Annales de l’institut Fourier (AIF)

Delange, Hubert

Sur les zéros réels des polynômes de Bernoulli

Kuhn, M. Gabriella

Random walks on free products

Ruiz, Alberto

Vega, Luis

ANNALES DE L’INSTITUT FOURIER
ALBERTO RUIZ
LUIS VEGA
Unique continuation for the solutions of
the laplacian plus a drift
Annales de l’institut Fourier, tome 41, no 3 (1991), p. 651-663
<http://www.numdam.org/item?id=AIF_1991__41_3_651_0>
© Annales de l’institut Fourier, 1991, tous droits réservés.
L’accès aux archives de la revue « Annales de l’institut Fourier »
(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-
nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-
tion commerciale ou impression systématique est constitutive d’une in-
fraction pénale. Toute copie ou impression de ce fichier doit conte-
nir la présente mention de copyright.
Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
Ann. Inst. Fourier, Grenoble
41, 3 (1991), G51-663
UNIQUE CONTINUATION FOR THE SOLUTIONS
OF THE LAPLACIAN PLUS A DRIFT
by A. RUIZ & L. VEGA
1. Introduction.
We consider solutions of the inequality
(1) |A<r)| < V(x)\Vu(x)\ xefl
where fl is an open connected set contained in R". We say that (1) lias
the unique continuation property if any solution which vanishes in an open
subdomain of Q must be identically zero in fL
Unique continuation properties for inequalities as (1) with singular
drift V have been treated in several works. Hormander [H] studied a general
case
(2) \Lu(x)\<V(x)\^u(x)\+W(x)[a(x)\,
where L is an elliptic Lipschitz coefficient operator. In particular, concer-
ning L^ spaces he proved unique continuation if q > n~ . This result
3n — 2
was extended to = —,— in [BKRS] when L is the Laplacian. Both proofs
are based on the so-called Carleman estimates which turn out to be false
i f( / < —^— (see [Je]).
Key-words : Unique continuation - Schrodingcr operators.
A.M.S. ClassiFication : 35L15 - 42B25.
652 A. RUIZ & L. VEGA
In this work we substitute L^ for the spaces of Morrey, also called
the Fefferman-Phong class for its recent use in the study of the eigenvalues
ofSchrodinger operators [FeP]. (See also [CS], [K], [ChR], [ChF], and [RV]
where these classes appear in the unique continuation context.)
We say that a function v ^ 0 locally in Z^ is in F^y a <^ n / p , if
there exists TQ > 0 such that
{ / \1^ 1(3) ||H|kp=sup ra[————[ ^) , x^^r<ro\<oo\^\B(x,r)\ JB(x,r) ) J
where B(x^r) denotes the ball of radius r centered at x and \B(x,r)\
its volume. This class corresponds to Lp^ in the classical notation for
\=ap-n. (See[St], [C], [P].)
We prove the unique continuation property for solutions u in Jf^(^),
the Sobolev space of functions in L2^), with two derivatives in Z/2^),
for the inequality (1) when the drift term is in F^ if a < 1 and
p^(n-2) /2( l -a ) .
Notice that it follows very easily from the definition of F^ that
L^ = F^ and therefore our result includes the previously known ones.
On the other hand, it also applies to functions with worse singularities.
For instance, assume that T : R71 —> R771 is linear and surjective and
consider w : Rm —^ R a function in Lq for q >, (2m 4- n — 2)/2, then
the function v{x) = w(Tx) satisfies the above conditions and (1) has the
unique continuation property. However, if m < n this function does not
Qy^ __ 2
need to be in L^ when p >, —_—. This result has interest in applications
to physics, where very often the singularities of potentials and drifts are on
manifolds of positive codimension.
The method we use is based on Carleman L2 weighted inequalities as
in [CS] and [ChR] and the diadic decomposition used in [RV]. There are
some technical complications since the linear Carleman weight <p{x) =7-3;
does not work and we need to use a parametrix, as in [Je], adapted to the
quadratic weight (p(x) == Xn + a^/2.
We would like to thank S. Chanillo for calling our attention to this
type of lower dimensional potential.
We, lastly, have some comments about the notation.
L2^) : The set of functions / such that
UNIQUE CONTINUATION FOR THE SOLUTIONS OF THE LAPLACIAN 653
\\f\\L-w= (hfW^vWdx} <oo.
I |a,;, : The norm of the Morrey space defined in (3).
A : The Fourier Transform in IR71.
A' : The Fourier transform with respect to x ' where x = {x' ,Xn} €
R77-1 x R.
\A '' Characteristic function of the set A.
D ' . - i ( Q / Q x ^ . . ^ Q I Q x ^ .
2. Statement of the results.
THEOREM 1. — Let u be a solution of (1) in H'^(^) and V a
nonnegative function in F^ for p ^ (n - 2)/2(1 - a), a < 1, with the
further hypothesis of \\\V [ [ \a,p being sufficiently small if p = (n-2)/2(l-a).
Then ifu vanishes in an open subset ofQ, u must be zero in Q.
The proof of Theorem 1 relies on the following Carleman estimate.
THEOREM 2. — There exist constants c > 0, depending only on
n, and Ao > 0 depending only on n and TQ in (3), such that for (p(x) =
x,, + 4/2 and V in F^\ a < 1,
(4) ||e^V^||^(v) ^ A^|||V|||,Je^An||^-i)
holds for any C°° function u supported on R71"1 x [-1/2,1/2] and any
\ > XQ, where
/3=( (a - l )+(7z-2) /2p) /2 .
3. The Proofs.
Proof of Theorem 1 from Theorem 2. — We follow well-known
arguments.
1. As in [KRS], [BKRS], we may reduce by reflection, rotations,
dilations and translations to the case where u = 0 out of the unit ball
654 A. RUIZ & L. VEGA
B((0,. . . , 0, -1), 1) centered at the point (0, . . . , 0, -1) and it suffices to
prove that in this case u must be zero in a neighborhood of the origin.
2. Take u(x)rj(\x\) = g(x) where rj is a C°° function such that 77 = 1 in
[0, e / 2 ] , 77 = 0 in [e, oo), for e > 0 to be chosen later on. We may substitute
V by
W(x) = XB^e)(x) (v(x) + ——\ , 6 > 0,7 < a,
and so we deal with a drift bounded below.
Since W is in F^ we use (4) and write :
II e^g^w) < C^(\\\V\\^p+6)\\e^Ag\\^-^
<. C^(\\\V\\\^ + ̂ )(1|^A^||^-^^^ ̂
+ || e^V^u\\mw-^
where the last term on the right-hand side of the above inequality is
bounded using known Sobolev inequalities (see [FeP], [ChF]). If x is in
supp^n(B(0,^)\B(0,£/2)) there exists ^(e) < 0 such that y(x) < ̂ . Then
if f3 < 0 and X big or |||V|||a,p < 1/2G, we obtain
I I ^V^II^dyY,,,,,,^,) < C'A^IIIVIII.^+^e^llA^II^^-i).
The proof would be finished if the last term is finite, but
Ap = (Ai])u + V77 • VIA + rjAu, rj € C^°\
hence
\\^g\\mw-^ <. C ̂  ̂ u^w-^ < C'Mfr^
\a\<2
since W~\x) < M.
The following elementary lemma is the fundamental tool which allows
us to go beyond the L^ class and to prove the Carleman inequality for
Morrey spaces.
LEMMA 3. — For x € IR" and f € L^W1), define the operator 1C
by
/Cf(x)= f K ( x , x - y ) f ( y ) d y
n
where suppA'(a:, •) is contained in a box R = TT[a^6,]. Using R as a
1=1
fundamental piece, construct by translations of R a grid of boxes {Ry}y^'
such that the interiors of Ry are mutually disjoint and R71 = I J^ . Then
v
ICT^(V) <. C||A"||^(R,^R,,) fsup / V\ \\f\\wv-^-
\ v JR,, )
UNIQUE CONTINUATION FOR THE SOLUTIONS OF THE LAPLACIAN 655
Proof. — Define fy = \R,,f. Because of the assumption about the
support of K we trivially have
E XsuppK(x^)(x - y) \R, (y) < ff\
Then
/ \ 2
IIWi.(V)= ^ E^\ v ) L'^V)
<52ny / \W2v(x)dx
„ J R,,
^52»||Jq|i«(R„^,.)(sup^ v) |̂|/.||?
and b}" Cauchy-Schwarz inequality
^52n||J<||l-.(R,.,R,.)(sup/' v) Ell^ll'i2^-1)-
\ ^ J R,, / y
Proof of Theorem 2. — As we said, we use a parametrix of the
Carleman perturbation as in [Je] and [BKRS]. In the sequel see these
references for the claimed properties.
The substitution u = ve~^ reduces (4) to
\\(D+i\(l+Xn)N)v\\^v)
^c(p,7i,ro,a)|||y|||,,^|||D+zA(l4-a;n)JV|^||^^-i)
where N = (0, . . . ,0,1), supp V C R71-1 x (-1/2,1/2) and D =
—i{Q/9x\,..., Q/9xn)- Our aim is to take a left inverse of
\D + i\(l + x,,)N[2 = ̂  D] - (—— - (1 + xn)\\ ,
which is a differential operator with variable coefficients with respect to
the last variable Xn' Denote by A\ the Fourier transform with respect to
the n — 1 first variables. Thus
(\D + Z-A(1 + Xn^vf (^/) = fK'l2-^ - {1-^XnW2} ^ ^\Xn)
QXr,
^(Xn^'WXn^') S (^n,0
9where ̂ (^,^) = |^ |+(-1) J(^—-(l+a;n)A),j = 1,2. A simple change
OXn
of variable reduces the problem to finding an inverse to the one-variable
656 A. RUIZ & L. VEGA
differential operator — - z. (The details can be found in [Je] p. 124.)
CiZ
Therefore a left inverse of \D-{-i\(l+Xn)N\'2 is given by B'^Xn, D)Bi (xn, D)
where Bj(xn,D), j = 1,2, is the pseudodifferential operator with symbol
(5) Pj^^) = (-l)^-1^! + Xn) + (-l)^-1^!^-1^)
with A2 =s,x= (x ' . xn ) , ^ = (^n), x € R71-1 x [-1/2,1/2], ^ € R71 and
(6) a00 \ pOQb(z, rj)=V2 e-^-^ds ) e-^-^2^2)/2- / e-52/2-5^-^,
^ , ^ € R .
From (6) and the property b{-z,r]) = b(z,rf) it is easy to obtain
r)13 <97
(7) 0^0^^ < ̂  + '52(1 4" tz'n) + ("1)JK/1 ~ ̂ )-1-H-1/?1
for any multi-index a,/? € N71.
Therefore, it is enough to prove
(8) \\T,B^x^D)B^Xn,D)u\\L2(v) ̂ c^M^^)
with supp u C R71-1 x (-1/2,1/2), Tj = 9/9^, 1 <, j < n and
Tn+lU = A(l + Xn)u.
From (7) it follows very easily that if 1 <, j <, n + 1 then TjB'z is
a Calderon - Zygmund operator with constants which depend only on Ao.
(See [Jo], theorem on p. 68.)
Now for V C F^ 0 <a <n , l < p <, n/a consider V = (MV^ )1^1
where M is the Hardy -Littlewood maximal function and 1 < pi < p. Then
V(x) ^ V(x), V is an Ai weight and V € F^P with \\\V\\\^p ^ c\\\V\\\a^
(See [ChF], Lemma 1. Observe the notation there is different and that the
proof still holds for the case 0 < p < n.)
Therefore we have
\\T,B^u\\^v)<.c\\u\\^.
It will be enough then to prove
(9) ll^i^llL(v)<^|||y|||,,,||^||^^-i)
for V C F^ as in Theorem 2 (notice that V~1 ^ V~1) and
B,u(x^xn) = s-1 [ b{s(l + Xn) - s-11$'|, s-1 ̂ e^ f(Qd^
UNIQUE CONTINUATION FOR THE SOLUTIONS OF THE LAPLACIAN 657
with x = (x'\Xn).
Consider now a bump function ^o € C§°(R) such that ^o(x) = 1
if \x\ < 1 and vanishes if \x\ >, 2. For k = 1,2,..., [Igs] + 1 define
T T X
^k{x) = ^o(^) - ̂ o(^rr) and ^oo (aQ = 1 - ̂ o(^ig^i)- Call for
k = 0 , . . . , [Ig 5] + 1 or k = oo
(10) bk(z^)=^(\z-iri\)b(z,rj),
(11) P^(x^)=S^bk(s(l+Xn)-S-l\^S-l^)x=(xf,Xn)^=^^n)
and B^ the pseudodifferential operator with symbol p^g. Notice that by
construction
[ig^l+i
Y, P^s+PTs^P^
k=0
and therefore (9) will follow from
IÎ IÎ (V) ^ c2-^A^|||V|||,^h||^(v-i)(12)
ll^i^ll^(V) ^ ̂ |||y|||^M^(v-i)
with e > 0 and k = 0,1,..., [\gs] + 1.
Let us first deal with B^. From (7) and (11) we have
o^_y_ oo < c^
Qx^Q^18 ~ (52+|$|)l+lal+^l'
Consider p^ = (s2 + |$|)p^ and -^i? tlle associated pseudodifferential
operator. Then B^ is as TjB^ and can be treated in the same way.
Therefore we have to prove
(13) ll^ll^(y)^c^|||y||k,h||^(v-i)
for V C F ,̂ 2/3 = (a - 1) + n—2, a < 1 and (Jf)7^^) = 5-2^(5-20^(0Ip
where <^($) = (1 + |$[)~1- Then it is well known that cj) behaves as [a;]1"71
when x is close to the origin and decays faster than any polynomial at
00
infinity. Therefore we can write ^{x) = V^ ^jW with supp cj)j C B(0,2-7)
j=-oo
and
fll^-lk00^2"^"^ i f j ^ o
lll^||L-^c^2-^ i f j^O.
Thus we have
II ̂ "Ikw ^ c^"-1) ̂  l|$,(s2.) * u\\^(v)
658 A. RUIZ & L. VEGA
and by Lemma 3 and the estimate for ||^,HL^
,,,,,,-,,(^^..->(.up^F)+Sc,.2-»<(s,,^v))l|<.||,.,,.,,
where {Q^},^z} is a grid made by cubes of volume (s~'22j)n. On the other
hand
sup/ y^^-^r-iiiyin^.
y JQ\.
Putting all the estimates together and choosing m > n — a we obtain, if
a < 1
II^HL^V) ^ c^-^IIIVIII^ill^ll^^.-^
which is better than (13).
Let us consider now p^ for k = 0,1,..., [\gs] + 1. Recall that B^f
is given by
^UW = f A'(.T,,, x - y)f(y) dy x = (^ Xn)
where
(1^) K(x^y)= [p'l^Qe^dfi
and 7^, as in (11). Notire that K depends on k and s but for simplicity of
the notation we have dropped the index.
First we need an upper bound of the kernel A".
LEMMA 4. — Define K as in (14). Then for any natural number m
there exists a constant c,,, > 0 such that
(15)
|A'(.r,,^)| $ c,,6>2n-32A:(l+|2A:^TU)-l(l+|^!//|)-(^^^^^^^
where y = (.//, //„) € R" and \Xn\ < 1/2.
Proof; — Assume first that |^,| <, (2k~ls)~l and \y'\ <, (2k~ls)~l.
Then we can write using cylindrical coordinates $ = (|$'|^,$n), ^ € 5"~2
K ( x ^ y ) ^ f e^'d |$'|"-2
^Q\ •7|$«|<•2A+25 V2^-35<||$'|-52(l+.r«)|<2/••+^
• f A^^)^1^^1"^^^^!)^.J s " - ' 2 /
Tlierefore
IA^.T, , y)\ < c22^2^-1) sup sup / p^(xn, ̂ e^^^ da^
^, ^/100<|$'|^10062 J S " - 2
<c22ks2{n-l)(2ks)-l(l + ̂ Vl)-1"-2)/2
UNIQUE CONTINUATION FOR THE SOLUTIONS OF THE LAPLACIAN 659
where the last step is a consequence of (7), (11) and the well-known
estimates of the Fourier transform of a smooth measure on the sphere (see
[Se] p. 233).
The general case is proved by integration by parts in (16) with respect
to the variables $„ and |$'| and using again (7).
Hereafter we shall follow very closely the argument in [T]. The result
will be obtained by interpolation between two estimates. The first one will
be an L2^"1) - L'2 (a;) estimate (uj will be eventually V^ and the second
one an L'2 — L2 bound. In order to obtain the first part we shall need first
an upper bound of the kernel. This suggests to cut K into pieces according
to its level values. Consider as before the bump function ^o C C§°(R) such
that ^o(x) = 1 if \x\ < 1 and vanishes if |a;| ^ 2. Then for j = 1,2,... we
define ^j(x) = ^o(^j) - ̂ 0(^1-) and
(17) K,(xn,y^yn) = ̂ yWx^y'^n).
00
Then K = ̂  Kj and from the Lemma 4 we have
j=o
(18)
f |A'j| < c^2"-^-^^^!^ ̂ •s^l"1)-1 i f 0< j ^ [Igs] 4- 1 - k
\\Kj\ ̂ c^s2tt-3^1n2k-^l^)2-1n(k+•i\l\2ksyn\1n)~l \!j>[\gs]+k
and supp Kj C {(^^H ^ 2j+'2s~2}. Then we have the following
lemma.
LEMMA 5. — For j = 0,1,..., define Kj as in (17). Then
(19) liy^(^^-2/V(2/)^ll^(^Q^T,||||y|||S^
where
^ ^ f c^^-'^77-2^)/2 ifO ^ j<, [\gs] + 1 - k
rj \ c^52a^-22^n-•2a^/2(2A•+^-l)-m ifj ^ [lg5] + 1 - k.
Proof. — We want to apply Lemma 3 but notice that the kernel
Kj has compact support only in y ' variables. Therefore we have to cut Kj
according to level sets with respect to the yn variable. Define for d = 0,1,...
Kjd{xn,y',yn) = Kj^n^y'^y^xiAVn)
where \i^ is the characteristic function of the set Id and Id = {yni^ <:
2ksyn <, 2rf+l} if d + 0 and Jo = {2/n,M <: (2^)-1}. Notice that
00
Kj = y ^ Kjd and the support of Kjd is contained in a box of dimension
d=0
^0 A. RUIZ & L. VEGA
2^5-2 x • . . x V^s-2 x 2d-ks-l. We can apply Lemma 3 to K^ where
the grid has the dimensions we have just mentioned. Call {R^} the boxes
of that grid. Therefore, if 0 ^ j ^ [Igs] + 1 - k, the left hand side of
inequality (19) is bounded by \\Kjd\\L» fsup,J^ VP\ But (18) gives
\ jd /
\\K,d\\L^ $c^2-7n^2n-32^(2-2).
On the other hand, decomposing each R^ in cubes with height equal to
2^+25-2
..p/^^L^,,/ y.\
v JR^ 2^+25-2 ^ y JQ^ )
< c^"2^^2^^"0^"'1^""^!!'̂ !?— Ill • llla,p
where {Q^}^ is a grid made by cubes of volume 2n^+2)5-2n. Putting both
estimates together and summing up in d we obtain the desired result in
this case.
Now consider j ^ [\gs] + 1 - k. Then by (18)
ll̂ rilk- < Cn^-3^^"^ ̂ 2-^+^2-^
and
sup [ vy < sup / yp if 2d-ks~l < 2j+2s~2
^ JR-i'1 v JQ^ ~
-tf j d
r f>d(t>k^~l ( r \
^j^vp ̂  -^ sup^vp) i{2d~ks~l ̂  y+2s-2-
\ j d /
Using both estimates in Lemma 3 and summing up in d we obtain the
desired bound also in this case.
Finally we need the following lemma
LEMMA 6. — For j = 0,1,..., define Kj as in (17). Then
(20) liy^(^^-2/)/(2/)^||^(R.-i^_i^/^^cT^|/»^
where
r ' = [ 2js~2 if^ ^ P^l + 1 - k3 \c2-ks-l ifj^[lgs]+l-k.
Before proving the lemma notice that a standard interpolation theo-
rem together with the estimates (19) and (20) give us
\\JKj{xn,x-y)f(y)dy\\L^y^cr^\\f\\^y-^
UNIQUE CONTINUATION FOR THE SOLUTIONS OF THE LAPLACIAN 661
where
,,^r c52(Q-l)2^(l-a)+(n-2)/2p)|||y|||^,p i f o ^ j ^ [ig5] +1 - k
rj [ c,ns2a~l~hj^-a)2~fc(l~^\2^ss-l)-u^ ifj ̂  [lg5] + 1 - fc.
Then summing up in j we obtain (take m > — — a)
z
f K(xn,x-y}f(y)dy\\^.(v) ̂  cs^-^-^'-^^V^M^v-.)
where 7^ is given in (14) and 2/3 = a — 1 + ———. Therefore, the proof of
2p
Theorem 2 is finished as long as we prove Lemma 6.
Proof of Lemma 6. — Using the definitions of Kj and K in (17) and
(14) respectively we have
I K,{xn,x -y)f{y)dy = I ̂ s^x'-y))f{y) f p^Xn^e^-^ d^
= ffWq{xn^e1^ drj
where
(21) qj(Xn,rf,rin) = fpU^^n)^-^^ (^rf) d^ •
Notice that the pseudodifferential operator defined by qj is of variable
coefficients just with respect to Xn' Then using PlancherePs theorem we
have
I /fWqix^e^ J|
II-7 II^R"-1^-!^,!^))
/•1/2 || f ( f ̂  \ , , ||2
= / / / fW^q^ri'^e^-drjn e" •" drf\\ dx,,
J - l / 2 \\J \J / l l2,2(R•— l)
r y1 /2 r - 2
= / / f(r^',r^n)q(xn,rf,r]n)eix^"dr]n dxndrj'.
JR..-I J-l/2 J
Therefore it is enough to prove ([Jo], p. 68)
A3 W
(22) —9^ ^CT,'(I+|^|)-^ -7=0,1,2.
Qx-n "Sn
Consider first ^ = 7 = 0 . From the definition of i))j we have
(23)
-̂<.(iy)
=^-2)n-l^(^)-(2^-2)n-l,'o(^).
662 A. RUIZ & L. VEGA
On the other hand, recall that the support of p^(a'n^'^n) is contai-
ned in the cylinder ||$'| - ̂ (l + Xn)\ < 2k^ls, Therefore if 2~js2 <: 2ks
(i-e., j > [\gs] + 1 -- k) by Young's inequality (11), (7) and (22) we have
ML- <. IbU -̂21^ i'o (-7)!!^ < c2-ks-l.s
.,/ oo .// ' A
H^"^2 ^ 2^5 write ^o"1' z . ^om where
m=i
iom (^ =?0 (^) (X{|^|<2-}($') -^{^l^-i}^)) .
We shall prove (22) for m == 0 being the general case a consequence of the
A'
exponential decay of ^ p. Thus
ll^|l^<lb?J|L-(2^-2)n~lll ?oolk-
. {vol(supp ̂ (a;,, .,^,) n {|^| ^ 2-^2}) ^ cVs-\
If ^ ^ 0, or 7 7^ 0 we use the above procedure together with (7). Notice
that we only have to compute derivatives with respect to ̂  where the right
decay is obtained.
BIBLIOGRAPHIE
[BKRS] BARCELO, KENIG, RUIZ, SOGGE, Weighted Sobolev inequalities and unique
continuation for the Laplaciaan plus lower order terms, 111. J. of Math., 32,
n.2 (1988), 230-245.
[C] S.CAMPANATO, Proprieta di inclusione per spazi di Morrey, Ricerche Mat., 12
(1963), 67-896.
[CS] S. CHANILLO, E. SAWYER, Unique continuation for A+V and the C. Fefferman-
Phong class, preprint.
[ChR] F. CHIARENZA, A. RUIZ, Uniform L2 weighted inequalities, Proc. A.M.S., to
appear.
[ChF] F. CHIARENZA, M. FRASCA, A remark on a paper by C. Fefferman, Proc.
A.M.S., (Feb. 1990), 407-409.
[FeP] C. FEFFERMAN, D.H. PHONG, Lower bounds for Schrodinger equations,
Journees Eqs. aux deriv. partielles, St. Jean de Monts, 1982.
[GL] N. GAROFALO, F.H. LIN, Unique continuation for elliptic operators; a geometric
variational approach, Comm. Pure Appl. Math., 40 (1987), 347-366.
[H] L. HORMANDER, Uniqueness theorem for second order differential operators,
Comm. PDE, 8 (1983), 21-64.
[Je] D. JERISON, Carleman inequalities for the Dirac and Laplace operators and
unique continuation, Adv. In Math., 63 (1986), 118-134.
UNIQUE CONTINUATION FOR THE SOLUTIONS OF THE LAPLACIAN 663
[Jo] J.L. JOURNE, Calderon-Zygmund operators, pseudo-differential operators, and
the Cauchyintegral of Calderon, Lecture Notes in Math., Springer Verlag,
1983.
[K] C. KENIG, Restriction theorems, Carleman estimates, uniform Sobolev inequa-
lities and unique continuation. Harmonic Analysis and PDE'S, Proceedings
El Escorial 1987, Springer Verlag, 1384, (1989), 69-90.
[P] J. PEETRE, On the theory of Ly^ spaces, J. Funct. Anal., 4(1969), 71-87.
[RV] A. RUIZ, L VEGA, Unique continuation for Schrodinger operators in Morrey
spaces, preprint.
[St] G. STAMPACCHIA, L^^-spaces and interpolation, Comm. on Pure and Appl.
Math., XVII (1964), 293-306.
[Se] E. STEIN, Oscillatory integrals in Fourier Analysis. In : Beijing Lectures in
Harmonic Analysis, Princeton Univ. Press, 112 (1986), 307-355.
[T] P. TOMAS, A restriction theorem for the Fourier transform, Bull. AMS, (1975),
477-478.
Manuscrit recu Ie 26 juillet 1990,
revise Ie 5 decembre 1990.
A. RUIZ & L. VEGA,
Departamento de Matematicas
Universidad Autonoma de Madrid
28049 Madrid
(Espagne).


Unique continuation for the solutions of the laplacian plus a drift

Blümlinger, M.

Obata, N.

Permutations preserving Cesàro mean, densities of natural numbers and uniform distribution of sequences

Moerdijk, Ieke

Classifying toposes and foliations

Nadirashvili, Nikolai S.

ANNALES DE L’INSTITUT FOURIER
NIKOLAI S. NADIRASHVILI
Metric properties of eigenfunctions of the
Laplace operator on manifolds
Annales de l’institut Fourier, tome 41, no 1 (1991), p. 259-265
<http://www.numdam.org/item?id=AIF_1991__41_1_259_0>
© Annales de l’institut Fourier, 1991, tous droits réservés.
L’accès aux archives de la revue « Annales de l’institut Fourier »
(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-
nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-
tion commerciale ou impression systématique est constitutive d’une in-
fraction pénale. Toute copie ou impression de ce fichier doit conte-
nir la présente mention de copyright.
Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
Ann. Inst. Fourier, Grenoble
41, 1 (1991), 259-265
METRIC PROPERTIES OF EIGENFUNCTIONS
OF THE LAPLACE OPERATOR ON MANIFOLDS
by Nikolai S. NADIRASHVILI
In this note, we prove two theorems which express a quasi-symmetry
relation between the positive and the negative part of the distribution
function of an eigenfunction of the Laplace operator on a Riemannian
manifold.
1. An estimate of the volume of a domain on which
an eigenfunction of the Laplace operator on a Riemannian surface
has constant sign.
Let M be a two-dimensional compact real analytic Riemannian
manifold, ^1,^25 • • .-eigenfunctions of the Laplace operator on M, Ai^ =
\iUi.
THEOREM. — There exists a positive constant C which depends on
M such that, for every i = 1,2,...,
vol{rc e M,Ui(x) > 0} > C .
The proof is based on the two following lemmas.
Let / be a bounded function, continuous on [0,1]. Let us denote by
N(f) the number of changes in the sign of the function / on [0,1].
Key-words : Beltrami-Laplace operator - Eigenfunctions - Nodal sets.
A.M.S. Classification : 58G25.
260 NIKOLAI S. NADIRASHVILI
LEMMA 1. — Let fn be a sequence of non-zero continuous functions,
defined on R, with values in R, with support in [0,1] and assume that
N{fn) is bounded by some fixed number N.
Then there exists a subsequence ni of N (n^ —> oo), real numbers
0m, such that: On, .fm converges as i —^ oo, for the (weak-)topology of the
space of distributions V to a non-zero distribution of order less than N.
Remark. — First, we recall that a distribution is said of order less
than N if it is a sum of derivatives of order less than N of Radon measures.
Moreover, if P is a polynomial of degree N and p, a Radon measure, the
set of all T 6 V satisfying PT = /A is an TV-dimensionnal affine subspace
of the space of all distributions of order less than N.
Proof of Lemma 1 (suggested by Y. Colin de Verdiere). — Let
N
Pn = TT (^ —Xk) be a sequence of polynomials of degree N such that Pn-fn
k=l
is > 0. By renormalisation and taking a subsequence, we may assume that
L Pn-fn = I? ^na^ Pn-fn converges to a probability measure fi and that
Pn converges to a polynomial P of degree exactly N.
Let To e V be such that : PTo = /A.
Let Tn = fn - To, then we get :
l imPn.Tn=0.
We introduce now the following decomposition of the space of distri-
butions :
P' = Zp C W ,
where Zp = {r € T>'\PT = 0}, and W is a topological complement of Z p .
W is a complement to Zp^ if n is big enough. Now we can write
uniquely :
Tn= Zn+Wn ,
where Zn e Zp^ and Wn € W. Now Pn-^n —> 0 and we deduce that Wn —^ 0,
because the multiplication by Pn is uniformly invertible in W.
Now Zn C Zp^ and Zp^ converges to Z p .
Two cases are possible :
First case : Zn is bounded and we can extract a convergent subse-
quence converging to 7i in Z p . Then TQ -{- Ti is not zero and we get the
conclusion.
METRIC PROPERTIES OF EIGENFUNCTIONS 261
Second case : Zn is unbounded; then there exists a sequence /3m —> 0
such that f3n, .Zn, converges to a non-zero distribution Ti and then :
/Wn.
converges to Ji.
Let us denote by B the unit disk in R2, S == QB, if / is a continuous
function on S then N(f) is the number of changes of sign of the function
/on S.
LEMMA 2. — Let u be a harmonic function in B which is continuous
in B, u \s= f, u(0) = 0. Let N(f) =k<oo. Define
Gy, = {x e B , u(x) > 0} .
Then mes Gu > C, where constant C > 0 is dependent on k .
Proof. — Let us assume the contrary. This means that there exists a
sequence of harmonic functions Un in B, Un\s = /n, Un(0) = 0, N(fn) < k,
mes Gu^ —> 0, n —> oo. According to lemma 1 there exists a real valued
sequence a^n and a subsequence fn^ such that, Omfn^ -^ f 1=- 0 in the
sense of distributions. From the convergence of the distributions Omfn^ on
S it follows that in an arbitrary compact subdomain of B the convergence
of functions Q-^Un^ is uniform. Let OmUn^ -> U in B. From [1] it follows
that U ^ 0 in B if / ^ 0 on S. We have (7(0) = 0. From the assumption
mes Gu^ —^ 0, n -^ oo, it follows that U < 0 in B. Equality (7(0) = 0 and
inequality U < 0 in B contradicts the maximum principle for harmonic
functions.
Proof of the theorem.
1. Let us denote by Bf, x e M, r, the geodesic circle on M with
centre x and radius r.
There is a constant Co > 0, such that for every e > 0 there exists
points xi ...XN € M, N > Co/e2, such that the circles B^2...B^N
mutually have no intersections.
2. There exists a constant ro, such that for every x € M, 0 < r < ro,
Bf is diffeomorphic to a disk.
3. There is a constant C\ > 0, such that for all x € M, A > 0 in the
circle B^ r. there exists a positive solution of the equation AZA -h Xu = 0.
4. Let x ^ M, A > 0, r ^ l / C ^ y / X < ro, u is a solution of the
equation AIA -h \u == 0 in JSjf. Then there exists a diffeomorphism h of the
262 NIKOLAI S. NADIRASHVILI
unit disk B on Bf, /i(0) = x, and a function s in B, 0 < s < oo, such
that s.u(h) is a harmonic function in B (by a representation theorem in
quasiconformal mapping theory, [2]). From the compactness of M it follows
that the Jacobian of the mapping h is uniformly bounded.
5. There is a constant C^ > 0, such that for all x € M, A > 0 in the
circle ^x /- every solution u ̂  0 of the equation AZA + \u = 0 changes
its sign [3].
6. Let Ui be an eigenfunction, Az^ = A^ on M, 7 is a nodal line of
the function Ui. For a two-dimensional real analytic manifold the following
estimate is true, [4],
length 7 < CsV^z
where constant 63 > 0 is dependent on M.
7. Let Ui be an eigenfunction, Ai^ == A^ on M. According to 1 we
can choose circles Bf1 ... B^ with £ = 2/62 \/Az. We have TV > CoCj\i/4.
According to 5 there exist points yn G B^, n = 1 . . . N , such that
Ui(yn) = 0.
According to 6 at least N / 2 points y k ^ - ' - V k j , J > N / 2 , from the set
{yn} have the following property : for all j = 1... J there exist r^,
1 1
< r, < ——=- ,
2CiV\i J C^V\^v-/l v ^% ^1  ^z
such that restriction of the function uj on QB^3 has no more than
8CiC3
C^Co
zeros.
According to 4 and lemma 2 for all j = 1... J
mes{a; € B?^ , ui(x) > 0} > C^2 .
We have J > Co/2e2 and so the theorem is proved.
2. An estimate of the relation between the positive
and the negative extrema of an eigenfunction.
Let M be a n-dimensional compact smooth Riemannian manifold,
^1,^2,.. .-eigenfunctions of the Laplace operator on M, Az^ = \iUi.
METRIC PROPERTIES OF EIGENFUNCTIONS 263
THEOREM 2. — There exists a positive constant C which depends
only on n and a positive constant N which depends on M such that, for
every i > N,
supUi(x)
— < M < r
C" |inf^)| ' •
M
by
We denote by Br C R71 the ball centered at 0 of radius r.
In Br we consider a uniformly elliptic second order operator L defined
^=E^(^)' (2J)
zj J
where Ozj is a symmetric positive definite matrix in Br. If the eigenvalues
of the matrix ||a^(^)|| lie on the segment [e'^e], e > 1 we say that the
operator L has an ellipticity constant e.
LEMMA 3. — Let u be a solution of the equation
a(x)Ly + \u == 0
in the ball B\, I/A < a(x) < A, A > 0, L is an elliptic operator with the
ellipticity constant e, A is a constant such that \\\ < C. Let us assume that
u(xo) > 0 and that there exists XQ € ^1/2 with u{xo) = 0. Then
\miu\ > Su(0) ,
B\
where the constant 6 > 0, 6 = 6{n, A, e, C).
Proof.
1. We shall prove Lemma 2 under the assumption that \ = 0. Denote
<^i =sup{0,n |aBi}
</?2 =inf{0,n \QBi} -
Let ui, ^2 be the solutions of the following Dirichlet problems :
Lu\ =0 in B\, u\ \QB^.= ^i?
L^2 =0 in f?2 , Z&2 |^Bi= ^2-
Then, ̂  = ^i +112, lAi > 0 in f?i, 14 (0) > u(Q). From the Harnack inequality
[5] it follows that there exists a constant 8 > 0, 8 = 6(n, e) such that
Ul \B1/2> ^l(O) .
264 NIKOLAI S. NADIRASHVILI
Since u{xo} = 0, XQ € Bi/2, then
infy?2 < -^i(O) < -6u(0) .
2. Let A ^ 0. Let us make a cylindric extension of the functions
u(x),a(x) and the operator L in the new coordinate Xn^-i- After this
extension we shall keep the notations u, a, L. Denote
v =uevxxn+l
clearly the function v is a solution of the elliptic equation
Q^v
aLv + -^-.— = 0 .
^n+l
Now the statement of Lemma 3 follows from the assertion 1 to the function
v in the unit ball in IR71"^1.
Proof of Theorem 2.
1. There are constant d = Ci(M) > 0,C2 = C^M) > 0 such that
for all x 6 M, A > 62 any solution of the equation Ait + \u = 0 in the ball
B^^ change its sign.
2. There exists a constant N > C^, N = N(M), such that for all
x 6 M there exists a diffeomorphism
^c^^^i^
such that the equation A'u -h \u = 0 in B^ , r- viewed in the ball Bi has
2Gi / v A
the form
a(a:)Ln + \'u = 0 (2.2)
where L is an elliptic operator of the type (2.1), e = 2, A = 2, |A| < C =
C(n) > 0. We can obtain such a diffeomorphism d if we introduce in the ball
B3. r. a normal coordinate system. Applying Lemma 3 to the solution
u of the equation (2.2) we obtain the statement of Theorem 2.
BIBLIOGRAPHY
[1] J.-P. LIONS, E. MAGENES, Problemes aux limites non homogenes et application,
vol. 1, Dunod, Paris, 1968.
[2] L. BERS, F. JOHN, M. SCHECHTER, Partial differential equations, Providence, R.I,
1974.
[3] J. BRUNING, Uber Knoten von Eigenfunnktionen des Laplace-Beltrami Operators,
Math. Z., 158 (1978), 15-21.
METRIC PROPERTIES OF EIGENFUNCTIONS 265
[4] H. DONNELLY, C. FEFFERMAN, Nodal sets of eigenfunctions on Riemannian
manifolds, Invent. Math., 93 (1988), 161-183.
[5] D. GILBARG, N.S. TRUDINGER, Elliptic partial differential equations of Second
order, Second Edition, Springer, 1983.
Manuscrit recu Ie 15 fevrier 1991.
N.S. NADIRASHVILI,
Universitat Bielefeld
Fakultat fur Mathematik
Universitat Strasse
Postfach 8640
4800 BIELEFELD
(Allemagne).


Metric properties of eigenfunctions of the Laplace operator on manifolds

Elkinani, Abdellah

Fonctions harmoniques opérant sur les algèbres de Banach involutives

Rovira, Josep Gonzalez

Equations of hyperelliptic modular curves

Flicker, Yuval Z.

Regular trace formula and base change for $GL(n)$

Parreau, François

Ergodicité et pureté des produits de Riesz

Not a member yet