Spectral asymptotics for Robin problems with a discontinuous coefficient

0.24 MB·English

by  Gerd Grubb

SPECTRAL ASYMPTOTICS FOR ROBIN PROBLEMS
WITH A DISCONTINUOUS COEFFICIENT
Gerd Grubb
1
1 Department of Mathematical Sciences, Copenhagen University, Universitetsparken
0
5, DK-2100 Copenhagen, Denmark. E-mail [email protected]
2

n Dedicated to the memory of M. Sh. Birman (1928–2009)
a
J

7 Abstract. Thespectralbehaviorofthedifferencebetweentheresolventsoftworealizations
2
A and A of a second-order strongly elliptic symmetric differential operator A, defined by
e1 e2
different Robin conditions νu=b γ u and νu=b γ u, can in the case where all coefficients
] 1 0 2 0
P are C∞ be determined by use of a general result by the author in 1984 on singular Green
A operators. We here treat the problem for nonsmooth b . Using a Krein resolvent formula,
i
h. we show that if b1 and b2 are in L∞, the s-numbers sj of (Ae1 −λ)−1 −(Ae2 −λ)−1 satisfy
t sjj3/(n−1) ≤ C for all j; this improves a recent result for A = −∆ by Behrndt et al., that
a P sp < ∞ for p > (n−1)/3. A sharper estimate is obtained when b and b are in Cε for
m j j 1 2
some ε > 0, with jumps at a smooth hypersurface, namely that s j3/(n−1) → c for j → ∞,
[ j
with a constant c defined from the principal symbol of A and b2−b1.
3 As an auxiliary result we show that the usual principal spectral asymptotic estimate for
v
pseudodifferential operators of negative order on a closed manifold extends to products of
7
pseudodifferential operators interspersed with piecewise continuous functions.
9
9
0
.
9
0
Introduction.
0
1 Consider a second-order strongly elliptic symmetric operator
:
v
i n
X
(0.1) A = − ∂ (a ∂ u)+a u
r j jk k 0
a X
j,k=1
on a bounded smooth domain Ω ⊂ Rn, and denote by A , A , resp. A, the realizations in
γ ν
L (Ω) defined by the Dirichlet condition γ u = 0, the Neumann condition νu = 0, resp. a
2 0
e
Robin condition νu−bγ u = 0 with b real. Here γ u = u| , and
0 0 ∂Ω
n
(0.2) νu = n γ (a ∂ u),
j 0 jk k
X
j,k=1
1991 Mathematics Subject Classification. 35J40, 47G30, 58C40.
Key words and phrases. Elliptic boundary value problem; Robin condition; spectral asymptotics; re-
solvent difference; Krein formula; piecewise continuous coefficient; pseudodifferential boundary operator.
Typeset by AMS-TEX
1
2 GERD GRUBB
the conormal derivative, with ~n = (n ,...,n ) denoting the interior normal to ∂Ω. It
1 n
is a classical result of Birman [B62], shown also for exterior domains, that the difference
between the resolvents of the Robin realization and the Dirichlet realization is compact
and has the spectral behavior, for large negative λ,
(0.3) s (A−λ)−1 −(A −λ)−1 j2/(n−1) ≤ C for all j;
j γ
(cid:0) (cid:1)
e
here sj(T) denotes the j-th eigenvalue of (T∗T)21 (the j-th s-number or singular value of
T), counted with multiplicities. This was shown assuming merely that b ∈ L (∂Ω).
∞
For the situation where all coefficients are C∞, the estimate was later improved to an
asymptotic estimate
(0.4) s (A−λ)−1 −(A −λ)−1 j2/(n−1) → c for j → ∞;
j γ
(cid:0) (cid:1)
e
this follows from Grubb [G74], Sect. 8 (with generalizations to higher-order operators),
and Birman and Solomiak [BS80] (including exterior domains).
The paper [G84] gave tools to extend (0.4) to nonselfadjoint situations (also for exterior
domains by a cutoff technique), by showing that for any singular Green operator G on Ω
of order −t < 0 and class 0,
(0.5) s (G)jt/(n−1) → c(g0) for j → ∞;
j
here G belongs to the calculus of pseudodifferential boundary operators, introduced by
Boutet de Monvel [B71] and further developed in [G84], [G96]; c(g0) is a constant derived
from the principal symbol g0. In fact, the resolvent difference in (0.4) is a singular Green
operator of order −2 and class 0, when all coefficients are C∞.
Considering another resolvent difference, J. Behrndt, M. Langer, I. Lobanov, V. Lo-
toreichik and I. Popov showed in a recent paper [BLLLP10], on the basis of a theory of
quasi-boundary triples by J. Behrndt and M. Langer [BL07], that when A = −∆ (hence
νu = γ u = n γ ∂ u) and b is a real function in L (∂Ω), the difference between the
1 j j 0 j ∞
resolvent of APand the resolvent of the Neumann realization A satisfies an estimate with
ν
2 replaced by 3, for λ in the intersection of resolvent sets ̺(A)∩̺(A ):
ν
e
e
(0.6) (A−λ)−1 −(A −λ)−1 ∈ C for p > 3/(n−1);
ν p
e
hereCp denotesthespaceofcompactoperatorsT withsingularvaluesequences(sj(T))j∈N ∈
ℓ ; the Schatten class of order p. (Besides real b, also functions with a fixed sign on Imb
p
were treated.)
In the case where b ∈ C∞(∂Ω), this follows from (0.5) since (A−λ)−1 −(A −λ)−1 is
ν
a singular Green operator of order −3, leading to the stronger estimate:
e
(0.7) s (A−λ)−1 −(A −λ)−1 j3/(n−1) → c for j → ∞;
j ν
(cid:0) (cid:1)
e
this was noted also in [G10a], Cor. 8.4 and Ex. 8.5.
The result of [BLLLP10] is more general by treating nonsmooth b. Their main result
Theorem 3.5 is proved in a formulation where the boundary condition is γ u = Θγ u for
0 1
an operator Θ, but it is explained in their Remark 3.7 how this can be made to include
ROBIN PROBLEMS 3
general conditions γ u = bγ u by use of the relations point of view of [BL07]; more on this
1 0
in [BL10].
The main purpose of the present paper is to show spectral asymptotics estimates as in
(0.7) for nonsmooth b. First we show (in Theorem 2.2) that an upper bound
(0.8) s (A−λ)−1 −(A −λ)−1 j3/(n−1) ≤ C for all j,
j ν
(cid:0) (cid:1)
holds for any complex be∈ L (∂Ω); this implies (0.6), for general A as in (0.1). As a
∞
corollary, a similar estimate holds for (A −λ)−1 −(A −λ)−1, when the A are defined
1 2 i
by boundary conditions νu = b γ u with b ∈ L (∂Ω).
i 0 i ∞
e e e
Next, we show (in Theorems 2.5 and 3.4) that asymptotic estimates hold when the b
i
are piecewise slightly better that continuous. Since asymptotic estimates are not additive,
we must aim directly for (A −λ)−1 −(A −λ)−1.
1 2
Theorem 0.1. Assume theat b1, b2 andeb2 −b1 are piecewise Cε on ∂Ω for some ε > 0,
having jumps at C∞ hypersurfaces. Then
(0.9) s (A −λ)−1 −(A −λ)−1) j3/(n−1) → c for j → ∞,
j 1 2
(cid:0) (cid:1)
where c is defined fromethe principalesymbol of A and b −b .
2 1
It suffices in fact that b , b and b −b are piecewise in Hr(∂Ω) for some r > 0 and
1 2 2 1 p
p > (n−1)/r, see the details below.
For the proof of (0.8) the method is, as in [BLLLP10], an application of functional
analysis, building on a general theory of extensions (here Grubb [G68]) together with
known facts on elliptic boundary value problems. The proof of (0.9) in the nonsmooth
situationsdraws on methods andresults for pseudodifferential boundary operatorsin[G84]
and a result on restricted kernels of pseudodifferential operators by Laptev [L77, L81].
As an auxiliary result of independent interest we show (Theorem 3.3) that a product of
classical pseudodifferential operators of negative order on a closed manifold, interspersed
withpiecewisecontinuousfunctionshavingjumpsatasmoothhypersurface, hasaprincipal
spectral asymptotics estimate as in the smooth case. Moreover, we extend (0.4) to b ∈
L (∂Ω) (Theorem 2.4).
∞
Some spectral estimates for resolvent differences in interior and exterior domains have
been described recently by Malamud in [M10], and spectral asymptotics have been shown
in [G10]; both papers treat higher-order operators but do not aim for the special bounds
obtained here. Let us also mention that we do not here address the question of nonsmooth
domains, as e.g. in Gesztesy and Mitrea [GM09, GM09a, GM10] and Abels, Grubb and
Wood [AGW10], [G08], and their references.
To keep the paper short, some introductory material found in other sources will not be
repeated here.
The main details of the extension theory [G68]–[G74] have been recalled and explained
in several recent papers [BGW09], [G08], [G10a]; resulting Krein-type resolvent formulas
are shown in [BGW09].
Sobolev spaces are recalled in numerous places. The basic facts we shall need on these
and other function spaces such as Besov and Bessel-potential spaces, are recalled e.g. in
[AGW10], Sect. 2.
The calculus of pseudodifferential boundary operators is explained in Boutet de Monvel
[B71] and in [G84], [G96], [G09].
4 GERD GRUBB
1. The Robin realization.
Let Ω be a bounded smooth subset of Rn with boundary ∂Ω = Σ, and let
n
(1.1) a(u,v) = (a ∂ u,∂ v)+(a u,v),
jk k j 0
X
j,k=1
be a sesquilinear form with coefficients in C∞(Ω) such that the associated second-order
operator(0.1)isformallyselfadjointandstronglyelliptic. Weassumemoreoverthata(u,u)
is real for u ∈ H1(Ω) and (with c > 0, k ≥ 0)
(1.2) a(u,u) ≥ ckuk2 −kkuk2, for u ∈ H1(Ω).
1 0
This holds if the matrix (a (x))n is real, symmetric and positive definite and a (x) is
jk j,k=1 0
real, at each x ∈ Ω.
Let b ∈ L (Σ), and define the sesquilinear form a by
∞ b
(1.3) a (u,v) = a(u,v)+(bγ u,γ v) .
b 0 0 L2(Σ)
Since kγ uk2 ≤ c′kuk2 ≤ εkuk2 +C(ε)kuk2 for any ε, we infer from (1.2) that
0 L2(Σ) 43 1 0
(1.4) Rea (u,u) ≥ c kuk2 −k kuk2, for u ∈ H1(Ω),
b 1 1 1 0
where c < c is close to c and k ≥ k is a large constant.
1 1
The sesquilinear form a on V = H1(Ω) in H = L (Ω) defines a realization A of A by
b 2
Lions’ version of the Lax-Milgram lemma (as recalled e.g. in [G09], Ch. 12), with domain
e
(1.5) D(A) = {u ∈ H1(Ω)∩D(A ) | (Au,v) = a (u,v) for all v ∈ H1(Ω)}.
max b
e
The operator A is closed, densely defined with spectrum in a sectorial region in
{Reλ ≥ −k }, and its adjoint A∗ is the analogous operator defined from
1 e
e
(1.6) a∗(u,v) = a(v,u)+(bγ u,γ v) .
b 0 0 L2(Σ)
In particular, when b is real, A is selfadjoint.
It will be useful to observe:
e
Lemma 1.1. For any small θ > 0 there is an α ≥ 0 such that the spectrum of A is
contained in the region
e
(1.7) M = {z ∈ C | |Imz| ≤ θ(Rez +α), Rez ≥ −k }.
θ,α,k1 1
Proof. Let K = kImbk . From the inequalities for a (u,u) we see that for u ∈ H1(Ω),
L∞(Σ) b
|Ima (u,u)| = |Im(bγ u,γ u)| ≤ K(εkuk2 +C(ε)kuk2)
b 0 0 1 0
≤ Kεc−1(Rea (u,u)+k kuk2)+KC(ε)kuk2
1 b 1 0 0
= Kεc−1Rea (u,u)+(Kεc−1k +KC(ε))kuk2.
1 b 1 1 0
ROBIN PROBLEMS 5
This (together with (1.4)) shows that for u 6= 0, a (u,u)/kuk2 has its values in M ,
b 0 θ,α,k1
where θ = Kεc−1 can be taken arbitrarily small, α = Kεc−1k +KC(ε). The numerical
1 1 1
ranges of A and A∗ are contained in this set, which then also contains the spectra. (More
details for this kind of argument can be found in [G09], Sect. 12.4.) (cid:3)
e e
The Neumann-type boundary operator (0.2) enters in the “halfways Green’s formula”
(1.8) (Au,v)−a(u,v) = (νu,γ v) ,
0 L2(Σ)
for smooth u and v. It is known e.g. from [LM68] that γ and ν extend to continuous
1
mappings from H1(Ω)∩ D(Amax) to H−12(Σ), such that for u ∈ H1(Ω)∩ D(Amax), v ∈
H1(Ω), (1.8) holds with the scalar product over Σ replaced by the sesquilinear duality
between H−12(Σ) and H21(Σ). Then
(1.9) (Au,v)−a (u,v) = (νu,γ v) −(bγ u,γ u) ,
b 0 H−21(Σ),H21(Σ) 0 0 L2(Σ)
and hence
D(A) = {u ∈ H1(Ω)∩D(Amax) | νu = bγ0u in H−21(Σ)},
representing the Roebin condition νu−bγ u = 0.
0
For b = 0, the condition is νu = 0, defining what we call the Neumann realization
A ; it is selfadjoint with D(A ) ⊂ H2(Ω). It is well-known that when b is smooth, then
ν ν
D(A) ⊂ H2(Σ).
Lemema 1.2. When b ∈ L (Σ), the domain of A satisfies
∞
e
3
D(A) ⊂ H2(Ω)∩D(Amax).
e
Proof. When u ∈ D(A), then u ∈ H1(Ω) implies γ0u ∈ H12(Σ) ⊂ L2(Σ). Multiplication
by b is continuous on L (Σ), so bγ u ∈ L (Σ). Then also νu = bγ u is in L (Σ). By the
2 0 2 0 2
e
ellipticity of the Neumann problem, Au ∈ L2(Ω) with νu ∈ L2(Σ) imply u ∈ H23(Ω). (cid:3)
When b has some smoothness or piecewise smoothness, we can get more regularity: It is
known that when b is in the Bessel potential space Hr(Σ) with r > (n−1)/p, p ≥ 2, then
p
multiplication by b is continuous in Hs(Σ) for |s| ≤ r (cf. e.g. Johnsen [J95]). In relation
to H¨older spaces Cr and Besov spaces Br there are inclusions
p.q
(1.10) Cr+2δ(Σ) ֒→ Br+δ(Σ) ֒→ Br+δ(Σ) ֒→ Hr(Σ), any δ > 0,
∞,2 p,2 p
so also functions in these spaces preserve Hs(Σ) for |s| ≤ r. (A summary of the relevant
facts on function spaces is given e.g. in [AGW10], Sect. 2.) Note that any ε > 0 can be
included as an r +2δ by taking r ∈]0,ε[, δ = (ε−r)/2 and p > (n−1)/r.
When X(Σ) is a function space over Σ, we say that b is piecewise in X, when the
(n−1)-dimensional manifold Σ is a union Σ ∪···∪Σ of smooth subsets Σ with disjoint
1 J j
interiors (such that the interfaces are smooth (n − 2)-dimensional manifolds), and there
are functions b ∈ X(Σ), such that b equals b on the interior Σ◦, for j = 1,...,J.
j j j
It is well-known that multiplication by 1 is continuous on Hs(Σ) for all |s| < 1.
Σj 2
6 GERD GRUBB
Proposition 1.3.
1◦ Let b ∈ Hr(Σ) with r > (n − 1)/p, p ≥ 2 (it holds if b is in one of the spaces in
p
(1.10)). Then D(A) ⊂ H32+r(Ω) if r < 1, D(A) ⊂ H2(Ω) if r ≥ 1.
2 2
2◦ Let b be pieceewise in Hpr(Σ) with r > (ne−1)/p, p ≥ 2. Then D(A) ⊂ H32+r(Ω) if
r < 1, D(A) ⊂ H2−ε(Ω) for any ε > 0 if r ≥ 1.
2 2 e
Proof. As ealready noted, u ∈ H1(Ω) implies γ0u ∈ H21(Σ). In the case 1◦, multiplication
by b preserves Hs(Σ) for |s| ≤ r, so bγ0u ∈ Hmin{r,12}(Σ). Then also νu = bγ0u is in
Hmin{r,21}(Σ), and now Au ∈ L2(Ω) with νu ∈ Hmin{r,12}(Σ) imply u ∈ H23+r(Ω) if r < 21,
u ∈ H2(Ω) if r ≥ 1, by the ellipticity of the Neumann problem.
2
In the case 2◦, since b = J b 1 , multiplication by b maps Hr(Σ) into itself if
j=1 j Σj
r < 1, and into H12−ε, any εP> 0, if r ≥ 1. Completing the proof as under 1◦, we find
2 2
that u ∈ H23+r(Ω) if r < 1, u ∈ H2−ε(Ω) if r ≥ 1. (cid:3)
2 2
Let us regard A from the point of view of the general extension theory of [G68], as
recalled in [BGW09], [G08], [G10a].
e
WetaketheDirichletrealizationA asthereferenceoperator,assumedtohaveapositive
γ
lower bound. (Seen from the point of view of [G68], [BL07] uses instead the Neumann
realization A as the reference operator.) The operator A corresponds, by the general
ν
theory of [G68], to a closed densely defined operator T:V → W, where V and W are
e
closed subsets of Z = kerA , and D(T) is dense in V; and this in turn is carried over
max
by use of the homeomorphism γ0 : Z →∼ H−12(Σ), to a closed operator L : X → Y∗, with
domain D(L) dense inX, where X and Y areclosed subspaces ofH−12(Σ). Here X = γ0V,
Y = γ W and D(L) = γ D(T) = γ D(A).
0 0 0
Proposition 1.4. The operator L : Xe→ Y corresponding to A by [G68] has X = Y =
H−21(Σ), and acts like b−P0 with a domain contained in H1(Σ). When b is real, L is
γ,ν e
selfadjoint as an unbounded operator from H−12(Σ) to H21(Σ).
Proof. Besides the description referred to above, we shall use the observations on operators
defined by sesquilinear forms worked out in [G70] (and partly recalled in [G09], Ch. 13.2,
see in particular Th. 13.19). Since the domain of a (u,v) equals H1(Ω), T is defined from
b
a sesquilinear form t(z,w) with domain H1(Ω)∩Z dense in Z, and hence V = W = Z. It
follows that X = Y = H−21(Σ), and L is densely defined and closed as an operator from
H−21(Σ) to H12(Σ). The adjoint L∗ is of the same type and corresponds to A∗. When b is
real, A is selfadjoint as noted above; then L is selfadjoint.
e
In the interpretation of the extension theory, A represents the boundary condition
e
e
γ u ∈ D(L), Γu = Lγ u;
0 0
where Γu = νu−P0 γ u, so Lγ u = νu−P0 γ u when u ∈ D(A). (Pλ is the operator
γ,ν 0 0 γ,ν 0 γ,ν
mappingDirichletboundaryvaluestoNeumannboundaryvaluesforsolutionsof(A−λ)u =
e
0; more on this below.) Since the functions in D(A) also satisfy νu = bγ u, we see that L
0
acts like
e
Lϕ = (b−P0 )ϕ.
γ,ν
ROBIN PROBLEMS 7
By Lemma 1.2, D(A) ⊂ H23(Σ), so D(L) = γ0D(A) ⊂ H1(Σ). (cid:3)
When we replace Aeby A− λ, where λ is in the reesolvent set ̺(A ) of A , we get for
γ γ
the corresponding operator Lλ:
Lλ acts like b−Pλ , with D(Lλ) = D(L) ⊂ H1(Σ).
γ,ν
For λ ∈ ̺(A )∩̺(A), there holds a Krein resolvent formula (shown in [BGW09], Th.
γ
3.4):
e
(1.11) (A−λ)−1 = (A −λ)−1 +Kλ(Lλ)−1(Kλ¯)∗.
γ γ γ
e
Here Kλ is the Poisson operator for the Dirichlet problem, i.e. the solution operator
γ
Kλ:ϕ 7→ u for the problem
γ
(A−λ)u = 0 on Ω, γ u = ϕ on Σ;
0
it maps Hs−12(Σ) → Hs(Ω) continuously for all s, and the adjoint (Kγλ)∗ maps e.g. L2(Ω)
1
to H2(Σ).
We can use this to show a spectral estimate for (A − λ)−1 − (A − λ)−1, going via
ν
differences with the Dirichlet resolvent. The argumentation is not the same as that of
e
[BLLLP10], which uses a Krein formula based on the Poisson operator for the Neumann
problem.
The spectrum of A is contained in a positive halfline [c ,∞[, and the spectrum of A
γ 0 ν
is contained in a larger halfline ] − k,∞[, cf. (1.2). For λ ∈ C\] − k,∞[, the Dirichlet-
to-Neumann operator Pλ = νKλ is a homeomorphism from Hs(Σ) to Hs−1(Σ) for all
γ,ν γ
s ∈ R, with inverse Pλ , the Neumann-to-Dirichlet operator. Then we can write
ν,γ
(1.12) Lλϕ = (b−Pλ )ϕ = (bPλ −1)Pλ ϕ, for ϕ ∈ D(L).
γ,ν ν,γ γ,ν
Since Pλ is of order −1, it is compact in L (Σ). Then bPλ −1 is a Fredholm operator
ν,γ 2 ν,γ
in L (Σ), as noted also in [BLLLP10]. If λ is such that: (1) Lλ is invertible (from D(L)
2
to H21(Σ)), (2) bPνλ,γ −1 is invertible in L2(Σ), then the inverse of Lλ must coincide with
the inverse of (bPλ −1)Pλ on H21(Σ).
ν,γ γ,ν
For bPλ − 1, we get invertibility as follows: We have as a simple application of the
ν,γ
principles in [G96] (cf. Th. 2.5.6, (A.25–26)) that
kPγλ,νϕkHs,µ(Σ) ≃ kϕkHs+1,µ(Σ), kϕkHs−1,µ(Σ) ≃ kPνλ,γϕkHs,µ(Σ),
uniformly in µ = |λ|21 for λ → ∞ on rays in C \ R+; this holds since Pγλ,ν is parameter-
elliptic of order 1 and regularity +∞ on the rays in C\R . In particular, one has on such
+
a ray {λ = µ2eiη} with η ∈]0,2π[, for s ∈ [0,1] and µ ≥ 1,
kPνλ,γϕkHs(Σ) +hµiskPνλ,γϕkL2(Σ) ≤ Cmin{kϕkHs−1(Σ),hµis−1kϕkL2(Σ)},
8 GERD GRUBB
sothenormofPλ inL (Σ)isO(hµi−1)ontheray. Takeµ solargethatkbPλ k ≤
ν,γ 2 0 ν,γ L(L2(Σ))
δ < 1 for µ ≥ µ , then bPλ −1 is invertible as an operator in L (Σ) for µ ≥ µ , with a
0 ν,γ 2 0
bounded inverse (bPλ −1)−1:
ν,γ
∞
(1.13) (bPλ −1)−1 = −1− (bPλ )k, converging in L(L (Σ)).
ν,γ ν,γ 2
X
k=1
Then b−Pλ has an inverse
γ,ν
(1.14) (b−Pλ )−1 = Pλ (bPλ −1)−1.
γ,ν ν,γ ν,γ
For Lλ we know from the extension theory that Lλ is bijective from D(L) to H12(Σ) if
and only if λ ∈ ̺(A). It follows from Lemma 1.1 by a simple geometric consideration that
for each ray {λ = µ2eiη} with η ∈]0,2π[, there is a µ such that such that λ ∈ ̺(A) for
e 1
µ ≥ µ .
1
e
For µ ≥ max{µ ,µ }, both (1) and (2) are satisfied, so then
0 1
(1.15) (Lλ)−1 = (b−Pλ )−1 = Pλ (bPλ −1)−1 on H12(Σ).
γ,ν ν,γ ν,γ
We note in particular that
(1.16) D(Lλ) = {ϕ ∈ H1(Σ) | (b−Pλ )ϕ ∈ H21(Σ)},
γ,ν
for such λ. Now D(L) = D(Lλ), and P0 −Pλ is bounded from H−21(Σ) to H21(Σ) (cf.
γ,ν γ,ν
[BGW09], Rem. 3.2), so we conclude that
(1.17) D(L) = {ϕ ∈ H1(Σ) | (b−P0 )ϕ ∈ H12(Σ)}.
γ,ν
It follows moreover that (1.16) holds for all λ ∈ ̺(A ).
γ
This shows the main part of:
Theorem 1.5. The domain of L satisfies (1.17), and it is also described by (1.16) for any
λ ∈ ̺(A ).
γ
On each ray in C\R , λ is in ̺(A) and (1.15) holds for |λ| sufficiently large. For such
+
λ,
e
(1.18) (A−λ)−1 −(A −λ)−1 = KλP (bPλ −1)−1(Kλ¯)∗.
γ γ ν,γ ν,γ γ
e
Proof. The statements before formula (1.18) were accounted for above, and the formula
follows by insertion of (1.15) in (1.11). (cid:3)
ROBIN PROBLEMS 9
2. Spectral estimates.
Spectral estimates for resolvent differences will now be studied. A classical reference for
the basic concepts is the book of Gohberg and Krein [GK69]; some particularly relevant
facts were collected in [G84], supplied with additional results. We shall include a short
summary here:
For p > 0, the space C is the Schatten class of compact linear operators T (in a Hilbert
p
space H) with singular value sequences (sj(T))j∈N ∈ ℓp, and Sp denotes the quasi-normed
space of compact operators T with s (T) = O(j−1/p); here S ⊂ C for all ε > 0.
j p p+ε
The rules shown by Ky Fan [F51]
(2.1) s (T +T′) ≤ s (T)+s (T′), s (TT′) ≤ s (T)s (T′),
j+k−1 j k j+k−1 j k
imply that C and S are vector spaces, and that a product rule holds:
p p
(2.2) Sp ·Sq ⊂ S1/(p−1+q−1), Cp ·Cq ⊂ C1/(p−1+q−1).
Moreover, the rule
(2.3) s (ATB) ≤ kAks (T)kBk
j j
implies that S and C are preserved under compositions with bounded operators. We
p p
mention two perturbation results:
Lemma 2.1.
1◦ If s (T)j1/p → C and s (T′)j1/p → 0 for j → ∞, then s (T + T′)j1/p → C for
j 0 j j 0
j → ∞.
2◦ If T = T + T′ for each M ∈ N, where s (T )j1/p → C for j → ∞ and
M M j M M
s (T′ )j1/p ≤ ε for j ∈ N, with C → C and ε → 0 for M → ∞, then s (T)j1/p →
j M M M 0 M j
C for j → ∞.
0
The statement in 1◦ is the Weyl-Ky Fan theorem (cf. e.g. [GK69] Th. II 2.3), and 2◦ is
a refinement shown in [G84], Lemma 4.2.2◦.
We also recall that when Ξ is a compact n′-dimensional smooth manifold (possibly with
boundary) and T is a bounded linear operator from L (Ξ) to Ht(Ξ) for some t > 0, then
2
T ∈ Sn′/t as an operator in L2(Ξ), with
′
(2.4) s (T)jt/n ≤ CkTk ,
j L(L2,Ht)
C depending only on Ξ and t. See [G84], Lemma 4.4ff. for references.
The Poisson operator Kλ is continuous from Hs−21(Σ) to Hs(Ω) for all s ∈ R, and its
γ
adjoint Kλ∗ is a trace operator of class 0 and order −1 in the pseudodifferential boundary
γ
operator calculus, hence is continuous from Hs(Ω) to Hs+12(Σ) for s > −1. Then the
2
composition Kλ∗Kλ is continuous from L2(Σ) to H1(Σ), so in view of (2.4), Kλ∗Kλ ∈
γ γ γ γ
S and hence Kλ ∈ S , as operators in L (Σ). The singular numbers of Kλ∗
n−1 γ (n−1)/(1/2) 2 γ
have the same behavior. Moreover, since Pλ is a pseudodifferential operator of order −1
ν,γ
on Σ, it lies in S when considered as an operator in L (Σ).
n−1 2
10 GERD GRUBB
Theorem 2.2. Let b ∈ L (Σ). For any λ ∈ ̺(A)∩̺(A ),
∞ ν
e
(2.5) (A−λ)−1 −(A −λ)−1 ∈ S .
ν (n−1)/3
e
Proof. First assume thatλliesso far out ona ray inC\R that thestatements inTheorem
+
1.5 are valid.
Applying (1.18) to our A and also to the case b = 0 (the Neumann realization), we find
by subtraction:
e
(A−λ)−1 −(A −λ)−1 = (A−λ)−1 −(A −λ)−1 −((A −λ)−1 −(A −λ)−1)
ν γ ν γ
(2.6) e = KeλPλ [(bPλ −1)−1 +1]Kλ¯∗
γ ν,γ ν,γ γ
= KλPλ (bPλ −1)−1bPλ Kλ¯∗.
γ ν,γ ν,γ ν,γ γ
The last expression is composed of the operator Kλ in S , the adjoint of Kλ¯ with
γ (n−1)/(1/2) γ
the same property, two factors Pλ in S and the bounded operators (bPλ −1)−1 and
ν,γ n−1 ν,γ
b, so it belongs to S , by (2.2).
(n−1)/3
Now let λ′ be an arbitrary number in ̺(A) ∩ ̺(A ). We use the following refined
ν
resolvent identity as in [BLLLP10]:
e
(2.7) (S −λ′)−1 −(T −λ′)−1
= (1+(λ′ −λ)(T −λ′)−1)((S −λ)−1 −(T −λ)−1)(1+(λ′ −λ)(S −λ′)−1),
valid for λ,λ′ ∈ ̺(T) ∩ ̺(S). Applying it to S = A and T = A for λ as above and
ν
λ′ ∈ ̺(A)∩̺(A ), we find that (A−λ′)−1 −(A −λ′)−1 is a composition of an operator
ν ν e
in S with two bounded operators, hence lies in S , as was to be shown. (cid:3)
(n−e1)/3 e (n−1)/3
There is an obvious corollary:
Corollary 2.3. Let b ,b ∈ L (Σ), and denote the corresponding realizations of Robin
1 2 ∞
conditions νu = b γ u resp. νu = b γ u by A resp. A . For any λ ∈ ̺(A )∩̺(A ),
1 0 2 0 1 2 1 2
e e e e
(2.8) (A −λ)−1 −(A −λ)−1 ∈ S .
1 2 (n−1)/3
e e
Proof. Write (A −λ)−1 −(A −λ)−1 as the difference between (A −λ)−1 −(A −λ)−1
1 2 1 ν
and (A −λ)−1 −(A −λ)−1, then the result follows from Theorem 2.2 (and (2.7)) since
2 e γ e e
S is a vector space. (cid:3)
p
e
Formula (1.18) also allows us to show a spectral asymptotics estimate for (A−λ)−1 −
(A −λ)−1 that was obtained in the smooth case for selfadjoint realizations and negative
γ
e
λ in Grubb [G74], Sect. 8, and Birman and Solomiak [BS80]. In the former paper it is
shown, also for 2m-order problems, that the operator is, on the orthogonal complement of
its nullspace, isometric to an elliptic pseudodifferential operator on Σ of order −2m (which
has the asserted spectral asymptotics); in the latter paper exterior domains are included.