Estimates for weighted eigenvalues of fourth-order elliptic operator with variable coefficients
We investigate the Dirichlet weighted eigenvalue problem for a fourth-order elliptic operator with variable coefficients in a bounded domain in $R^n$. We establish a sharp inequality for its eigenvalues. It yields an estimate for the upper bound of the $(k + 1)$-th eigenvalue in terms of the first...
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| author | Sun, He-Jun Сун, Хе-Джун |
| author_facet | Sun, He-Jun Сун, Хе-Джун |
| author_sort | Sun, He-Jun |
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| datestamp_date | 2020-03-18T19:36:24Z |
| description | We investigate the Dirichlet weighted eigenvalue problem for a fourth-order elliptic operator with variable coefficients in a bounded domain in $R^n$.
We establish a sharp inequality for its eigenvalues. It yields an estimate for the upper bound of the
$(k + 1)$-th eigenvalue in terms of the first $k$ eigenvalues.
Moreover, we also obtain estimates for some special cases of this problem. In particular, our results generalize the Wang -Xia
inequality (J. Funct. Anal. - 2007. - 245) for the clamped plate problem to a fourth-order elliptic operator with variable coefficients. |
| first_indexed | 2026-03-24T02:30:09Z |
| format | Article |
| fulltext |
UDC 517.9
He-Jun Sun (College Sci., Nanjing Univ. Sci. and Technol., China)
ESTIMATES FOR WEIGHTED EIGENVALUES
OF FOURTH-ORDER ELLIPTIC OPERATOR
WITH VARIABLE COEFFICIENTS*
ОЦIНКИ ЗВАЖЕНИХ ВЛАСНИХ ЗНАЧЕНЬ
ЕЛIПТИЧНОГО ОПЕРАТОРА ЧЕТВЕРТОГО ПОРЯДКУ
IЗ ЗМIННИМИ КОЕФIЦIЄНТАМИ
We investigate the Dirichlet weighted eigenvalue problem for a fourth-order elliptic operator with variable
coefficients in a bounded domain in Rn. We establish a sharp inequality for its eigenvalues. It yields an
estimate for the upper bound of the (k + 1)-th eigenvalue in terms of the first k eigenvalues. Moreover, we
also obtain estimates for some special cases of this problem. In particular, our results generalize the Wang – Xia
inequality (J. Funct. Anal. – 2007. – 245) for the clamped plate problem to a fourth-order elliptic operator with
variable coefficients.
Дослiджено задачу Дiрiхле про зваженi власнi значення для елiптичного оператора четвертого порядку
iз змiнними коефiцiєнтами в обмеженiй областi iз Rn. Встановлено точну нерiвнiсть для її власних
значень, з якої випливає оцiнка для верхньої межi (k + 1)-го власного значення через першi k власних
значень. Також отримано оцiнки для цiєї задачi у деяких окремих випадках. Зокрема, нашi результати
узагальнюють нерiвнiсть Ванга – Ксi (J. Funct. Anal. – 2007. – 245) для затиснутої пластини на випадок
елiптичного оператора четвертого порядку iз змiнними коефiцiєнтами.
1. Introduction. As we know, there have been some remarkable results about estimates
for eigenvalues of elliptic operators with constant coefficients such as the Laplacian ∆,
the biharmonic operator ∆2 and so on. For the Dirichlet Laplacian problem, we refer
to [2, 3, 6, 8, 10, 18, 20] and the excellent survey [1] of Ashbaugh. Let us give a
brief survey on some results about the Dirichlet eigenvalue problem of the biharmonic
operator (also called the clamped plate problem):
∆2u = λu, in Ω,
u|∂Ω =
∂u
∂ν
∣∣∣∣
∂Ω
= 0,
(1.1)
where Ω is a bounded domain in Rn with piecewise smooth boundary ∂Ω and ν denotes
the outward unit normal vector to ∂Ω. Let λr be the r-th eigenvalue of problem (1.1).
In their pioneering work [16], Payne, Pólya and Weinberger proved
λk+1 − λk ≤
8(n+ 2)
n2
1
k
k∑
r=1
λr. (1.2)
In 1984, Hile and Yeh [11] generalized (1.2) to
n2k3/2
8(n+ 2)
(
k∑
r=1
λr
)−1/2
≤
k∑
r=1
λ
1/2
r
λk+1 − λr
*This work was supported by the National Natural Science Foundation of China (Grant No. 11001130) and
the NUST Research Funding (Grant No. 2010ZYTS064).
c© HE-JUN SUN, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 7 999
1000 HE-JUN SUN
by using an improved method of Hile and Protter [8]. In 1990, Hook [9], Chen and Qian
[4] independently obtained
n2k2
8(n+ 2)
≤
k∑
r=1
λ
1/2
r
λk+1 − λr
k∑
r=1
λ1/2
r .
In 2006, Cheng and Yang [5] established a universal inequality
λk+1 ≤
1
k
k∑
r=1
λr +
[
8(n+ 2)
n2
]1/2
1
k
k∑
r=1
[
λr(λk+1 − λr)
]1/2
.
This also gave an affirmative answer for a question introduced by Ashbaugh in [1]. In
2007, Wang and Xia [19] further derived a sharper inequality
k∑
r=1
(λk+1 − λr)2 ≤ 8(n+ 2)
n2
k∑
r=1
(λk+1 − λr)λr (1.3)
for an n-dimensional complete minimal submanifold in a Euclidean space.
Elliptic operators with variable coefficients are also very important in analysis and
applications (see [7, 12]). However, there have been fewer references on estimates for
eigenvalues of elliptic operators with variable coefficients. To the author’s knowledge,
Hook [9], Qian and Chen [15], Sun [17] considered second order elliptic operators with
variable coefficients and obtained some inequalities of eigenvalues.
For simplicity, we use the following notations:
Di =
∂
∂xi
, Dij =
∂2
∂xi∂xj
.
In this paper, we are concerned about the Dirichlet weighted eigenvalue problem of
fourth-order elliptic operator
∑n
i,j=1
Dij
(
aij(x)Dij
)
with variable coefficients aij(x),
which is described by
n∑
i,j=1
Dij
(
aij(x)Diju
)
= Λρu, in Ω,
u|∂Ω =
∂u
∂ν
∣∣∣∣
∂Ω
= 0,
(1.4)
where ρ is a positive continuous function (also called the density) on Ω and the functions
aij(x) = aji(x) ∈ C2(Ω) for i, j = 1, . . . , n.
A prototype of fourth-order elliptic operator
∑n
i,j=1
Dij
(
aij(x)Dij
)
is the bihar-
monic operator ∆2. Namely, if ρ(x) ≡ 1 and aij(x) ≡ 1 for i, j = 1, . . . , n simulta-
neously, problem (1.4) becomes problem (1.1). Moreover, since the weight function ρ
denotes the density in applications, problem (1.4) have more applications. For exam-
ple, weighted estimates are intelligent in some filtering and identification problems (cf.
[13, 14]).
The goal of this paper is to obtain some estimates for eigenvalues of problem (1.4).
In Theorem 2.1, we establish a sharp inequality for its eigenvalues. Noticing that (2.1) is
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 7
ESTIMATES FOR WEIGHTED EIGENVALUES OF FOURTH-ORDER . . . 1001
a quadratic inequality of Λk+1, we give an estimate for the upper bound of the (k+1)-th
eigenvalue Λk+1 in terms of the first k eigenvalues in Theorem 2.2. From these results,
we can find the influences of variable coefficients aij(x) and the weight function ρ(x) on
estimates of eigenvalues of problem (1.4). Furthermore, we derive some brief estimates
for eigenvalues of some special cases in Corollaries 2.1 – 2.3. Our results generalize and
extend some previous results for the clamped plate problem. In particular, inequality
(1.3) of Wang and Xia [19] is a corollary of Theorem 2.1.
2. Results and their proofs.
Theorem 2.1. Let Λr be the r-th weighted eigenvalue of problem (1.4). Denote
by
σ =
(
minx∈Ω ρ(x)
)−1
, τ =
(
maxx∈Ω ρ(x)
)−1
and
ζ = maxx∈Ω
[∑n
l=1
|Dl(all(x))|2
]1/2
.
Suppose that the functions aij(x) satisfy:
0 < ξ ≤ aij(x) ≤ η, i, j = 1, . . . , n,
where ξ and η are two positive constants. Then we have
k∑
r=1
(Λk+1 − Λr)2 ≤
≤ 8(n+ 2)σ2η
n2τ2ξ
k∑
r=1
(Λk+1 − Λr)Λr +
8σ
9
4 ζ
n2τ2ξ3/4
k∑
r=1
(Λk+1 − Λr)Λ3/4
r . (2.1)
Proof. Denote by ur the r-th weighted orthonormal eigenfunction of problem (1.4)
corresponding to eigenvalues Λr, r = 1, 2, . . . , k. Namely ur satisfies
n∑
i,j=1
Dij
(
aij(x)Dijur
)
= Λrρur, in Ω,
ur|∂Ω =
∂ur
∂ν
∣∣∣∣
∂Ω
= 0,
∫
Ω
ρurus = δrs.
Let x = (x1, . . . , xn) be the Cartesian coordinate functions of Rn. We define the
trial functions ϕrl by
ϕrl = xlur −
k∑
s=1
blrsus, for r = 1, . . . , k, and l = 1, . . . , n,
where
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 7
1002 HE-JUN SUN
blrs =
∫
Ω
ρxlurus = blsr.
Then, for r, s = 1, . . . , k, and l = 1, . . . , n, it is easy to check∫
Ω
ρϕrlus = 0 (2.2)
and ∫
Ω
ρϕrlxlur =
∫
Ω
ρϕ2
rl.
Hence the Rayleigh – Ritz inequality in variation method reads as
Λk+1 ≤
∫
Ω
ϕrl
∑n
i,j=1
Dij
(
aij(x)Dijϕrl
)
∫
Ω
ρϕ2
rl
. (2.3)
Since
n∑
i,j=1
Dij [aij(x)Dij(xlur)] =
=
n∑
i,j=1
δilDij
(
aij(x)Djur
)
+
n∑
i,j=1
δjlDij
(
aij(x)Diur
)
+
n∑
i,j=1
δilDj
(
aij(x)Dijur
)
+
+
n∑
i,j=1
δjlDi
(
aij(x)Dijur
)
+
n∑
i,j=1
xlDij
(
aij(x)Dijur
)
=
= 2
n∑
i=1
Dil
(
ail(x)Diur
)
+ 2
n∑
i=1
Di
(
ail(x)Dilur
)
+ xlΛrρur,
we have ∫
Ω
ϕrl
n∑
i,j=1
Dij
(
aij(x)Dijϕrl
)
=
=
∫
Ω
ϕrl
[
2
n∑
i=1
Dil
(
ail(x)Diur
)
+ 2
n∑
i=1
Di
(
ail(x)Dilur
)
+
+xlΛrρur −
k∑
s=1
blrsΛsρus
]
=
= 2
∫
Ω
ϕrl
n∑
i=1
[
Dil
(
ail(x)Diur
)
+Di
(
ail(x)Dilur
)]
+ Λr
∫
Ω
ρϕ2
rl. (2.4)
Substituting (2.4) into (2.3), we obtain
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 7
ESTIMATES FOR WEIGHTED EIGENVALUES OF FOURTH-ORDER . . . 1003
(Λk+1 − Λr)
∫
Ω
ρϕ2
rl ≤
≤ 2
∫
Ω
xlur
n∑
i=1
[
Dil
(
ail(x)Diur
)
+Di
(
ail(x)Dilur
)]
+
k∑
s=1
blrsc
l
rs, (2.5)
where
clrs = −2
∫
Ω
us
n∑
i=1
[
Dil
(
ail(x)Diur
)
+Di
(
ail(x)Dilur
)]
= −clsr.
Using integration by parts, we deduce
Λrb
l
rs =
∫
Ω
xlus
n∑
i,j=1
Dij
(
aij(x)Dijur
)
=
=
∫
Ω
ur
{
2
n∑
i=1
[
Dil
(
ail(x)Dius
)
+Di
(
ail(x)Dilus
)]
+ λsρxlus
}
=
= −clsr + Λsb
l
rs.
It yields
clrs = (Λr − Λs)b
l
rs. (2.6)
At the same time, we have
2
∫
Ω
xlur
n∑
i=1
[
Dil
(
ail(x)Diur
)
+Di
(
ail(x)Dilur
)]
=
= −2
∫
Ω
ur
n∑
i=1
Di
[
ail(x)Dil
(
xlur
)]
+ 2
∫
Ω
xlur
n∑
i=1
Di
(
ail(x)Dilur
)
=
= −2
∫
Ω
[
ur
n∑
i=1
Di
(
ail(x)Diur
)
+ urDl
(
all(x)Dlur
)
+ urall(x)Dllur
]
= wrl,
(2.7)
where
wrl = 2
∫
Ω
[
n∑
i=1
ail(x)|Diur|2 + 2all(x)|Dlur|2 +Dl
(
all(x)
)
urDlur
]
.
Substituting (2.6) and (2.7) into (2.5), we arrive at
(Λk+1 − Λr)
∫
Ω
ρϕ2
rl ≤ wrl +
k∑
s=1
(Λr − Λs)(b
l
rs)
2.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 7
1004 HE-JUN SUN
Using (2.2), we have
−2(Λk+1 − Λr)2
∫
Ω
ϕrlDlur =
= −2(Λk+1 − Λr)2
∫
Ω
√
ρϕrl
(
1
√
ρ
Dlur −
√
ρ
k∑
s=1
dlrsus
)
≤
≤ δr(Λk+1 − Λr)3
∫
Ω
ρϕ2
rl +
Λk+1 − Λr
δr
∫
Ω
(
1
√
ρ
Dlur −
√
ρ
k∑
s=1
dlrsus
)2
=
= δr(Λk+1 − Λr)3
∫
Ω
ρϕ2
rl +
Λk+1 − Λr
δr
∫
Ω
1
ρ
|Dlur|2 −
k∑
s=1
(dlrs)
2
, (2.8)
where the constants δr form a decreasing sequence of positive numbers. At the same
time, using integration by parts again, we get
−2
∫
Ω
ϕrlDlur = −2
∫
Ω
xlurDlur + 2
k∑
s=1
blrsd
l
rs =
∫
Ω
u2
r + 2
k∑
s=1
blrsd
l
rs, (2.9)
where
dlrs =
∫
Ω
usDlur = −dlsr.
Substituting (2.9) into (2.8) and taking sum on r from 1 to k, we obtain
k∑
r=1
(Λk+1 − Λr)2
∫
Ω
u2
r + 2
k∑
r,s=1
(Λk+1 − Λr)2blrsd
l
rs ≤
≤
k∑
r,s=1
δr(Λk+1 − Λr)2(Λr − Λs)(b
l
rs)
2 −
k∑
r,s=1
1
δr
(Λk+1 − Λr)(dlrs)
2+
+2
k∑
r=1
δr(Λk+1 − Λr)2
∫
Ω
n∑
i=1
ail(x)|Diur|2 + 2
∫
Ω
all(x)|Dlur|2+
+
∫
Ω
Dl(all(x))urDlur
+
k∑
r=1
1
δr
(Λk+1 − Λr)
∫
Ω
1
ρ
|Dlur|2. (2.10)
Because {δr}∞r=1 is is a decreasing sequence, one can get
k∑
r,s=1
δr(Λk+1 − Λr)2(Λr − Λs)(b
l
rs)
2 ≤ −
k∑
r,s=1
δr(Λk+1 − Λr)(Λr − Λs)
2(blrs)
2.
(2.11)
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 7
ESTIMATES FOR WEIGHTED EIGENVALUES OF FOURTH-ORDER . . . 1005
Using (2.11) and
2
k∑
r,s=1
(Λk+1 − Λr)2blrsd
l
rs = −2
k∑
r,s=1
(Λk+1 − Λr)(Λr − Λs)b
l
rsd
l
rs,
we can eliminate the unwanted terms in both sides of (2.10) and obtain the following
general inequality:
k∑
r=1
(Λk+1 − Λr)2
∫
Ω
u2
r ≤
≤ 2
k∑
r=1
δr(Λk+1 − Λr)2
∫
Ω
n∑
i=1
ail(x)|Diur|2 + 2
∫
Ω
all(x)|Dlur|2+
+
∫
Ω
Dl(all(x))urDlur
+
k∑
r=1
1
δr
(Λk+1 − Λr)
∫
Ω
1
ρ
|Dlur|2. (2.12)
Taking sum on l from 1 to n in (2.12), we have
n
k∑
r=1
(Λk+1 − Λr)2
∫
Ω
u2
r ≤
≤ 2
k∑
r=1
δr(Λk+1 − Λr)2
∫
Ω
n∑
i,l=1
ail(x)|Diur|2 + 2
∫
Ω
n∑
l=1
all(x)|Dlur|2+
+
∫
Ω
n∑
l=1
Dl(all(x))urDlur
+
k∑
r=1
1
δr
(Λk+1 − Λr)
∫
Ω
1
ρ
n∑
l=1
|Dlur|2. (2.13)
Now we need to calculate and estimate some terms in (2.13). It is easy to find that
0 < τ = τ
∫
Ω
ρu2
r ≤
∫
Ω
u2
r ≤ σ
∫
Ω
ρu2
r = σ. (2.14)
Combining ∫
Ω
ur
n∑
i,j=1
Dij(aijDijur) = Λr
∫
Ω
ρu2
r = Λr
and ∫
Ω
ur
n∑
i,j=1
Dij(aijDijur) =
=
∫
Ω
n∑
i,j=1
aij |Dijur|2 ≥ ξ
∫
Ω
n∑
i,j=1
|Dijur|2 = ξ
∫
Ω
|∆ur|2,
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 7
1006 HE-JUN SUN
we have ∫
Ω
|∆ur|2 ≤ ξ−1Λr.
Hence it yields
∫
Ω
n∑
l=1
|Dlur|2 =
∫
Ω
|∇ur|2 ≤
∫
Ω
u2
r
∫
Ω
|∆ur|2
1/2
≤ σ1/2ξ−1/2Λ1/2
r , (2.15)
where ∇ denotes the gradient operator. At the same time, since
n∑
l=1
Dl(all(x))urDlur ≤
≤ |ur|
[
n∑
l=1
|Dl(all(x))|2
]1/2 [ n∑
l=1
|Dlur|2
]1/2
≤ ζ|ur||∇ur|,
we have∫
Ω
n∑
l=1
Dl(all(x))urDlur ≤ ζ
∫
Ω
|ur||∇ur| ≤ ζ
∫
Ω
u2
r
∫
Ω
|∇ur|2
1/2
≤
≤ ζσ3/4ξ−1/4Λ1/4
r . (2.16)
Substituting (2.14), (2.15) and (2.16) into (2.13), we obtain
nτ
k∑
r=1
(Λk+1 − Λr)2 ≤ σ1/2ξ−1/2
{
k∑
r=1
1
δr
(Λk+1 − Λr)σΛ1/2
r +
+2
k∑
r=1
δr(Λk+1 − Λr)2
[
(n+ 2)ηΛ1/2
r + ζ(σξ)1/4Λ1/4
r
]}
. (2.17)
Then, putting
δr =
2
∑k
r=1
(Λk+1 − Λr)σΛ1/2
r
[
(n+ 2)ηΛ1/2
r + ζ(σξ)1/4Λ1/4
r
]
∑k
r=1
(Λk+1 − Λr)2
1/2
2
[
(n+ 2)ηΛ
1/2
r + ζ(σξ)1/4Λ
1/4
r
]
in (2.17), we have
nτ
[
k∑
r=1
(Λk+1 − Λr)2
]1/2
≤
≤ 2σξ−1/2
{
2
k∑
r=1
(Λk+1 − Λr)Λ1/2
r
[
(n+ 2)ηΛ1/2
r + ζ(σξ)1/4Λ1/4
r
]}1/2
.
Therefore, (2.1) holds.
Theorem 2.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 7
ESTIMATES FOR WEIGHTED EIGENVALUES OF FOURTH-ORDER . . . 1007
Inequality (2.1) is sharp. It implies an estimate for the upper bound of Λk+1. In fact,
(2.1) is a quadratic inequality of Λk+1. Solving it, we can obtain an upper bound of
Λk+1 in terms of the first k eigenvalues.
Theorem 2.2. Under the assumptions of Theorem 2.1, we have
Λk+1 ≤
1
k
[
(1 + E)
k∑
r=1
Λr + F
k∑
r=1
Λ3/4
r
]
+
+
{[
(1 + E)
1
k
k∑
r=1
Λr + F
1
k
k∑
r=1
Λ3/4
r
]2
− 1
k
[
(1 + 2E)
k∑
r=1
Λ2
r + 2F
k∑
r=1
Λ7/4
r
]}1/2
,
where
E =
4(n+ 2)σ2η
n2τ2ξ
and F =
4σ
9
4 ζ
n2τ2ξ3/4
.
From (2.1), we can find the influences of variable coefficients aij(x) and the weight
function ρ(x) on estimates of eigenvalues of problem (1.4). Besides the lower and upper
bounds of aij(x), estimate (2.1) depends on ζ = maxx∈Ω
[∑n
l=1
|Dl(all(x))|2
]1/2
.
Namely if A(x) = (aij(x))n×n denotes the (n× n)-matrix with components aij(x), it
depends on the diagonal elements of the matrix A(x). In particular, when all(x) are all
constants for l = 1, . . . , n, it holds ζ = 0. Therefore, for this special case, we have the
following corollaries.
Corollary 2.1. Under the assumptions of Theorem 2.1, if all(x) are all constants
for l = 1, . . . , n, we have
k∑
r=1
(Λk+1 − Λr)2 ≤ 8(n+ 2)σ2η
n2τ2ξ
k∑
r=1
(Λk+1 − Λr)Λr. (2.18)
Remark 2.1. It is not difficult to find that inequality (1.3) in [19] is a corollary of
Theorem 2.2.
Since (2.18) is also a quadratic inequality of Λk+1, we can get an estimate for the
upper bound of Λk+1 in terms of the first k eigenvalues.
Corollary 2.2. Under the assumptions of Corollary 2.1, we have
Λk+1 ≤
[
1 +
4(n+ 2)σ2η
n2τ2ξ
]
1
k
k∑
r=1
Λr +
{[
4(n+ 2)σ2η
n2τ2ξ
1
k
k∑
r=1
Λr
]2
−
−
[
1 +
8(n+ 2)σ2η
n2τ2ξ
]
1
k
k∑
s=1
(
Λs −
1
k
k∑
r=1
Λr
)2}1/2
. (2.19)
Then, using (2.19) and the Cauchy – Schwarz inequality
k∑
r=1
Λ2
r ≥
1
k
(
k∑
r=1
Λr
)2
,
we can get a weaker but more explicit inequality.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 7
1008 HE-JUN SUN
Corollary 2.3. Under the assumptions of Corollary 2.1, we have
Λk+1 ≤
[
1 +
8(n+ 2)σ2η
n2τ2ξ
]
1
k
k∑
r=1
Λr.
1. Ashbaugh M. S. Isoperimetric and universal inequalities for eigenvalues // Spectral Theory and Geometry
(Edinburgh, 1998). London Math. Soc. Lect. Notes. – Cambridge: Cambridge Univ. Press, 1999. – P. 95 –
139.
2. Ashbaugh M. S. Universal eigenvalue bounds of Payne – Pólya – Weinberger, Hile – Protter and H. C. Yang
// Proc. Indian Acad. Sci. (Math. Sci.). – 2002. – 112, № 1. – P. 3 – 30 .
3. Ashbaugh M. S., Hermi L. A unified approach to universal inequalities for eigenvalues of elliptic
operators // Pacif. J. Math. – 2004. – 217, № 2. – P. 201 – 219.
4. Chen Z. C., Qian C. L. Estimates for discrete spectrum of Laplacian operator with any order // J. China
Univ. Sci. Tech. – 1990. – 20, № 3. – P. 259 – 266.
5. Cheng Q.-M., Yang H. C. Inequalities for eigenvalues of a clamped plate problem // Trans. Amer. Math.
Soc. – 2006. – 358, № 6. – P. 2625 – 2635.
6. Cheng Q.-M., Yang H. C. Bounds on eigenvalues of Dirichlet Laplacian // Math. Ann. – 2007. – 337,
№ 1. – P. 159 – 175.
7. Davies E. B. Uniformly elliptic operators with measurable coefficients // J. Funct. Anal. – 1995. – 132,
№ 1. – P. 141 – 169.
8. Hile G. N., Protter M. H. Inequalities for eigenvalues of the Laplacian // Indiana Univ. Math. J. – 1980.
– 29, № 4. – P. 523 – 538.
9. Hook S. M. Domain independent upper bounds for eigenvalues of elliptic operator // Trans. Amer. Math.
Soc. – 1990. – 318, № 2. – P. 615 – 642.
10. Harrell II E. M., Stubbe J. On trace identities and universal eigenvalue estimates for some partial
differential operators // Trans. Amer. Math. Soc. – 1997. – 349, № 5. – P. 1797 – 1809.
11. Hile G. N., Yeh R. Z. Inequalities for eigenvalues of the biharmonic operator // Pacif. J. Math. – 1984. –
112, № 1. – P. 115 – 133.
12. Kurata K., Sugano S. A remark on estimates for uniformly elliptic operators on weighted Lp spaces and
Morrey spaces // Math. Nachr. – 2000. – 209, № 1. – P. 137 – 150.
13. Ljung L. Recursive identification // Stochastics Systems: Math. Filtering and Identification and Appl. –
1981. – P. 247 – 283.
14. Maybeck P. S. Stochastic models, estimation, and control III. – New York: Acad. Press, 1982.
15. Qian C. L., Chen Z. C. Estimates of eigenvalues for uniformly elliptic operator of second order // Acta
Math. Appl. Sinica. – 1994. – 10, № 4. – P. 349 – 355.
16. Payne L. E., Pólya G., Weinberger H. F. On the ratio of consecutive eigenvalues // J. Math. and Phys. –
1956. – 35. – P. 289 – 298.
17. Sun H. J. Yang-type inequalities for weighted eigenvalues of a second order uniformly elliptic operator
with a nonnegative potential // Proc. Amer. Math. Soc. – 2010. – 138, № 8. – P. 2827 – 2837.
18. Sun H. J., Cheng Q.-M., Yang H. C. Lower order eigenvalues of Dirichlet Laplacian // Manuscr. math.
– 2008. – 125, № 2. – P. 139 – 156.
19. Wang Q. L., Xia C. Y. Universal bounds for eigenvalues of the biharmonic operator on Riemannian
manifolds // J. Funct. Anal. – 2007. – 245, № 1. – P. 334 – 352.
20. Yang H. C. An estimate of the difference between consecutive eigenvalues. – Preprint IC/91/60 of ICTP,
Trieste, 1991.
Received 24.09.09
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 7
|
| id | umjimathkievua-article-2780 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:30:09Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/da/5f5a2fcd00bffbd1efddf4df5c0c0bda.pdf |
| spelling | umjimathkievua-article-27802020-03-18T19:36:24Z Estimates for weighted eigenvalues of fourth-order elliptic operator with variable coefficients Оцiнки зважених власних значень елiптичного оператора четвертого порядку iз змiнними коефiцiєнтами Sun, He-Jun Сун, Хе-Джун We investigate the Dirichlet weighted eigenvalue problem for a fourth-order elliptic operator with variable coefficients in a bounded domain in $R^n$. We establish a sharp inequality for its eigenvalues. It yields an estimate for the upper bound of the $(k + 1)$-th eigenvalue in terms of the first $k$ eigenvalues. Moreover, we also obtain estimates for some special cases of this problem. In particular, our results generalize the Wang -Xia inequality (J. Funct. Anal. - 2007. - 245) for the clamped plate problem to a fourth-order elliptic operator with variable coefficients. Дослiджено задачу Дiрiхле про зваженi власнi значення для елiптичного оператора четвертого порядку iз змiнними коефiцiєнтами в обмеженiй областi iз $R^n$. Встановлено точну нерiвнiсть для її власних значень, з якої випливає оцiнка для верхньої межi $(k + 1)$-го власного значення через першi $k$ власних значень. Також отримано оцiнки для цiєї задачi у деяких окремих випадках. Зокрема, нашi результати узагальнюють нерiвнiсть Ванга – Ксi (J. Funct. Anal. – 2007. – 245) для затиснутої пластини на випадок елiптичного оператора четвертого порядку iз змiнними коефiцiєнтами Institute of Mathematics, NAS of Ukraine 2011-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2780 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 7 (2011); 999-1008 Український математичний журнал; Том 63 № 7 (2011); 999-1008 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2780/2315 https://umj.imath.kiev.ua/index.php/umj/article/view/2780/2316 Copyright (c) 2011 Sun He-Jun |
| spellingShingle | Sun, He-Jun Сун, Хе-Джун Estimates for weighted eigenvalues of fourth-order elliptic operator with variable coefficients |
| title | Estimates for weighted eigenvalues of fourth-order elliptic operator with variable coefficients |
| title_alt | Оцiнки зважених власних значень елiптичного оператора четвертого порядку iз змiнними коефiцiєнтами |
| title_full | Estimates for weighted eigenvalues of fourth-order elliptic operator with variable coefficients |
| title_fullStr | Estimates for weighted eigenvalues of fourth-order elliptic operator with variable coefficients |
| title_full_unstemmed | Estimates for weighted eigenvalues of fourth-order elliptic operator with variable coefficients |
| title_short | Estimates for weighted eigenvalues of fourth-order elliptic operator with variable coefficients |
| title_sort | estimates for weighted eigenvalues of fourth-order elliptic operator with variable coefficients |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2780 |
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