Weyl's spectral asymptotic formula for Dirichlet Kohn-Laplacian
Gespeichert in:
| Datum: | 1999 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут прикладної математики і механіки НАН України
1999
|
| Schriftenreihe: | Нелинейные граничные задачи |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/169265 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Weyl's spectral asymptotic formula for Dirichlet Kohn-Laplacian / Yu.A. Alkhutov, V.V. Zhikov // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 8-12. — Бібліогр.: 3 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-169265 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1692652020-06-10T01:26:25Z Weyl's spectral asymptotic formula for Dirichlet Kohn-Laplacian Alkhutov, Yu.A. Zhikov, V.V. 1999 Article Weyl's spectral asymptotic formula for Dirichlet Kohn-Laplacian / Yu.A. Alkhutov, V.V. Zhikov // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 8-12. — Бібліогр.: 3 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/169265 en Нелинейные граничные задачи Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| format |
Article |
| author |
Alkhutov, Yu.A. Zhikov, V.V. |
| spellingShingle |
Alkhutov, Yu.A. Zhikov, V.V. Weyl's spectral asymptotic formula for Dirichlet Kohn-Laplacian Нелинейные граничные задачи |
| author_facet |
Alkhutov, Yu.A. Zhikov, V.V. |
| author_sort |
Alkhutov, Yu.A. |
| title |
Weyl's spectral asymptotic formula for Dirichlet Kohn-Laplacian |
| title_short |
Weyl's spectral asymptotic formula for Dirichlet Kohn-Laplacian |
| title_full |
Weyl's spectral asymptotic formula for Dirichlet Kohn-Laplacian |
| title_fullStr |
Weyl's spectral asymptotic formula for Dirichlet Kohn-Laplacian |
| title_full_unstemmed |
Weyl's spectral asymptotic formula for Dirichlet Kohn-Laplacian |
| title_sort |
weyl's spectral asymptotic formula for dirichlet kohn-laplacian |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/169265 |
| citation_txt |
Weyl's spectral asymptotic formula for Dirichlet Kohn-Laplacian / Yu.A. Alkhutov, V.V. Zhikov // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 8-12. — Бібліогр.: 3 назв. — англ. |
| series |
Нелинейные граничные задачи |
| work_keys_str_mv |
AT alkhutovyua weylsspectralasymptoticformulafordirichletkohnlaplacian AT zhikovvv weylsspectralasymptoticformulafordirichletkohnlaplacian |
| first_indexed |
2025-07-15T04:01:29Z |
| last_indexed |
2025-07-15T04:01:29Z |
| _version_ |
1837684059650654208 |
| fulltext |
WEYL’S SPECTRAL ASYMPTOTIC FORMULA
FOR DIRICHLET KOHN–LAPLACIAN
c© Yu.A.Alkhutov, V.V.Zhikov
1. Introduction
The counting function N(λ) = N(λ, Ω) of the Dirichlet Laplacian on a bounded open
set Ω ⊂ Rd is a defined as the number of eigenvalues less than a given λ. The problem
of the asymptotic behaviour of the counting function as λ → +∞ has been extensively
studied by mathematicians and physicists almost 100 years. The first mathematical
results in this direction belong to H.Weyl who showed in 1911 that for domains with
smooth boundaries
N(λ, Ω) ∼ (2π)−dωd|Ω|d/2 as λ → +∞, (1)
where ωd is the volume of the unit ball in Rd. Formula (1) was then extended to
arbitrary open sets in Rd with finite volume and generalized to higher-order elliptic
operators with constant coeficients, see [1], [2].
Our aim is the corresponding spectral asymptotics for the Kohn-Laplacian ∆H asso-
ciated to the Heisenberg group. This operator ∆H is of Hörmander type, not strongly
elliptic, and invariant with respect to translations on the Heisenberg group. More ex-
actly, we consider the eigenvalue problem
−∆Hu = λu, u|∂Ω = 0, (2)
where Ω is a bounded domain in the odd-dimensional space R2n+1, and get the asymp-
totic formula
N(λ, Ω) ∼ Cn|Ω|λds/2, ds = 2(n + 1). (3)
The exponent ds (> d = 2n+1) depends on n and we say ds is the spectral dimension
relative to our problem.
2. Preliminaries
Let us recall the definition of the operator ∆H in dimension d = 3. The definitions
and statements in any odd dimension d = 2n + 1 will be given later.
Consider the two linear operators X1 and Y 1:
X1 =
∂
∂x
+ 2y
∂
∂z
, Y 1 =
∂
∂y
− 2x
∂
∂z
This research was partially supported by the Russian Foundation for Basic Research under grants
No 96 - 01 - 00443 and No 96 - 01 - 00503
and introduce the gradient ∇H by
∇H = (X1, Y 1) = σ∇,
where ∇ is the standard gradient: ∇ =
(
∂
∂x
, ∂
∂y
, ∂
∂z
)
, and σ is the following matrix
σ =
(
1 0 2y
0 1 −2x
)
.
Then the operator ∆H is given by
∆H = (X1)2 + (Y 1)2 =
=
∂2
∂x2 +
∂2
∂y2 + (4y2 + 4x2)
∂2
∂z2 + 4y
∂2
∂z∂x
− 4x
∂2
∂z∂y
= div(σT σ∇),
where
σT σ =
1 0 2y
0 1 −2x
2y −2x 4y2 + 4x2
.
The operator ∆H is elliptic (i.e. σT σξ · ξ ≥ 0 for any ξ ∈ R3) but clearly not strongly
elliptic, because the first eigenvalue of σT σ is zero and the rank of σT σ is two in every
point. However we have the following condition on commutator
[X1, Y 1] = X1Y 1 − Y 1X1 = −4
∂
∂z
. (4)
As a consequence of (4), ∆H is an Hörmander type operator, and enjoys nice properties
like hipoellipticity, subelliptic estimates, the maximum principle, Poincaré’s inequality.
The space R3 becomes a group if the group law + define as following:
for vectors ξ = (x, y, z), ξ′ = (x′, y′, z′) we set
ξ′ + ξ = (x + x′, y + y′, z + z′ − 2(x′y − xy′)).
Notice that ξ + ξ′ 6= ξ′ + ξ and the Lebesgue measure is invariant with respect to
these right or left translations. The operator ∆H is invariant with respect to the left
translations, i.e. for fixed ξ′,
∆H(u(ξ′ + ·)) = (∆H(u))(ξ′ + ·).
The similar definitions can be given for any odd-dimensional space R2n+1. Let ξ =
(x1, x2, . . . xn, y1, y2, . . . yn, z) = (x, y, z), where x, y ∈ Rn. Consider the operators
Xj =
∂
∂xj
+ 2yj ∂
∂z
, Y j =
∂
∂yj
− 2xj ∂
∂z
, j = 1, 2, . . . n,
and set
∇H = (X1, X2, . . . Xn, Y 1, Y 2, . . . Y n),
∆H =
n∑
j=1
(Xj)2 + (Y j)2.
Then all properties of ∆H remain the same.
3. Counting function of the Dirichlet Kohn-Laplacian
Let Ω be a bounded domain in R2n+1.
We denote by D0
H(Ω) the closure of C∞0 (Ω) with respect to the norm
∫
Ω
|∇Hv|2dξ +
∫
Ω
v2dξ
1/2
.
The Poincaré inequality
∫
Ω
v2dξ ≤ C(Ω)
∫
Ω
|∇Hv|2dξ for any v ∈ C∞0 (Ω),
and Lax-Milgram lemma give the unique solvability of the problem:
{ −∆Hu = f in Ω,
u|∂Ω = 0,
(5)
where f ∈ L2(Ω), i.e. the existence and the uniqueness of a function u ∈ D0
H(Ω) such
that ∫
Ω
∇Hu · ∇Hϕdξ =
∫
Ω
fϕdξ for any ϕ ∈ C∞0 (Ω).
Consider the collection of all the solutions of problem (5) for f varying in L2(Ω).
This set is a domain of −∆H as a positive self-adjoint operator in L2(Ω). By definition
we have ∫
Ω
(−∆H)uϕdξ =
∫
Ω
∇Hu · ∇Hϕdξ for any ϕ ∈ D0
H(Ω).
Remark that the inverse operator (−∆H)−1 is compact. It is clearly from the following
subelliptic estimate:
‖ϕ‖H1/2(Ω) ≤ C
∫
Ω
|∇Hv|2dξ +
∫
Ω
v2dξ
1/2
for any ϕ ∈ C∞0 (Ω),
where ‖ · ‖H1/2(Ω) is the classical H1/2 norm.
So for any bounded open set, the spectrum of −∆H consists of a countable sequence
of positive eigenvalues λj(Ω)(j = 1, 2, . . . ):
0 < λ1(Ω) ≤ λ2(Ω) ≤ . . . ≤ λj(Ω) ≤ . . . , λj(Ω) →∞ as j →∞.
Definition. Let λ be a given positive number. We denote by L(λ) = N(λ, Ω) the
number of eigenvalues less than λ.
The function N(λ,Ω) is called the counting function of the Dirichlet Kohn-Laplacian
on Ω.
Let us formulate the main result.
Theorem. Assume Ω is measurable in the sense of Jordan. Then asymptotic
relation (3) holds with
Cn =
1
(n + 1)Γ(n + 1)(4π)n+1
∞∫
0
(
Θ
shΘ
)n
dΘ,
where Γ(α) is the Euler gamma-function.
Clearly that it is sufficient to proof formula (3) for smooth domains Ω only.
4. Sketch of the proof
We apply Carleman’s analytic aproach or ”parabolic equation method”.
Let K(ξ, ξ′, t) be a fundamental solution associated to the parabolic operator
∂
∂t
−∆H .
One can prove the following properties:
K(t, ξ, ξ) = K(t, 0, 0) =
1
(4πt)n+1
∞∫
0
(
Θ
shΘ
)n
dΘ ÷ An
tn+1 (6)
0 ≤ K(t, ξ, ξ′) ≤ c1
tn+1 exp(−c2
t
ρ2(ξ, ξ′)), (7)
where c1, c2 > 0 and
ρ(ξ, ξ′) =
=
[
((x− x′)2 + (y − y′)2)2 + (z − z′ − 2(x′ · y − x · y′))2]1/4
.
By G(ξ, ξ′, t) (ξ, ξ′ ∈ Ω) denote a Green function of the parabolic problem
{
∂u
∂t
−∆Hu = 0 in Ω× (0,∞),
u|∂Ω = 0.
Then G is continuous on Ω×Ω× (0,∞); moreover, from estimate (7) and the maxi-
mum principle we have
G(ξ, ξ, t) ≤ Γ(ξ, ξ, t) for any ξ ∈ Ω, t > 0,
Γ(ξ, ξ) ≤ G(ξ, ξ) + c(δ)t if ξ ∈ Ω and ρ(ξ, ∂Ω) ≥ δ > 0.
It follows that
An |Ω|
tn+1 =
∫
Ω
Γ(ξ, ξ, t)dξ ∼
∫
Ω
G(ξ, ξ, t)dξ as t → +0.
Let ϕj(ξ) be a eigenfunction corresponding to the eigenvalue λj and normalized by∫
Ω
ϕ2
jdξ = 1. Then we have
ϕj ∈ C∞(Ω),
∑
λj<λ
ϕ2
j (ξ) ≤ Cλn+1,
G(ξ, ξ′, t) =
∞∑
j=1
e−λjtϕj(ξ)ϕj(ξ′).
As a result, we obtain the important relation
∞∫
0
e−λtdN(λ) =
∞∫
0
G(ξ, ξ, t)dξ ∼ An |Ω|
tn+1 as t → +0. (8)
Now it is sufficient to apply the classical Tauberian theorem of Hardy-Littelwood.
Tauberian Theorem (see [3]). Assume that N(λ) is a nondecreasing function on
[0,∞) and
∞∫
0
e−λtdN(λ) < ∞ for any t > 0.
Then the relations
N(λ) ∼ cλα as λ →∞ (α > 0),
∞∫
0
e−λtdN(λ) ∼ αΓ(α)c
tα
as t → +0
are equivalent.
Now from (6), (8) we get asymptotic formula (3).
References
1. Weil H., Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichun-
gen (mit einer Anwendung auf die Theorie Hohl- raumstrahlung), Math. Ann. 71 (1912), 441-479.
2. Hörmander L., The Analysis of Linear Partial Differential Operators, vol. III, Springer Verlag, 1985.
3. Feller W., An Introduction to Probability Theory and its Applications, vol. II, John Wiley & Sons
Inc., New York-London-Sydney-Toronto, 1971.
Vladimir State Pedagogical University,
Department of Math.,
prospect Stroiteley 11,
Vladimir, 600024, Russia
E-mail: alkhutov@vgpu.elcom.ru, zhikov@vgpu.elcom.ru
|