|2dx1dx2 = Z
Σ1R
vR(∂12vR) dx2.
Using the relation(−∂12−∂22−λR)vR = 0, we deduce formula (2.11), hence formula (2.10).
Remark2.7. 1) Formula (2.10) takes also the form
∂RλR= Z
ΣR
vR(∂n2vR) dτ, R >0. (2.12) 2) Since for anyR >0,vRbelongs toH01(ΣR), thus satisfiesk∂τvRk2L2(ΓR)≥π2kvRk2L2(ΓR), formula (2.10) implies the inequality
∂RλR≥ Z
ΣR
(π2−λR)|vR|2dτ, R >0. (2.13) AsvRis not identically0on the Neumann boundaryΣR, the above inequality implies:
the function R 7→λR is increasing on (0,∞). (2.14) 3) The functionR 7→ λRis analytic on (0,∞), but has no extension as an analytic (nor evenC1) function on the closed interval[0,∞).
LetwRbe the derivative∂RvR. OnΓR, the eigen-equations forvR,vR+h yield (−∆−λR+h)(vR+h−vR) = (λR+h−λR)vR.
The functionvR+h−vR satisfies the zero Dirichlet conditions on∂ΓR\ΣR. OnΣR, we can write like in the proof above
∂n(vR+h−vR)
ΣR =∂nvR+h
ΣR −∂nvR+h
ΣR+h
Dividing byhand lettingh→0, we deduce thatwRis solution of the mixed problem
(−∆−λR)wR = (∂RλR)vR in ΓR,
wR = 0 on ∂ΓR\ΣR,
∂nwR = −∂n2vR on ΣR.
(2.15) We note that formula (2.12) is the compatibility relation for the existence of a solution to the mixed problem (2.15). Moreover, the normalizationR
ΓR|vR|2dx1dx2 = 1implies the relation Z
ΓR
wRvRdx1dx2 =−1 2
Z
ΣR
|vR|2dτ . (2.16)
Lemma 2.8. With the notations of Lemma2.5 andω = p
π2−λ1(Γ), the derivativewR = ∂RvR satisfies the estimates
kwRkH1(ΓR) ≤C e−Rω, R ≥1. (2.17) Proof. The solutionwRof (2.15)-(2.16) is unique and can be written as
wR=µRvR+w⊥R, with Z
ΓR
wR⊥vRdx1dx2 = 0 and µR=−1 2
Z
ΣR
|vR|2dτ.
Recall that the variational space associated with Problem (2.15) is the form domainV =Dom(QΓR) and denote byV0 its dual space. LetfRdenote the right hand side of (2.15). Let us prove that
kwR⊥kH1(ΓR)≤KkfRkV0 for K = 1 +π2
ω2 . (2.18)
Indeed, ifh·,·iV0,V denotes the duality pairing, for anyε ∈(0,1)we have kw⊥RkH1(ΓR)kfRkV0 ≥ hfR, wR⊥iV0,V =k∇wR⊥k2L2(ΓR)−λ1(ΓR)kwR⊥k2L2(ΓR)
≥εk∇w⊥Rk2L2(ΓR)+ (1−ε)λ2(ΓR)−λ1(ΓR)
kw⊥Rk2L2(ΓR). Using Lemma2.1, we getλ1(ΓR)< λ1(Γ)andλ2(ΓR)−λ1(ΓR)≥π2−λ1(Γ) =ω2, hence
εk∇wR⊥k2L2(ΓR)+ (ω2−επ2)kw⊥Rk2L2(ΓR)≤ kw⊥RkH1(ΓR)kfRkV0.
Choosingεso thatε=ω2−επ2, we obtain (2.18). From (2.18) we deduce (still using the condensed notationλRforλ1(ΓR))
kwRkH1(ΓR) ≤ |µR| kvRkH1(ΓR)+kwR⊥kH1(ΓR)
≤ kvRk2L2(ΣR)kvRkH1(ΓR)+KkfRkV0
≤p
1 +λRkvRk2L2(ΣR) +KkfRkV0
≤√
1 +π2kvRk2L2(ΣR)+KkfRkV0. (2.19) As the duality pairing betweenfR∈V0 and anyg ∈V satisfieshfR, giV0,V =R
ΓR(∂RλR)vRgdx− R
ΣR∂n2vRgdτ, we get
|hfR, giV0,V| ≤ |∂RλR|kgkL2(ΓR)+k∂n2vRkL2(ΣR)kgkL2(ΣR). (2.20) Using Lemma1.3that provides a uniform estimate ofkgkL2(ΣR)bykgkH1(ΓR) (here the assumption R≥1comes into play), we find that (2.20) becomes
|hfR, giV0,V| ≤
|∂RλR|+Ck∂n2vRkL2(ΣR)
kgkH1(ΓR) and we obtain
kfRkV0 ≤ |∂RλR|+Ck∂n2vRkL2(ΣR). (2.21) Combining estimates (2.19) and (2.21) with formula (2.12) yields
kwRkH1(ΓR) ≤√
1 +π2kvRk2L2(ΣR)+K
kvRkL2(ΣR)k∂n2vRkL2(ΣR)+Ck∂n2vRkL2(ΣR) .
The application of Lemma2.3ends the proof.
3. FICHERA LAYER: FINITENESS OF DISCRETE SPECTRUM
This section is devoted to the proof of our main theoretical result, that is Theorem1.2 that de- scribes the essential spectrum of the Dirichlet Laplacian LΛ on the Fichera layer and states the finiteness of its discrete spectrum. We also exhibit a lower bound for the whole spectrum ofLΛ.
