Ames W.F., Harrel E.M., Herod J.V. (editors). Differential Equations with Applications to Mathematical Physics
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On
Unique Continuation
Theorem
for
Uniformly
Elliptic Equations with
Strongly
Singular
Potentials
Kazuhiro Kurata
Gakushuin University
Abstract
We prove unique continuation properties for solutions
of
uni-
formly elliptic equations:
-div(A(z)Vu)
+
b(z)
-
Vu
+
(V(z)
+
lV(z))u
=
0
with Lipschitz continuous
A(z)
and singular
b(z),
W(z)
and
V(z).
The principal assumptions on
b(z),
W(z)
and
V(z),
in our the-
orems, are
V,(2V
+
z
.
VV)-,W+,(~ZIW+)~
E
Qt(Q),
IW-(z)l
I
C/lz12,1b(z)I
5
C/lzl
for
some constant
C
>
0,
where
Qt(0)
=
K,(0)
+
Ft(0)
for
some
1
<
t
5
n/2, V-
=
max(0,-V),V+
=
max(0,
V).
Here
IC,(Q)
is the Kato class and
Ft(Q)
is the Fefferman-
Phong
class.
I
Introduction
We consider the second order uniformly elliptic equation with real
coefficients:
LU
=
-div(A(z)V~)
+
b(z)
*
VU
+
V(Z)U
+
W(Z)U
=
0
(1)
Differential Equations
with
Copyright
@
1993
by
Academic Press, Inc.
Applications to Mathematical
All
rights
of
reproduction in any
form
reserved.
Physics
ISBN
0-12-056740-7
201
202
Kazuhiro Kurata
in
a
domain
R
c
Rn
(n
2
3). Here
A(z)
=
(uij(z))l<i,jsn
is
a
symmetric matrix which satisfies, for some
X
E
(0,1] and
r
>
0,
n
xIt12
5
C
aij(z)titj
I
X-'M~,
z
E
0,
t
E
R"
9
(2)
(3)
i,j=1
iaij(x)
-
aij(y)l
I
rlz
-
YI,
i,j
=
1,2,-,n
z,Y
E
a,
and
b(z)
=
(6i(z))l<isn.
The followiiig two types of unique con-
tinuation property
for
solutions of
(1)
are well known
for
bounded
coefficients
b,
V,
14'.
(W) Let u
E
WtL(R)
be
a
weak solution of Lu
=
0
in
R
and u
=
0
(S)
Let u
E
14'/$!R)
be
a
weak solution of Lu
=
0
in
R
and
u vanishes of infinite order at
a
point
z,
E
R
in the sense
SB,(Z,)
u2dy
4
0
as r
--f
0
for every
m
>
0
at
a
point
x,
E
0,
then
u
=-
0
in
R.
Recently these results are extended to various classes of
unbounded coefficients. When
A(x)
E
(6ij),
see e.g.
[lo],
[7], [12],
[3],
[2]. In particular, Jerison and Kenig [7] showed the property
(S)
for
W
E
LyL:(R)
(b,V
E
0)
and Stein [12] extended this result to
the weak-Lni2 class. Sawyer
[lo]
and Fabes, Garofalo and Lin
[3]
studied it for
W
of tlie I<ato class
Kn(R)
and Chanillo and Sawyer
[2] for
14'
in tlie Fefferman and Phong class
Ft
with
t
>
(n
-
1)/2.
As
is well-known (cf.
[9]),
in general, tlie Holder continuity of
the coefficients
aij(x)
does not suffice for solutions of
(1)
to have
the property (W). Therefore, tlie regularity condition
(3)
is
optimal.
Under general conditions (2), (3), tlie unique continuation theorem
for
(1)
was shown under different assumptions on
b
and
V,W
by
[l],
Hormander proved tlie property (W) for
(l),
when
n
>
4,
V
=
0,W
E
LFoc(Rn),
p
2
(4n
-
2)/7 and
b
E
L&,(R"),
q
>
(3n
-
2)/2;
(S)
at the origin, when
n
2
3,
V
=
O,lb(z)I
5
C/1z1'-6,
IW(z)I
I
C/1x12-6
for some
6
>
0.
When
A(z)
E
C"(R),
Sogge
[ll]
proved
(S)
for
(l),
if
lb(z)I
5
C/l~l'-~
for some
6
>
0
and
V
=
0,W
E
w-L"'~ (see also [13]).
on some open subset
R'
of
R,
then u
=
0
in
R.
[GI,
[41
and
[51.
1
oc
Unique Con
t
in
ua
t
ion
Theorem
203
The standard approach for the unique continuation problem is
to
establish an appropriate Ca.rleman estimate and almost all results
was shown by this method
([l], [2],
[GI,
[7],
[lo],
[12],
[ll],
[13]).
Garofalo and Lin found
a
new a.pproach to this problem and par-
tially improved the result of
[GI:
they proved
(S)
at the origin, when
n
2
3,
v
=
O,lb(.)I
I
Cf(1~1)/1.1,
IW(.)l
I
Cf(l.l)/l.I2
with
so'"
f(t)/tdt
<
+oo.
In this paper we shall extend these results
to
several directions
under an additional assumption on the quantity
(2V
+
z
.
VV)-. In
particular, we shall generalize the results of
[4],
[5]
and show
(W)
for
(1)
with
b
=
0
under the assumptions
(i)
V,I.lIVVI,W+,(la:IW+)2
E
Ii,(R)
t
F~(R)
for some
1
<
t
5
n/2, (ii)
ITY-(Z)~
5
6/lx12
for
sufficientlly small
6
>
0,
(iii) certain smallness condition on
V-;
(S)
under additional technical conditions. We also deal with the case
b
+
0
and basically our assumption
on
b
is the same as in 151.
2
Main
Results
To state our results we first recall the definitions of
Iin(Q)
and
Ft(R).
V
E
Lio,(Rn)
is said to be of the
Kate
class
Kn
if
where
BT(5)
=
{y
E
Rnllz
-
yI
<
r}
for
r
>
0.
For
1
5
t
5
n/2,
V
E
Lfo,(R")
is said to be of the Fefferman and Phong class
Ft
if
We say
V
E
Kn(R)
(resp.
V
E
Ft(s2))
if
xnl'
E
Iin
(resp. xnV
E
Ft),
where
XQ
is the characteristic function
of
R.
We note that
Fn/2
=
L"12(R")
C
F;
C
F,
for
1
6
s
6
2
6
71/2 and weak-L"12(R")
C
Ft
for every
t
E:
1,n/2);
1'
E
K,(R)
implies
V
E
Fl(R); and that
L"12(R)
and
K,(R)
are incomparable for
n
2
3.
For
1
<
t
5
n/2, we
define the function space
Qt(R)
by
Qt(R)
=
{V
=
V1
+
V2;
V1
E
We introduce some functional spa.ces.
204
Kazuhiro Kurata
Kn(R),
V2
E
&(a)}
and set
v=
vl+
vz
EQ
t
(0)
q(r;
2;
V)
=
inf
{qK(r;
XE,(z)nRVl)+
II
xE,(z)nnV2
IIF~)
For
1
<
t
5
n/2
and
E
>
0,
V
E
Qt(R)
is said
to
be in
M(Q;t,e)
if;
(6)
(a)
V-
satisfies limr,o(supzER
q(~;
x;
V-))
5
E.
(b) For every
z,
E
R7
there exists
ro
>
0
such that
Ix-z,lIVV(z)l
E
Qt(Br,(xo)
n
0).
For
b,
V,
(for
W,
see
REMARK
2),
we wsume
ASSUMPTION
(A.1):
(i)
V
E
M(R;
t,
E)
for
a
sufficiently small
E
=
~(n,
t,
A,
r).
(ii) For every
x,
E
R
lim
q(r;
x,;
(2V
+
(x
-
2,)
-
VV)-)
=
0,
r+O
k-ca
liin
q(r,;
2,;
V-x(V->k))
=
0.
(7)
(iii) When
b(x)
+
0,
for every
x,
E
52,
there exists
f
:
(O,r,)
+
R+
and
C
>
0
such that
f
is iiondecreasing on
(0,
r,),
lim,,o
f
(r)
=
0,
and for every
x
E
BTO(xo)
n
52
To
obtain the property
(S)
for
L,
we require an additional
ASSUMPTION
(A.2):
For every
zo
E
R
Theorem
1
Suppose that
(A.l)
and
(A.2)
are satisfied. Then
L
has
the property
(S)
in
R
for
lV:L(R)-solutions.
Unique Con
t
in ua
t
ion
Theorem
205
Theorem
2
Suppose that
(A.l)
is
satisfied. Then
L
has the fol-
lowing property
(PI
in
52
for 1V:L(R)-soZutions: If
u
E
W;;(R)
is
a
weak solution of
(1)
in
S2
and
satisfies, for some
x,
E
R
and
A,a
>
0,
Lo,
u2dz
=
o(exp(-$))
(10)
as
T
+
0,
then
u
3
0
in
0.
To
obtain the property
(W)
for
L,
we can weaken our conditions.
ASSUMPTION
(C):
(i)
V
E
A4(52;t,c)
for some
1
<
t
5
n/2
and
a
sufficiently small
=
+,
t,
A,
r).
(ii)
When
b(x)
f
0,
for every
z,
E
52,
there exist
r,
>
0,
C
>
0
and
a
sufficiently small
c(n,
A,
I?)
>
0
such that
Theorem
3
Suppose that
(2),
(3)
and
Assumption
(C)
is
satisfied.
Then
L
has the property
(14')
in
R
for lV/$(R)-solutions.
We should mention several remarks on Theorems
1,
2
and
3.
REMARK
1:
We obtain Theorems
1,
2
and
3
by
strong quantita-
tive estimates; for example, under (A.l) and
(A.2),
for weak solutions
u
of
(l),
we have
for
0
<
r
5
r*/2,
where
C1
depends on
u,n,t,A,I',
and the local
properties
of
b,
V,
and
14'
at
x,
and
r*(< r,)
on
n,
t,
A,
I',
and
the local properties
of
b,
V,
and
1V
at
2,.
For
the details, see
[8,
Theorem
1.1, 1.2,
l.G].
REMARK
2:
When
1V
+
0,
the property
(W)
for
L
also holds un-
der
W+,(lx-x,lTV+)*
E
Qt(B,,(z,)nS2)
and
lIV-(z)l
5
-j
Ix-xol
for
a
sufficiently small
S(
n,
A,
I?)
>
0;
(P)
under lim,.,o
q(
r;
so;
W+
+
((2
-
6
n,XJ
206
Kazuhiro Kurata
Z,IW+)~)
=
0
and
lW-(z)l
5
Cw;
I-zo
(S)
under
Ji'"(q(r;
x,;
W+)
+Jq(r;
xo;
(Iz
-
~,llY+)~))/r
dr
<
+oo.
REMARK
3:
For solutions
u
E
Wfg(R),
the condition
limk,,
~(r,;
x,;
V-x(v-,k))
=
0
in
(A.l)
can be removed, because
we do not need
(STEP
2)
in the proof of Theorems (see section
3).
REMARK
4:
When
2V
+
(z
-
z,)
-
VV
2
0
a.e.
x
E
B,.o(xO),
then the property
(S)
is satisfied without the smallness assumption
lim,+o suprEn q(~;z;V-)
5
E
on V- in some special case (see
[8,
Theorem
1.51).
However, in general, this smallness condition cannot
be removed (see
[14]).
REMARK
5:
For
A(z)
=
(&j),
the condition (b) in the definition
of
M(R;
t;
6)
can be relaxed by
(z
-
2,)
-
VV(z)
E
Qt(BTo(zO))
n
R).
Theorems
1, 2, 3
extend the results in
[4], [5]
which assumes
V
z
0
and stronger pointwise condition (see section
1)
on
W
in
our terminology; Theorem
1
is
a
partial extension of the result in
[2]
to general
A(z);
the property (P) is studied in
[5]
and
[3],
and
Theorem
2
extends their results to tlie operator
L
with more singular
V.
Let us clarify how do our theorems extend the previous works by
using the following example.
EXAMPLE
1:
Let
V(z)
=
Ii'hl-
-
L,v,,
X
where
1i'6,L7,6,7
2
0
are constants. When
y
>
0,
V satisfies
(A.l)
and
(A.2)
and Theorem
1
yields the property
(S)
for general
A
and
b
satisfying
(2)-(3),
(8)
with
f(t)/t
dt
<
too. The results in
[4], [5]
only assure (W) (see
[5,
Theorem
1.31)
for general elliptic equations
with this potential
V
in the case
0
5
6,7
5
1;
the ones in
[2]
are
applicable for
6,7
2
0
and sufficiently small
Lo,li'o,
but those are
restricted in the special case
A(z)
=
(bij)l<i,j<n
and
b
3
0
(cf.
[ll]).
EXAMPLE
2:
Consider the operator
L
=
-
Egl
AZi
+
V
in
R
C
Rn
and V
=
-xEl
&,,
where
zi,R;
E
R",
u
2
3,
i
=
l,-.-,N
(N
2
1)
and
n
=
vN.
Since
V,
(z
-
2,)
.VV
E
I<,
we can apply
Theorem
3,
and Theorem
2
for solutions
u
E
IY&(R)
of
Lu
=
0.
However, for
N
>
1
previous results do not yield unique continuation
property for
L.
$9
Unique Continua tion
Theorem
207
3
Sketch
of
the
Proof
of
Theorems
The basic idea of the proof is to combine Garofalo and Lin's varia-
tional method and the inequality:
for V
E
Ft
(1
<
t
5
n/2)
and for every
T
>
0,y
E
R",
and
u
E
Cm(Rn).
See
[8]
for the extension of this inequality and its proof.
(STEP
1)
First we use the geometric reduction procedure
of
[l].
We
fix
a
point
x,
in
0,
then the eqution
(1)
is reduced to
where
M
is the Riemannian manifold
(BTo(xo),
G)
for sufficiently
small
T,
>
0
and
bAf
=
G(b/JTj),V,
=
V/,/ij,
WM
=
W/JTj,
and
p(z)
is
a
Lipschitz function satisfying
p(z,)
=
1,
I%(.,
t)l
5
AI,
1-11
5
p(x)
5
1-12
for some positive constants
Al,pl,
and
1-12
which depend
only on
n,X,
and
I'.
Here G(x)
=
(gij(~))~~j,j~,-,
is
determined by
A(z)
as
follows:
..
j;j(
x)
=
arJ(
z)(detA(z))'l(n-2),
gij
(z)
=
P(
z)gij(
z),
where
g
=
Idet(G)I,
A-'(x)
=
(aij(z))l<i,jsn,
(@)
=
(g)ij-',
and
divM,
VM
are the intrinsic divergence and gradient
in
the metric
G.
Note that
~(x)
is the geodesic distance from
z,
to
x
in the metric gjj
(cf.
[l,
p.
4271).
Therefore, to prove Theorems we may assume
x,
=
0
and study
the local properties of solutions for the equation
(14)
on
M
=
(B,,,
G),
B,,
=
BTo(0).
208
Kazuhiro Kurata
(STEP
2)
We approximate the solution
u
of
(14)
by the ones of
the following boundary value problem:
-divM(pVMv)
+
(bk)M
*
VMV
+
(&)~v
-k
(1vk)~v
=
0
in
BR~,
v
=
u
on
~BR~,
(15)
where
fk,
for each
k
>
0,
is
defined by
fk(z)
=
f(z)
(if
lf(z)I
5
k),
=
k
(if
f(z)
>
k),
=
-k
(if
f(z)
<
-k)
for any function
f,
and
bk
=
((bj)k)l<j<n-
There exists
a
sufficiently small
R1
>
0
depending only on
n,
t,
A,
r,
and the local properties of
b
and
V
at
0
such that the problem
(15)
has
a
unique weak solution
v
=
uk
E
Wl;c(B~l)
and
uk
satisfies
lluk
-
u~~HI,~(B~,)
4
0
as b
-+
+oo
(see
[8,
Lemma
4.21).
22
(STEP
3)
Define
-
1
(2
-
VMU)1/V,jfU dvM,
T
J%
and
Hk(r),Ik(r)
and
N~(T)
by using
(lG),
(17)
for functions
bk,
Vk,
Wk,
and
uk
instead of
b,V,W,
and
u.
We use the following identity
(see
[8,
Lemma
4.11)
for
uk
obtained in
(STEP
2).
Let
u
E
1V2*2(Dr,)
satisfy
(14)
a.e. on
Bra.
Assume that
V,
W,
lbI2
E
Qt(0)
for some
1
<
t
5
n/2,
and there exists
r,
>
0
such that
IzllVV(z)I
E
Qt(Br,
n
0).
Then, for
a
sufficiently small
R1
=
Rl(n,A,r)
>
0
and for a.e.
T
E
(O,R1),
we have
n-2
I'(
T)
=
---I(
T')
+
2
puz
dSM
-f
J(
T;
w)
+
&(
T)
T
r
Unique Continuation
Theorem
209
where
up
=
(X/IXI)
*
VMU,
InO(r)I
5
~(SB,
1vM1U2
dvM
+
D(r))*
(STEP
4)
Let
(A.l)
and
(A.2)
be satisfied. By using
(13),
[3,
Lemma
1.11
and
(18),
we compute
Ni(r)/Nk(r)
and obtain
T E
(O,R1),
k
=
1,2,...,
for
some
C,C,
>
0,
where
~(r;
U)
=
~(r;
0;
U)
and
O(T)
=
q(r;
(2V
-t
X
*
vv)-)
+
q(r;
W+)
+
Jq(r;
(I.lw+)2).
(STEP
5)
By using
-$(log(*))
=
2F
-I-
O(
l),
we have
Taking
k
+
+oo
of
(20),
we obtain
where
C2
depends oiily
011
n,
t,
A,
I'
and
C3
on
n,
t,
A,
r,
and the local
properties
of
b,
V,
and
IV
at
0.
This implies Theorem
1.
Theorem
2
and Theorem
3
can be proved
in
the same way.
Bibliography
[l]
N.
Aronszajn,
A.
Krzywicki and
J.
Szarski,
A
unique continua-
tion theorems
for
the exterior differential forms
on
Riemannian
manifolds,
Ark. Mat.,
4,
1962,
p.
417-453.