Funkcialaj Ekvacioj, 35 (1992) 571-580
Remarks on Irregular Open Sets and Its Application
to the Eigenvalue Distribution
By
Yoichi MIYAZAKI
(Nihon University, Japan)
0. Introduction
In the study of the eigenvalue distribution of elliptic operators in a
bounded domain
many authors have assumed various assumptions
in
concerning the regularity of the boundary of . We discuss the relationship
between these assumptions and give a new equivalent definition of the
Minkowski dimension in Section 1.
In Section 2 we show that some of the above assumptions on the boundary
are unnecessary in deriving the asymptotic formula for the eigenvalue
distribution. In the last section we give another proof of Lapidus’ result
concerning the eigenvalue distribution for elliptic operators with the Dirichlet
boundary condition in a bounded domain having the irregular boundary.
$R^{n}$
$¥Omega$
$¥Omega$
1. The Minkowski dimension
Let
be a bounded open set in
and $p>0$ we put
$¥Omega$
$R^{n}$
.
Let
$¥Gamma=¥partial¥Omega$
be its boundary.
For
$¥epsilon>0$
$¥delta(¥mathrm{x})=$
dist
$(¥mathrm{x}, ¥Gamma)$
,
$ r=¥max$
$¥Gamma_{¥epsilon}=¥{x¥in¥Omega;¥delta(x)<¥epsilon¥}$
,
$¥{¥delta(¥mathrm{x});x¥in¥Omega¥}$
,
.
$I_{p}(¥epsilon)=¥int_{¥{x¥in¥Omega;¥delta(x)¥geqq¥epsilon¥}}¥delta(x)^{-p}dx$
For a Lebesgue measurable set $D$ in
we denote by $|D|$ its ( -dimensional)
Lebesgue measure. We investigate the relationship between the estimate of
and that of
. We begin with the following lemma.
$R^{n}$
$|¥Gamma_{¥epsilon}|$
$¥mathrm{n}$
$I_{p}(¥epsilon)$
Lemma 1. 1.
Let
$0<¥epsilon<r$
and $p>0$ .
It follows that
$I_{p}(¥epsilon)=p¥int_{¥epsilon}^{r}¥frac{|¥Gamma_{t}|}{t^{p+1}}dt-¥frac{|¥Gamma_{¥epsilon}|}{¥epsilon^{p}}+¥frac{|¥Omega|}{r^{p}}$
Proof:
By Fubini’s theorem we have
.
Yoichi MIYAZAKI
572
$I_{p}(¥epsilon)=¥int_{¥{x¥in¥Omega;¥delta(x)¥geqq¥epsilon¥}}dx¥int_{¥delta(x)}^{¥infty}pt^{-p-1}dt$
$=¥int_{¥epsilon}^{¥infty}dt¥int_{¥{X¥in¥Omega;¥epsilon¥leqq¥delta(x)<t¥}}pt^{-p-1}dx$
.
$=¥int_{¥epsilon}^{¥infty}¥frac{p¥{|¥Gamma_{t}|-|¥Gamma_{¥epsilon}|¥}}{t^{p+1}}dt$
Noting that
$|¥Gamma_{t}|=|¥Omega|$
when
we get the lemma.
$t>r$ ,
$¥mathrm{q}.¥mathrm{e}.¥mathrm{d}$
.
positive constants
In this and the next sections we denote by
independent of which may differ from each other. The next lemma follows
easily from Lemma 1.1.
$C$
$¥epsilon$
Lemma 1.2.
for some
$¥theta¥geqq 0$
Let $0<¥epsilon<¥min¥{r, 1/2¥}$ and $p>0$ .
implies the following estimate for
The inequality
$I_{p}(¥epsilon)$
$|¥Gamma_{¥epsilon}|¥leqq C¥epsilon^{¥theta}$
.
$0<I_{p}(¥epsilon)¥leqq¥left¥{¥begin{array}{l}C(0<p<¥theta)¥¥C|1¥mathrm{o}¥mathrm{g}¥epsilon|(p=¥theta)¥¥C¥epsilon^{¥theta-p}(p>¥theta).¥end{array}¥right.$
Proposition 1.3.
$0<¥epsilon<¥min¥{r,
(i)
(ii)
(iii)
Let
fixed.
1/2¥}$ , are equivalent.
$|¥Gamma_{¥epsilon}|¥leqq C¥epsilon^{¥theta}$
The following inequalities
$¥theta¥geqq 0$
for any
$¥epsilon$
,
.
$I_{p}(¥epsilon)¥leqq C¥epsilon^{¥theta-p}$
$I_{p}(¥epsilon)¥leqq C¥epsilon^{¥theta-p}$
for any
for some
$ p>¥theta$
.
$ p>¥theta$
.
Proof:
From Lemma 1.2 it follows that (i) implies (ii). It is clear that
(ii) implies (iii). We note that (i) is always valid when $¥theta=0$ since
. Hence it remains to prove that (iii) implies (i) when $¥theta>0$ .
Assume (iii). Then we have
$|¥Gamma_{¥epsilon}|¥leqq|¥Omega|$
$¥epsilon^{-p}|¥{¥mathrm{x}¥in¥Omega;¥frac{¥epsilon}{2}¥leqq¥delta(x)<¥epsilon¥}|¥leqq¥int_{¥{X¥in¥Omega;¥epsilon/2¥leqq¥delta(x)<¥epsilon¥}}¥delta(x)^{-p}dx$
$¥leqq I_{p}(¥frac{¥epsilon}{2})¥leqq C(¥frac{¥epsilon}{2})^{¥theta-p}$
which gives
$|¥{¥mathrm{x}¥in¥Omega;¥frac{¥epsilon}{2}¥leqq¥delta(¥mathrm{x})<¥epsilon¥}|¥leqq C¥epsilon^{¥theta}$
Replacing
$¥epsilon$
with
$¥epsilon/2^{k}$
.
in the above inequality, we have
573
Eigenvalue Distribution
$|¥{x¥in¥Omega;¥frac{¥epsilon}{2^{k+1}}¥leqq¥delta(x)<¥frac{¥epsilon}{2^{k}}¥}|¥leqq C(¥frac{¥epsilon}{2^{k}})^{¥theta}$
.
Hence we get
$|¥Gamma_{¥epsilon}|=¥sum_{k=0}^{¥infty}|¥{x¥in¥Omega;¥frac{¥epsilon}{2^{k+1}}¥leqq¥delta(¥mathrm{x})<¥frac{¥epsilon}{2^{k}}¥}|$
$¥leqq C¥epsilon^{¥theta}¥sum_{k=0}^{¥infty}2^{-¥theta k}¥leqq C¥frac{¥epsilon^{¥theta}}{1-2^{-¥theta}}$
from which (i) follows.
Remark 1. 1.
show that
$¥mathrm{q}.¥mathrm{e}.¥mathrm{d}$
In the same way as the proof of Proposition 1.3 we can
.
implies
$|¥Gamma_{¥epsilon}|¥leqq C¥epsilon^{¥theta}|¥log¥epsilon|$
$I_{¥theta}(¥epsilon)¥leqq C|¥log¥epsilon|$
$¥mathrm{d}$
.
Now we recall the Minkowski dimension. For $d¥geqq 0$ let
-dimensional upper Minkowski content relative to :
$¥ovalbox{¥tt¥small REJECT}_{d}(¥Gamma)$
be the
$¥Omega$
$¥epsilon^{-(n-d)}|¥Gamma_{¥epsilon}|$
$¥ovalbox{¥tt¥small REJECT}_{d}(¥Gamma)=¥lim_{¥epsilon¥rightarrow}¥mathrm{s}¥mathrm{u}¥mathrm{p}0$
The upper Minkowski dimension of
is defined by
$¥Gamma$
relative to
$¥Omega$
.
which we denote by
$D(¥Gamma)=¥inf¥{d ¥geqq 0;¥sim¥ovalbox{¥tt¥small REJECT}_{d}(¥Gamma)=0¥}=¥sup¥{d¥geqq 0;_{¥mathrm{C}}¥ovalbox{¥tt¥small REJECT}_{d}(¥Gamma)=¥infty¥}$
$D(¥Gamma)$
,
which is equivalent to
$(¥mathrm{i})$
$D(¥Gamma)=n-¥lim_{¥epsilon¥rightarrow}¥inf_{0}¥frac{1¥mathrm{o}¥mathrm{g}|¥Gamma_{¥epsilon}|}{1¥mathrm{o}¥mathrm{g}¥epsilon}$
Since
$¥Gamma_{¥epsilon}$
contains a ball of radius
$¥epsilon/4$
when
.
$0<¥epsilon<r$
, we have
$|¥{x¥in R^{n} ; |x|¥leqq 1¥}|(¥frac{¥epsilon}{4})^{n}¥leqq|¥Gamma_{¥epsilon}|¥leqq|¥Omega|$
from which it follows that
$0¥leqq D(¥Gamma)¥leqq n$
.
is the boundary of a bounded open set
Further using the fact that
from below.
can get the best estimate for
$¥Gamma$
$D(¥Gamma)$
Proposition 1.4.
Remark 1.2.
n?
$1¥leqq D(¥Gamma)¥leqq n$
If we replace
$¥Gamma_{¥epsilon}$
.
with
$¥tilde{¥Gamma}_{¥epsilon}=¥{¥mathrm{x}¥in R^{n} ; ¥delta(¥mathrm{x})<¥epsilon¥}$
$¥Omega$
, we
574
Yoichi MIYAZAKI
in (1.1), we obtain the definition of the (usual) upper Minkowski dimension
of 1:
$¥log|¥tilde{¥Gamma}_{¥epsilon}|$
$¥tilde{D}(¥Gamma)=n-¥lim_{¥epsilon¥rightarrow}¥inf_{0}¥overline{¥log¥epsilon}$
.
Lapidus [8] obtained the counterpart of Proposition 1.4 for
(1.2)
$¥tilde{D}(¥Gamma)$
:
$n-1¥leqq¥tilde{D}(¥Gamma)¥leqq n$
through the comparison with the Hausdorff dimension and the topological
dimension. Our proof of Proposition 1.4 below is also effective for
and
therefore it is an alternative proof of (1.2). We note that our notations
are defined in the opposite way to that of Lapidus [8].
and
$¥tilde{D}(¥Gamma)$
$¥tilde{D}(¥Gamma)$
$D(¥Gamma)$
Proof: Since the second inequality has been already proved, we have
only to show the first inequality. There exists
such that $¥delta(a)=r$ . For
a unit vector
we put
$ a¥in¥Omega$
$¥omega¥in R^{n}$
$l(¥omega)=¥inf¥{t>0;a+t¥omega¥in¥partial¥Omega¥}$
It is seen from the boundedness of
(1.3)
$¥Omega$
and the definition of
$ r¥leqq l(¥omega)<¥infty$
Let $0<¥epsilon<r/2$ .
follows that
Since
(1.4)
.
$ a+l(¥omega)¥omega¥in¥partial¥Omega$
and
.
$ a+t¥omega¥in¥Omega$
for
$¥{a+t¥omega;|¥omega|=1, l(¥omega)-¥epsilon<t<l(¥omega)¥}¥subset¥Gamma_{¥epsilon}$
Now let
$¥chi_{¥epsilon}(x)$
for
(1.4) we have
$¥chi_{¥epsilon}(x)=0$
that
$r$
$0<t<l(¥omega)$ ,
it
.
be the characteristic function of
:
for
, and
. Putting $x=a+t¥omega(t >0, |¥omega|=1)$ and using (1.3) and
$¥Gamma_{¥epsilon}$
$¥chi_{¥epsilon}(¥mathrm{x})=1$
$x¥in¥Gamma_{¥epsilon}$
$x¥not¥in¥Gamma_{¥epsilon}$
$|¥Gamma_{¥epsilon}|=¥int_{R^{n}}¥chi_{¥epsilon}(¥mathrm{x})d¥mathrm{x}=¥int_{|¥omega|=1}d¥omega¥int_{¥mathrm{o}}^{¥infty}¥chi_{¥epsilon}(a+t¥omega)t^{n-1}dt$
$¥geqq¥int_{|¥omega|=1}d¥omega¥int_{l(¥omega)-¥epsilon}^{¥mathrm{I}(¥omega)}t^{n-1}dt¥geqq¥int_{|¥omega|=1}d¥omega¥int_{r-¥epsilon}^{r}t^{n-1}dt$
$¥geqq S_{n}(r-¥epsilon)^{n-1}¥epsilon¥geqq S_{n}(¥frac{r}{2})^{n-1}¥epsilon$
where
$S_{n}$
is the area of the unit sphere in
$R^{n}$
.
Hence we get the first inequality.
$¥mathrm{q}.¥mathrm{e}.¥mathrm{d}$
.
Now we give another equivalent definition of the upper Minkowski
dimension relative to .
$¥Omega$
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Eigenvalue Distribution
Proposition 1.5.
Let
$p¥geqq 1$
Then we have
.
$D(¥Gamma)=n-p-¥lim_{¥epsilon¥rightarrow}¥inf_{0}¥frac{1¥mathrm{o}¥mathrm{g}I_{p}(¥epsilon)}{1¥mathrm{o}¥mathrm{g}¥epsilon}$
Proof:
.
Put
$¥theta=¥lim_{¥epsilon¥rightarrow}¥inf_{0}¥frac{1¥mathrm{o}¥mathrm{g}|¥Gamma_{¥epsilon}|}{¥mathrm{l}o¥mathrm{g}¥epsilon}$
,
$¥zeta=¥lim_{¥epsilon¥rightarrow}¥inf_{¥mathrm{o}}¥frac{1¥mathrm{o}¥mathrm{g}I_{p}(¥epsilon)}{1¥mathrm{o}¥mathrm{g}¥epsilon}$
.
Our asssertion is equivalent to
$¥theta=¥zeta+p$
Because of Proposition 1.4 and because
$0<¥epsilon<r/2$ we have
(1.5)
$0¥leqq¥theta¥leqq 1$
,
.
$(|¥Omega|-|¥Gamma_{r/2}|)r^{-p}¥leqq I_{p}(¥epsilon)¥leqq|¥Omega|¥epsilon^{-p}$
?
$p¥leqq¥zeta¥leqq 0$
when
.
First we will show
(1.6)
$¥zeta¥geqq¥theta-p$
When $¥theta=0$ , (1.6) is clear from(1.5).
we have
.
Assume
$¥theta>0$
.
For any
$p$
,
$ 0<¥rho<¥theta$
,
$|¥Gamma_{¥epsilon}|¥leqq¥epsilon^{¥theta-¥rho}$
for sufficiently small
$¥epsilon>0$
.
Since
$0<¥theta-¥rho¥leqq 1-¥rho<p$,
$I_{p}(¥epsilon)¥leqq C¥epsilon^{¥theta-¥rho-p}$
(1.7)
$¥theta¥geqq¥zeta+p$
$¥zeta=-p$ ,
$0<¥rho<¥zeta+p$ ,
,
Next we will show
from which (1.6) follows.
When
Proposition 1.3 gives
.
(1.6) is clear from (1.5). Assume
$¥zeta>-p$ .
For any
$¥rho$
,
we have
$I_{p}(¥epsilon)¥leqq¥epsilon^{¥zeta-¥rho}=¥epsilon^{(¥zeta+p-¥rho)-p}$
for sufficiently small
$¥epsilon>0$
.
Since
$0<¥zeta+p-¥rho<p$ ,
Proposition 1.3 gives
$|¥Gamma_{¥epsilon}|¥leqq C¥epsilon^{¥zeta+p-¥rho}$
from which (1.7) follows.
Combining (1.6) and (1.7) we get the proposition.
Remark 1.3.
by .
$¥tilde{¥Gamma}_{¥epsilon}$
$¥Gamma_{¥epsilon}$
$¥mathrm{q}.¥mathrm{e}.¥mathrm{d}$
The statements in this section are also valid if we replace
.
Yoichi MIYAZAKI
576
2. Assumptions on
$¥Omega$
In the study of the aymptotic formula for the eigenvalue distribution for
uniformly elliptic differential operators in a bounded domain
in
, various
assumptions on are imposed such as:
$¥Omega$
$R^{n}$
$¥Omega$
(HI)
(H2)
$¥Omega$
has the restricted cone property.
$|¥Gamma_{¥epsilon}|¥leqq C¥epsilon$
.
(H3)
$I_{1}(¥epsilon)=¥int_{¥{X¥in¥Omega;¥delta(x)¥geqq¥epsilon¥}}¥delta(x)^{-1}dx¥leqq C|¥log¥epsilon|$
(H4)
$¥int_{¥Omega}¥delta(¥mathrm{x})^{-p}dx<¥infty$
Here $0<¥epsilon<¥min¥{r, 1/2¥}$ .
replace
with $¥min$
since it follows that
.
for some , $0<p<1$ .
$p$
Conditions
remain the same if we
, which has been often used by many authors,
$(¥mathrm{H}2)-(¥mathrm{H}4)$
$¥{¥delta(x), 1¥}$
$¥delta(x)$
$¥min$ $¥{¥delta(x), 1¥}¥leqq¥delta(x)¥leqq(r+1)¥min$
We consider the relationship between
$¥{¥delta(x), 1¥}$
$(¥mathrm{H}1)-(¥mathrm{H}4)$
.
.
Lemma 2. 1.
(i) (HI) implies (H2).
(H2) implies (H3) and (H4).
$(¥mathrm{i}^{b}1)$
Proof: (i) is
due to Metivier [11] according to Lapidus [8]. Since
does not seem to be accessible, we give the proof briefly. It is known
if
possesses the restricted cone property
can be written
where
, $ i=1,2,¥ldots$ , $N$ has the boundary of Lipschitz class (Gagliard
see also [19] . Then (i) is easily seen from this fact, (ii) is due to Lemma
$¥Omega$
$¥Omega$
$¥Omega_{i}$
$)$
[11]
that
$¥Omega=¥bigcup_{i=1}^{N}¥Omega_{i}$
[5],
1.2.
$¥mathrm{q}.¥mathrm{e}.¥mathrm{d}$
.
From Lemma 2.1 we conclude the following.
(a) Tsujimoto [17] assumed (H2) and (H3), but it is sufficient to assume
only (H2).
(b) Pham The Lai [15], Robert [16] and the author [13] assumed
, but it is sufficient to assume only (HI).
(c) Agmon [2], Maruo & Tanabe [9], Maruo [10] and Tsujimoto [18]
assumed (HI) and (H4), but it is sufficient to assume only (HI).
$(¥mathrm{H}1)-(¥mathrm{H}3)$
577
Eigenvalue Distribution
3. Another proof of Lapidus’ result
Lapidus [8] obtained the remainder estimate of the asymptotic formula
-th order positive elliptic operators whose
for the eigenvalue distribution for
with
leading term has constant coefficients in a bounded domain in
fractal boundary. His result is a significant step towards the resolution of
the (modified) Weyl-Berry conjecture. Using Lemma 1.2 we can give another
proof of Lapidus’ result in case of the Dirichlet boundary condition, although
$>n$ .
At the same time we can extend his result to the operator
we assume
whose leading term has variable coefficients.
In order to state Lapidus’ result we recall the standard notations. Let
the Sobolev
. As usual we denote by
be a bounded domain in
.
the closure of
and by
space of order $m$ with the norm
containing
. Let
be a closed subspace of
. Let
in
$B[u, v]$ be a symmetric integro-differential sesquilinear form on $V¥times V$ :
$¥mathit{2}m$
$R^{n}$
$¥mathit{2}m$
$H^{m}(¥Omega)$
$R^{n}$
$¥Omega$
$H_{0}^{m}(¥Omega)$
$||_{m}$
$||$
$H^{m}(¥Omega)$
$V$
$H^{m}(¥Omega)$
$C_{0}^{¥infty}(¥Omega)$
$H_{0}^{m}(¥Omega)$
$B[u, v]=¥int_{¥Omega}¥sum_{|¥alpha|,|¥beta|¥leqq m}a_{¥alpha¥beta}(x)D^{a}u(x)¥overline{D^{¥beta}v(¥mathrm{x})}dx$
which satisfies
$a_{¥alpha¥beta}¥in L_{¥infty}(¥Omega)$
$(|a|, |¥beta|¥leqq m)$
,
$a_{a¥beta}¥in¥ovalbox{¥tt¥small REJECT}^{¥infty}(¥Omega)$
$(|¥alpha|=|¥beta|=m)$
,
and satisfies
$B[u, u]¥geqq¥delta||u||_{m}^{2}$
,
$¥delta>0$
, for any
.
$u¥in V$
which are infinitely
denotes the space of all functions in
Here
differentiable and whose derivatives are bounded in .
Let A be the self-adjoint operator associated with the variational triple
$¥{B, V, L_{2}(¥Omega)¥}$ .
We define
$¥Omega$
$¥ovalbox{¥tt¥small REJECT}^{¥infty}(¥Omega)$
$¥Omega$
$a(x, ¥xi)=¥sum_{|a|=|¥beta|=m}a_{a¥beta}(x)¥xi^{¥alpha+¥beta}$
,
$¥mu_{A}(x)=(2¥pi)^{-n}|¥{¥xi¥in R^{n} ; a(x, ¥xi)<1¥}|$
$¥mu_{A}(¥Omega)=¥int_{¥Omega}¥mu_{A}(¥mathrm{x})d¥mathrm{x}$
,
.
Let $N(t)$ be the number of the eigenvalues of A not exceeding . Then Lapidus
$(|¥alpha|=|¥beta|=m)$ are
obtained the following theorem when the coefficients
constants.
$t$
$a_{¥alpha¥beta}(x)$
Theorem A. Let $V=H_{0}^{m}(¥Omega)$ . Let $d¥in[n-1,
Then the following remainder estimate holds.
n]$
satisfy
$¥ovalbox{¥tt¥small REJECT}_{d}(¥Gamma)<¥infty$
.
Yoichi MIYAZAKI
578
(i) If $d¥in(n-1,$ ], then
$n$
as
$N(t)=¥mu_{A}(¥Omega)t^{n/2m}+O(t^{d/2m})$
$ t¥rightarrow¥infty$
.
(ii) If $d=n-1$ , then
as
$N(t)=¥mu_{A}(¥Omega)t^{n/2m}+O(t^{(n-1)/2m}¥log t)$
$ t¥rightarrow¥infty$
.
Remark 3. 1. Lapidus also obtained Theorem A when
replacing
with the (usual) upper Minkowski content:
$V¥neq H_{0}^{m}(¥Omega)$
,
$¥ovalbox{¥tt¥small REJECT}_{d}(¥Gamma)$
$¥tilde{¥ovalbox{¥tt¥small REJECT}}_{d}(¥Gamma)=¥lim_{¥epsilon}¥sup_{¥rightarrow 0}¥epsilon^{-(n-d)}|¥tilde{¥Gamma}_{¥epsilon}|$
and assuming that
$¥Omega$
possesses the
$‘‘(C^{¥prime})$
,
condition” (see [12]).
Lapidus’ proof of Theorem A is based on the min-max principle and the
approximation of
by the union of cubes. We shall give another proof of
$>n$ .
Theorem A when
Our proof is based on Lemma 1.2 and the
following estimates for the spectral function.
$¥Omega$
$¥mathit{2}m$
Lemma 3.1. Let $V=H_{0}^{m}(¥Omega)$ . Let
$>n$ .
Let $e(t, x, x)$ be the spectral
function of the operator A. Then there exists a constant $C>0$ such that
$¥mathit{2}m$
(3. 1)
$0¥leqq e(t, x, x)¥leqq Ct^{n/2m}$
(3.2)
,
$|e(t, x, x)$ $-¥mu_{A}(x)t^{n/2m}|¥leqq C¥delta(x)^{-1}t^{(n-1)/2m}$
hold
for $t>0$ .
Proof: According
to Tsujimoto [17] we have only to derive the estimate
for the resolvent kernel
(the integral kernel of (A
)) as follows:
$-¥lambda$
$G_{¥lambda}(x, y)$
(3.3)
for
$|G_{¥lambda}(x, y)|¥leqq C|¥lambda|^{n/2m-1}¥exp(-c|¥lambda|^{1/2m}|x-y|)$
$¥lambda¥in¥Lambda_{¥theta}=$
$¥{¥lambda¥in C;¥theta¥leqq¥arg¥lambda¥leqq 2¥pi-¥theta, |¥lambda|¥geqq C¥}$
with some
$¥theta¥in(0, ¥pi/2)$
where
and are positive constants.
This estimate was obtained by Tsujimoto [18, Lemma 5.1] when
possesses the restricted cone property. Looking through his proof we see that
(3.3) is also valid when satisfies the following conditions.
(v1) (the interpolation inequality)
$C$
$c$
$¥Omega$
$V$
$||u||_{k}¥leqq C||u||_{m}^{k/m}||u||^{1-k/m}¥mathrm{o}$
(v2) (Sobolev’s lemma)
continuous functions
$¥ovalbox{¥tt¥small REJECT}^{¥tau}(¥Omega)$
$V$
’
$0¥leqq k¥leqq m$
,
.
$u¥in V$
can be imbedded into the space of the Holder
for some
$¥tau>0$
and the inequality
$|u(¥mathrm{x})|¥leqq C||u||_{m}^{n/2m}||u||_{0}^{1-n/2m}$
,
$u¥in V$
579
Eigenvalue Distribution
holds.
.
for any
We note that for any arbitrary domain
Conditions
are satisfied
if $V=H_{0}^{m}(¥Omega)$ . Hence we get the estimate (3.3) and thus the lemma follows.
(v3)
$¥{e^{x¥eta}f(x);f¥in V¥}¥subset V$
$¥eta¥in C$
$¥Omega$
$(¥mathrm{v}¥mathrm{l})-(¥mathrm{v}3)$
$¥mathrm{q}.¥mathrm{e}.¥mathrm{d}$
.
Now that we have established Lemma 1.2 and Lemma 3.1, we can derive
Theorem A in the standard manner. In the following we denote by
positive
constants independent of
and
which may differ from each other. Let
$>n$ .
From the assumption it follows that
$C$
$t$
$¥epsilon$
$¥mathit{2}m$
(3.4)
$|¥Gamma_{¥epsilon}|¥leqq C¥epsilon^{n-d}$
for ,
$¥epsilon$
$0<¥epsilon<¥min¥{r$
,
$¥frac{1}{2}¥}$
.
Using (3.4), Lemma 1.2 and Lemma 3.1 we have
$|N(t)-¥mu_{A}(¥Omega)t^{n/2m}|=|¥int_{¥Omega}¥{e(t, x, x) -¥mu_{A}(x)t^{n/2m}¥}dx|$
$¥leqq¥int_{¥delta(x)<¥epsilon}¥{e(t, x, x)+¥mu_{A}(¥mathrm{x})t^{n/2m}¥}dx$
$+¥int_{¥delta(x)¥geqq¥epsilon}|e(t, x, x)$
$-¥mu_{A}(¥mathrm{x})t^{n/2m}|dx$
$¥leqq C|¥Gamma_{¥epsilon}|t^{n/2m}+C¥int_{¥delta(x)¥geqq¥epsilon}¥delta(x)^{-1}dx¥cdot t^{(n-1)/2m}$
$¥leqq C¥epsilon^{n-d}t^{n/2m}+¥left¥{¥begin{array}{l}C¥epsilon^{n-d-1}t^{(n-1)}/2m(¥mathrm{c}¥mathrm{a}¥mathrm{s}¥mathrm{e}(¥mathrm{i}))¥¥C|1¥mathrm{o}¥mathrm{g}¥epsilon|t^{(n-1)}/2m(¥mathrm{c}¥mathrm{a}¥mathrm{s}¥mathrm{e}(¥mathrm{i}¥mathrm{i})).¥end{array}¥right.$
Putting
$¥epsilon=t^{-1/2m}$
for sufficiently large
$t>0$ ,
we get
$|N(t)-¥mu A(¥Omega)t^{n/2m}|¥leqq¥left¥{¥begin{array}{l}Ct^{d}/2m¥¥Ct^{(n-1)}/2m¥mathrm{l}¥mathrm{o}¥mathrm{g}t¥end{array}¥right.$
$((¥mathrm{c}¥mathrm{a}¥mathrm{s}¥mathrm{e}(¥mathrm{i}))¥mathrm{c}¥mathrm{a}¥mathrm{s}¥mathrm{e}(¥mathrm{i}¥mathrm{i}))$
which is the desired result. Thus we complete the proof of Theorem A.
As stated in Remark 3.1 we must consider
in the case
according to Lapidus [8]. But from our proof of Theorem A we see that
Theorem A is valid without replacing
by
even in the case
if satisfies
.
$¥tilde{¥ovalbox{¥tt¥small REJECT}}_{d}(¥Gamma)$
$¥ovalbox{¥tt¥small REJECT}_{d}(¥Gamma)$
$V¥neq H_{0}^{m}(¥Omega)$
$V$
$V¥neq H_{0}^{m}(¥Omega)$
$¥tilde{¥ovalbox{¥tt¥small REJECT}}_{d}(¥Gamma)$
$(¥mathrm{v}¥mathrm{l})-(¥mathrm{v}3)$
Acknowledgement. The author wishes to thank Professor Toshihusa
Kimura for guidance and encouragement.
580
Yoichi MIYAZAKI
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nuna adreso:
School of Dentistry
Nihon University
Kanda-Surugadai
Chiyoda-ku, Tokyo 101
Japan
la
4-an
de
oktobro, 1990)
(Ricevita
(Reviziita la 1-an de aprilo, 1991)