Journal of Applied Mathematics and Stochastic Analysis 5, Number 4, Winter 1992, 375-380
NOTE ON THE INEQUALITIES OF J. KAZDAN ’
WILL Y. LEE
Rutgers University- Camden
Department of Mathematics
Camden, NJ 08102, USA
ABSTCT
In this note, we prove the Kazdan’s inequalities without using
what is called the Heisenberg uncertainty principle. Instead we prove it
using Garofalo-Lin inequality among other things.
Key words: Heisenberg uncertainty principle, unique continuation theorem, Garofalo-Lin inequality, Schwarz inequality, Poincare inequality.
AMS (MOS) subject damificatiotm:
Primary: 35, Secondary:
49.
1. INTRODUCqON
In [4], J. Kazdan has shown strong unique continuation theorem (Theorem 1.8 of [41)
(Lemma 2.4 of [4]). Several analytic as well
used to prove the main lemma. Among them are the following
whose proof is mainly based on his main lemma
as geometric inequalities were
inequalities:
There exist constants
C1, C2, C3, C4, C 5 and r0 such
that for all r E (0, r0)
() S Cff()(H(,) + D(r)) (j = 1,2)
1
r’"’ "
/
OB,.
V u 12dS <_ rB(r) + C3H(r
+ D(r)
(1)j
(2)
I3(r) < C4f(r)(H(r + D(r)+ v/rH(r)B(r))
(3)
I4(r) < Cf(r)(H(r) + D(r)+ rB(r)).
(4)
1Received: February, 1992.
Revised: July, 1992.
Printed in the U.S.A. (C) 1992 The Society of Applied Mathematics, Modeling and Simulation
375
WILL Y.LEE
376
Here f(r),I(r),I2, I3(r),H(r),D(r),B(r) are defined as follows: let f be
ro
function with
f(0)- 0 satisfying
f/(r-r
<c
o
and let u satisfy for n
a smooth increasing
> 3 the
differential
inequality with a and b constants:
Imu()l
blur)
<---lu()l +-af(r)
/uudV
Br
Ix(r
r .,1,
()
._ Bf puAudV
2
1
I3(r) rn-3
1
H(r) = r,_i
D(r)
u()l
n!-"a
(7)
/ uAudS
(8)
OB
/OB,. lulZdS
(9)
VuldV
(10)
Br
B()-ra,, ,
(ii)
OB,.
In his proof of inequalities (1)(2), Kazdan relies on what is called the Heisenberg
uncertainty principle (see [2], [31, & [4]):
f
Br
v<
/ =dS//B
OB
VldV, n>3_
(12)
r
r
dV
V w l2dW, n > 3
w2dS +
<
OB r
(13)
Br
where C and C are dimensional constants. Inequality
Indeed a straightforward computation shows that C’ =
(13)
is an easy consequence of
c_x, ,= A(.- + ) for any A > O.
(12).
Note
of the Inequalities of J. Kazdan
377
(:3) and (4), we prove
inequalities (1)5 and (2) without using the Heisenberg uncertainty principle (12)-(1:3). Instead
we use the following lemma which is the Garofalo-Lin inequality (see 4.11 of [2]) applied to the
operator L where
Lu = &u + b(x). V u + V(z)u = 0.
(14)
Here b(x) and V are majorized by with constants a and b:
Since there is nothing to comment on the proofs of inequalities
_
bf()
b() <
W()l
<
(15)
,,r2,
Let u E W 1,2
Then there exists
oc satisfy equation (14).
(0, 1) depending on n,b, V and u such that for a//r (0, r0)
Lemmn:
ro
a
small constant
(6)
Proof-.
First observe that
I z)dV
u((.).u)(
Br
Ib()l Vul( 2- Il=)dW
Br
u2(r 2
2)dv)l/2(
I(
Br
Br
-
)dV) 1/
(Schwarz inequality)
< II b I1 Lr20(fu2dV)’/(flVu 12dV) 1/2
Br
< c II b I L0(
Br
fBr v
u
zdV)
(Poincare inequality)
where C is a dimensional constant. Consequently we obtain
u(().-I
c
-II
Br
IIrJl
Br
u
ZdV.
(7)
Choose r0 so small that
(18)
B
Br
WILL Y.LEE
378
(17)-(18) then reveal that
Inequalities
V u)(r2Su(b(z).
Br
f u dV.
2)dV )_
x
(19)
Br
Secondly we have
)dV
u2dV.
V
Br
Br
Choose r 0 such that
o
Inequalities
(’*-- 2)/II v Ii L
(20)-(21) then show that
-
[
(21)
2)[ u2dV.
>_
B
/ (I
Br
Equation
+ ub(z). V u + Vu)(r
Vu
(23)
(22)
B
Finally integration by parts and equation
r
(20)
z
(14) give us the following identity:
[a)dV = r ] uadS
n
OB r
combined with inequalities
/OB u2dS > Brf
r
] uadV.
Br
(23)
(19) and (22)shows that
Vul2(,"- 112) dV
V udV
Sr
( 2)
Br
r
+n/
f dV
dV
Br
>_-/u2dV-(n-2)/u2dV+niudV
B
B
B
r
r
r
B
for all r
(0,%)
(18) and (21).
where r o is chosen to be the minimum of the right hand sides of inequalities
This completes the proof.
We give the proof of (I)lonly
Proof of (l)l:
IIl(r)l
as the proofs of
(1)2 and (2) are essentially the same.
5rf luJ IAuldV
B
lul
2
B
r2
lul+
Vul)dV
(by (,.5))
Note
< af(r)
nr
u2dS +
OB
a
+ hi2.
B
OB
Kazdan
379
v u lZdV) 1
(Lemma and Schwarz)
B
u2dS +
<_
where C
_
u2dV) I(
rn-
r"
of the Inequalities of J.
uaV +
(
B
Vu
v)
B
af(r)H(r)+f(r)H(r)+f(r)D(r)
(Lemma, (9)& (10))
Clf(r)(H(r + D(r))
This complete the proof of (I)i.
A simple computation shows inequalities (I)2, (2), (3) and (4) are satisfied with
C 2=a+2b, C 3=(n-2)+(n+2)7+C2f(r), C 4-a+(3b/2)V3 C 5=(3/2)C 4, where 3’
satisfies
0(r)< (n + 2)3’.
REFEreNCES
M.S. Baouendi and E.C. Zachmanoglou, "Unique continuation of partial differential
equations and inequalities from manifolds of any dimension", Duke Math. J. 45, (1978),
pp. 1-13.
[2]
N. Garofalo and F.H. Lin, "Monotonicity properties of variational integrals A weights
and unique continuations", Indiana Univ. Math. J. 35, (1986), pp. 245-268.
[3]
"Unique continuation for elliptic operators: A geometric-variational approach", Comm. Pure 8j Applied Math XL, (1987), pp. 347-366.
[4]
J.L. Kazdan, "Unique continuation in geometry", Comm. Pure 8J Applied Math. XLI,
(1988), pp. 667-681.
D. Gilbarg and N.S. Trudinger, "Elliptic Partial Differential Equations of Second
Order", Springer-Verlag, 1983.