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Title
C＾∞ Null-Solutions for Some Non-Fuchsian Operators with
C＾∞ Coefficients
Author(s)
MANDAI, Takeshi
Citation
[岐阜大学教養部研究報告] vol.[22] p.[95]-[100]
Issue Date
1986
Rights
Version
岐阜大学教養部 (Dept. Math., Fac. Gen. Educ., Gifu Univ.)
URL
http://repository.lib.gifu-u.ac.jp/handle/123456789/47595
※この資料の著作権は、各資料の著者・学協会・出版社等に帰属します。
95
C(゛) N ull- Solutions for Some N on- Fuchsian Operators
with C゛ Coefficients
Dedicated to Professor K .0 hta on his 60th birthday
T akeshi M A N DA I
Dept. M ath., Fac. Gen. Educ., Gifu Univ.
(Received Oct、13, 1986)
§0 . 1ntroduetion
Let t be an 田- th order partial differential operator near (0,0)∈ 瓦 ×召 5j. T he operator
t is called 原 F 麗隔㈲ e 戒出麗 i前川
(0
7) can bewritten asP=
w旅
Σ
餓加 d to ぼ the followi昭 two conditions hold.
aj,a(ぴ) tj DtDl with ら ,o≡1.
j 十¦ α¦ ≦尻
(亀αhavesuitablesmoothness.)
, ( ii)
aj,a(0,x) ≡Oif α≠0.
ln this artid e, we use the following notations.
(ぴ卜 ( 柵 ,‥・み ) are thevariables on R゛1+1.
G・,ぞ) = ( r,ぞI,…ふ ) are thedual variables.
∂
,= ∂
/∂/, 貼 = ∂
/≒ , ∂
χ= (∂
χ. …,∂
4 ), 五= ∂
ぶ几 = ∂が etc・,
μ= 一
泊h Dχj= 召貼ぴ= 1,…ぷ).
μl= £)釦 ‥刀穴 for multi-index α= (α1,…, 恥).
し ¦ ゛ α1+ ‥・+ α
n,
G= {xE R″
; し ¦< ε
} (ε> O).
M.S.Baouendi-C.Goulaouic(田 ,[2]) considered the Cauchy problem for operators of
F uchs type w ith real- analytic coefficients and show ed a Cauchy- K ovalevsky type the-
orem and a Holmgren type theorem. ln the ひ ) category, H .T ahara([51,[6]) considered
Fuchsian hyperbolic (jperators and studied the ひ w e11-posedness of the Cauchy problem
and so on. R oughly speaking, the strud ure of the Cauchy problem for operators of Fuchs
type is sim ilar to that of the non- charad eristic C auchy problem .
T he condition ( i ) is quite natural but the meaning of the condition ( ii ) is not clear.
H ence, it seems natural to ask w hat happens when the condition ( ii ) is violated.
0 ne
possibility is the existence oI C゛ null- solutions. A C∽ function zx near (0, 0) is called a
(グ U㎡ 1- SO臨ti皿 10r t at (0,0) if 乃7= Onear (0,0) and (0,0) E suppzx⊂ { ( ぴ ) け ≧O} . Let the
condition ( i ) issatisfied for theprincipal part of 沢 ln real-analytic case, there exists
a Cy nuII-solution if there eχists (ノ, α) such that プ十 ¦ 訓 = 肌 y< 肴
1, aj,a(0,0) ≠Oand ak,β(0.
-
x匝 Ofor 力十 ¦ β卜肌力く yl (S.0 uchi [4;T heorem 1.8]). ln this article, wewant to get a
similar result in (グ case.
T his reseai ch was partly supported by Grant-in- A id for Scientific Research (N 0. 61540094) , M inistry of
Education, Science and Culture.
T akeshi M A N DA I
96
ln§1, westatethemaintheoremバn〕 provethisthQorem, thbbehavior of bicharacteristic curves plays an important role, which is studied in §2. we prove the theorem in §3.
1n §4, therearestudiedverysimplecasestowhichour theorem doesnotapply.
§L. Statement of Theorem
W e consider the follow ing operator t in a neighborhood び of (0,0) ∈ 瓦 × 召 ;j :
7) =
_
Σ
j
十
¦
α
¦
aj,a( ぴ ) tj D 4D 1 十
=
斑
_
_
.
_
_
ノ
Σ
川
α
¦
臨 a( ぴ ) D4D 1 ,
<
斑
_
where ら,α, ゐ
j,αE C゛( 印 and a・ ,o≡L
`NVe assume the f0110wing tw o assumptions.
(A-1)
Theprincipal symbol 偏 of 7) isfactored asfollows. 八 ( び ; r,ぞ) = j H
(な= 1
ん(ぴ ;ぞ)), where{ん;j = 1府 , 田} arereal anddistind on び×(R ¥ ㈲ ).
(A-2)
Thereexist qE 狂 …, 副 and ぐoE R71¥ { O} such that λ. (O↓
O;ぞo)≠0.
T he following theorem is the aim of this article.
Theorem. Assttme (y1- 7) 凹ぱ(/1-2). Thm けhe托 aistsa (グ 11㎡1-solMtio71和yP at (0,0).
Remarks. (1) The assumption (A-2) isequivalent tothat there exists (J,a) such that y十
㈲ = 肌 j く m andα
j,a(0,0)≠0. Notethatweneednotassumethat のよ 0,x) ≡0
for ん刊 β卜肌ん< ノ.
(2) Notethat wemakeno assumption on lower order termsexceptthatthecoj
ダ
e f f i c
i e
n
t s
a
r
e
C 胞
T
h
e
e
s s e
n
t i a
l s
a
r
e
t h
e
b
e h
a
v
i o
r
o
f
b
i c
h
a
r
a
d
e
r
i s t i c
c
u
r v
e
s
s t a
-
ed below and the we11-posedness of the Cauchy problem in { ( ぴ ) E U ; t > 0} .
E xample. A typical example is 7:) = 陥一心 on 召 2. A C∽ function xx near (0,0) satisfies
乃7= O if and only if z7( ぴ ) = 貳 tび ) for some ひ ) fund ion g(s) near s= 0.
H ence,
if we take g such that OE suppg ⊂ { s≧O} , 瞰 en z7( ぴ ) = g( 友戸) is a Cや nuH-solution
for P at (0,0) .
§2 . Behavior of Bicharacteristic Curves
ln this section, we prove a key lemma which shows that some bicharacteristic curves
goawayfromズニOwhenZ→+ 0.
F ix ・7 and ぞoin the assumption (A - 2) and put λ≡ 几 .
Consider the follow ing system
of ordinary differential equations.
嗇=一
几(な;ぞ
)/け(X
o
)=0
(B
)¦
¥
首二/し( tぶミ)μ ぞ( yo) 二O・
.・
T he solution of this system is the bicharad eristic curve of P p21ssi贈 Z ( 4)) E ( 4),0;λ(0,0; ξo)
/4),ぐo).
L em ma l .
There eχist 如 si面 c 11Mm be拓 y< ダ such th H he加 ao抑泌 g hoはs.
j 加 α砂びΞ(Oy )μheyeeχ
isH 1,t2 sRch that
(α) O< ちく 4< 4),
C N ul】-So】utjons for Some N on- Fuchsian Operators wjth C゛ Coefficients
(ろ)
仇e so㈲ ion oチ (j ) eχists αud satiφ es lx( 1) ¦ ≦ y 知r れ≡[ ち, 4]] ,
(C)
レ巾) ¦ > yかy屹≡
[リ 11.
97
T he key pojnt is that y and ダ do not depend on 4).
P rりof.
Since O≠ λ(0,0; ぞo) ⊇ 公私 (0,0; ξo) ξo,j, we have 几 (0,0; ぞo) ≠ O. Put 島判几 (O,0; ぐo) ¦
> O and £ 1こ max{ 臨 (0,0;ぞo) ¦ , 扁ブ W e can take x゛ > O such that lん ( 咄ぞ) 一几 (0,0; ぐo) 鴎島 /
4 and l恥 ( ぴ ; ぞ) 一心 (0,0洽 )) ¦ ≦ 7L1/4 for μ≡[ O/ ] , ¦刈 ≦ ダ, ぽーぞol≦ y .
By the equations (B) , as long as は( X) ¦ ≦ ど and ぽ ( X) 一引 ≦ が, we have l羞収 ( /) 十ゐ(0,0;
ぞ
o
) (loが-lo匹)} ¦≦{ 隔(な;ぞ)一几(0,0;ぞ
o
)¦≦ム)/訂 andぽ (汐 )- ぐo一心(0,0;ぞo
) (loμ-lo匹)} ¦
1
≦7し1/4た
H ence, we obtain
(* )
¦ボ )十ふ(0,0;ξ
O
) (lOμ-lOg4)) ¦ ≦(秘/4) 110μ-lOg乱
ぽ(/)一心)一几(0,0;ぐO
) (lOμ-lOg4)) ¦ ≦(L小 )110昭一
lOg姉
T hese estimates imply lポ ) ¦ ≦ (5ね /4) 110μ -log4)l and lξ( /) 一首ol≦(51 1/4) 口oμ - log4) ¦ ・
Hence, ズ( り, ぞ(j) exist and し (川 ≦y, ぽ(/) - ぞol≦がfor μ≡[らゐ], wheTe t2= ちexp(-4汐511).
U sing the estimates ( * ) again, w e obtain し ( 川 ≧(3八 /4) 110叶一log峠
H ence, if w e put
4= 4)exp(-2ダ/511), thenwehavelズ(川 ≧3八岬(1011) for μ≡[ら,y
i]. Thus, theproof iscompleted by taking yく 3八丿/(10£ 1) レ
§3. Proof of Theorem
W e may assume that び= (- T, 口 × 召 ″ ( T > O).
(0, 刀 x Ry W etakepositivenumbers らが
N ote that P is strictly hyperbolic in
asir1Lemma L Theorem isprovedby three
stet) s・
T he first step is to construct singular solutions whose singular supports converge to
(0,0).
Lemma 2.
FOy 皿 y 柘旺 (Oy ) , 匝e花 eχist なE (0鳥 ) のld M旺涯 ((Oy ) × 刄 ) md th H he
jOHO面紹 hO吐
(α)
x
x= O卵 < t2,
(ろ)
釦 = O加 (0, 0 ×防/2,
( c)
(ち,0)E singsuppz
x.
Proof.
■
■
F ix 4)∈ (Oy ) . T ake 4 and ら in Lemma 1. By T heorem 26. 1.5 1n [31, we can
take 叩
(印≡C゛ (召) such that χぷ) = Oif 恣ち and χ1(/) こ 1 1f j≧4. Putχ= ? (χ1哨). Notethat/1
= Oif /< ら and singsupp/1∩( μ x U2. ・) ⊂[ ら, 4] ×召 .
上
Take χ2(x)∈ひ (R ) such that χ2(x) = 1 1f 岡 ≦y/2 and χ2(x) = Oif 岡 ≧7こ Since 7:) is
strictly hyperbolic in 口 > O} , w e can take z
・E y) ((Oy ) x j X ) such that 狸 = χ2yl on (Oy ) ×
一
R ゛l and z7= O i臼 < ら.
Since singsupp(χ2/1に [ ら測 × 防 , the bicharacteristic curve passing
Z(4)) doeSnot intersed W F(χ2/1) by Lemma 1. Hence, W Fひ争Z( 4)). Putting zxこ χlz
f1- びΞ
厦) ((Oy ) ×刄 ), we obtain z
7= Ofor Zくな良 = Oin (Oy ) ×防/2 and W Fg∋Z(位
T he next step is a (グーversion of L em ma 2.
T akeshi M A N DA I
98
L em ma 3.
励り = 1,2, … け here eχist ら = ( ら,舅) ∈ (Oy ) × 防 /3, ぢ ∈ (Oゐ ) and zらE C刎[Oy ]
×7X″
) SJ げぼ服到 IO面昭 hO吐
(α) ろ→(0,0) (プ→ (x)),
(り
回 O川
( じ)
吻(ろ) ≠0,
原∧
(j ) 月白= O加 (Oy ) ×防/3・
Proof.
T ake a sequence of positive numbers ゐ ∈ (Oy ) such that め→ O(ブ→ ∽) . A pply-
ing L em m a 2 for 4) = め , we can take ら ∈ の ((Oy ) × 刄 ) which satisfies
(α)
り = Oif 区め for some め ∈(0, ゐ),
(ゐ) P馬= Oin (Oy ) x G /2,
(0
(め,0)E singsuppvゴ・
W e may assume that suppvj ⊂ £ ×尺 for some compact set 瓦 ⊂ R y Put 凪 = ふ I Ve can
take応,4∈ ひ ((Oy ) ×召 ) which satisfy (a) gj.に j丿n が ((Oy ) ×召″) (ん→ (x)) for each y,
(b) gj.瓦= Oif /≦心/2 0r lズ¦≦り3 and(c) gj,瓦havecompact supports with resped tox.
Since 7) is strictly hyperbolic in 口 > O} , w e can take 吻 , z
, E C゛ ((Oy ) ×召 ) such that ? 吻 ,4
= 焉,ヵin(Oy ) ×£ ″and 的4 = Oif /≦め/2. Sincegj,k→万(ん→(x)), wehave扨j、k→ z
なinの
((Oy ) ×召 ) (ん→ ∽) for each プ. Sincesingsupp馬ヨ (め,0), thereexist 万
㈹ ) such that 妨,肖 )(ろ) ≠OaUd lろー(め,0) ¦ ≦1/y Thus, thelemmaisprovedby putting 吻
= 的,z
心) and が= ゐソ2,
T he final step is to construct a C゛ nu11- solution zf by means of 陶 in L emm a 3.
P roof of T heorem.
T ake 吻 as in L emm a 3 and extend them to (- 八丿) X R x
zas 吻 = 0
1f /< 0. χ
Vithout loss of generality, we may assume that & < が if ん> ブ. ln other w ords,
we assume 吻(み) = Oif j < たχ
Ve shall construct the desired Min thefrom a= Σ ら馬 where
ら are positive constants.
T ak e ら > O ( ソ= 1, 2 , … ) satisfying the follow ing tw o inequalities.
(1) ら≦2-川釧汁 1)-1(プ≧1).
(2)
ら≦2-j4-o 1司仇(み) ¦ ( 臨 (み) 卜 1)-1(1≦ん< y).
-
Herej ・1
レistheCj-norm on[一丿/2/ /2] x G.
By the first inequality, the sum Σ ¦臨吻 ¦に converges for any ん, since llら㈲に ≦ ¦¦ ら㈲し
≦ 2- j for プ≧ だ
H enceバ・= Σ ら匈 converges in ひ )( (づソ2y /2) × 防 ) .
T his zx satisfies that
乙
x= O if /≦ O and 月 7= O in (ゴソ2/ /2) × 防 /3.
By the second inequality and the assum ption that zら ( み ) = O if j くん, w e have 固み ) 卜
∽
∽
¦仇心(み)十 Σ ら侑(み) 副司仇(み) ¦ - Σ GD秘(み) 12-j4 -1= 良¦ 心(み) ¦/2> Oforany左
Hence, (0,0)E supp犯
§4 . Simple Cases to which Theorem Does Not Apply
ln this section, w e shall consider very simple cases to which our theorem does not
apply.
L et 77= 1 and Pこ陥 t十 α( ぴ ) ∂χ十ろ( ぴ ) , w here a, bE Cべ U ) for a neighborhood び
of (0,0) ∈召2 and α(ぴ ) isreal-valued. lf α(0,x) ≡0, then t is of Fuchs type and there
exist no C゛ null- solutions. lf とz(0,0) ≠ 0, then w e can apply our theorem to 沢 H ence, w e
(?
N u11-Solutions for Some N on- Fuchsian Operators with C゛ Coefficients
99
are concerned with the case where ど
7(0,0) = Obut α(0,x) ま0.
F oI゛ simplicity, w e assume that a( t, χ) ≡ α( x) does not depend on j and that χ= O is
an isolated zero of ど
z( x). For such operators, we have the following results.
Proposition.
( I ) がα(x) > OかyX> OaMd a(ズ) く O力yxく 0j 加刀tk 犯 aist肴oC 勺1戒1
-SOlHtiOUSかyP d (0,0)ご
(H) がα(ズ) < OかyX> O匹 α(ズ) > O粕7X< 0けhen theyeaistsa C゛れM匹solHtioR升)yP
が (0,0).
Examples.
(I )
Let ? = 隔十ズみ. A C∽function z
xnear (0,0) satisfieSRt= Ofor 乙> 0
1f and only if 副ぴ ) = g(xμ) for some C∽function g(5) on R .
lf zx( な )→ O when ( ぴ ) ap-
prC
)aches(0,0), then g≡Ohence g≡0.
( n) Let ? = 陥 - x覗. For a C∽function g(s) near s= O which satisfies OE suppg⊂
{s≧O}, thefunctiondefinedby x
x(ぴ)= g(な) when j≧Oand x
x(ぴ)= Owhen j≦Ois a C゛
nu11- solution for t at (0,0).
N ote that suppz7⊂ μ ≧ 0,x ≧ O} .
T here also exist C゛ nu11- so-
lutions whose supports are ind uded in { Z≧ 0,x≦ O} .
R emark.
E ven in ( I ) , there can exist ひ nu11- solutions.
A typical eχample is ? =
陥十α(x) ∂
x- (か+ 1). Thisoperator hasa ひ nu11-solution z
x defined by xx(ぴ ) = yo l for
丿≧O and x巾 ,ズ) = O for Z≦ 0.
T he essential difference betw een ( I ) and ( n ) is the behavior of charad eristic curves.
T he charad eristic curve of 7:) passing ( 4)ゐ ) is the solution of
(C
) 宗一于
α
(x),x(y
o
)=xo
.
From now on, let α(x) be defined on [- ら d ( じ> O) .
L emma 4.
( I )
Uれdeバ he assMm囲 oM 紬 ( I ) 可 Pyo知 s消皿 , 粕 y a砂 ( 4)両 ) ∈ { ( ぴ )∈ び ;
/> 0, 1刈≦d, がzにぷz
厨凹丿 (C) t四dStOOwhe筧t→十〇
.
( n) AssMmethd a(x) < O力に > O叫ホ a(x) > O力に < O)
Foyally (ら濁)∈{ (ぴ ) ∈ び;
/> 0,0< ズ< d ( 几却バ(ぴ)E Uj > 0,-c< xく叫)けhe粍 eχ
isH 1志 SHd thd
(α) 0< 石く 4 く 4),
(ろ) 服 SO㈲i皿絢 ) イ (C) a鐙s粕γtE [ 赳, to] ,
(○ しけ) ¦ > 図力バ ∈[リ 11.
Proof.
proved.
( I ) lf 陥= 0, then thesolution of (C) isx( /) ≡O and there is nothing to be
L et xo> O. lf x( 4) = O for some 4 ∈ (0, 4)), then x( り ≡ 0.
H ence x( /) exists and x( り
> O for μ
there exists j ) ≧ O. such that x( /)→ 茜 ( 卜→十〇) .
N ote that ズ( 0 ≧油 for に [0高 ] . A ssume that
油 > O. T hen w e have α(x) ≧ 訂 on (j ), じ] for a positive constant 訂 , hence there holds ダ ( /)
≧MyL Thisimpliesthatj巾)≦訂 (loμ-log4))十而 fo心 ≦4), hencex(り→-(x) け→ 十〇
). This
is a contradid ion.
T hus, w e obtain λ
7( 0 → O when /→ 十〇. W hen λ
i) く 0, a similar argument
goesweII.
( H) Assumethat α(x) < Ofor x> 0. 1f O< 痛< G thenthesolution x(j) of (C) satisfies
that x( /) ≧痛 for ぼ 4) as long as ズ( /) exists.
Since w e have ,7(x) ≦ 一肛 on (輿), d for a pos-
T akeshi M A N DA I
100
itive constant M μ here lholds x ( X) ≦ 一M tL
T his implies that x( j) ≧一訂 (loμ - Io仙 ) 十両) for
j≦ 4) as long as x( O exists. N oting that x( O is monotone decreasing, we obtain the desired result. ln the case that ど
z(x) > Ofor ズ< 0, we can prove similarly.
P roof of P roposition.
( I )
L et zx be a (グ null- solution for t at (0,0) . Fix ( 4)み ) ar-
bitrarily and considerlthe solution x( /) of (C) . Since zx( ぴ ( X)) = 哨( なり )) 十x ( /) z4 ( μ ( X)) =
-ゐ(ぴ(Z加 (ぴ(X))μ wehavez
x(ぴ( X)) = ylexp(恰 (s,ズ(s))7SdS) for someconstant え Let lろ
(ぴ)匯訂n
e
a
r(0,0). Th
e
川イ1ろ
(s
,ズ
(j))/辿に
j訂110μに
h
e
h
c
e匯(ぴ(顔≧レ川代O
nth
e
other hand, for any yV there exists a constant G such that 匯 ( ぴ ) ¦ ≦ ら川 H ence, wehave
ノ1= O w hich im plies zj ( 4),ぷ)) = 0. Since ( ち,4 ,) is arbitrary, the proof is completed・
( H)
lf x(Z) isthesolution of (C), then thebicharacteristiccurveof t passing (い b;-α
(而)/4),1) is(ぴ(/);-α(ズ(X))ぞ(X)/G け)) whereぐ(4))= 1 andざ(り= -ど(x(/))ぞ(/)μ Hence, by a
similar argument to that in §3, w e obtain a desired C゛ nuU- solution.
十
R eferenCeS
田
Baouendi,M.S.-Goulaouic,C., C4uchy problemswithcharaderisticinitial hypersurface, Comm. PI{粍
A卸 1,. A励 /・., 26 (1973), ¥455-475.
[2] …---, Cauchy problemswith multiplecharaderisticsinspacesof regular distributions, R臨S. Ay
ldh.
SMnJeys, 29 (1974), 72-78.
[3] H6rmander,L・, Tk Audysisof L泌cαyPa眉d D坦旨e戒這I OpemloysI V, Springer-Verlag, Berlin-Heidelberg- N ew Y ork- T oky0, 1985.
一
国
Ouchi,S。 Existenceof singular solutionsandnuUsolutionsfor linear partial differential operatorsぶ
尺じ. 心 . 咄畝で
乃切 , & 友 /1, 皿 7肋., 32 (1985), 457-498.
[5] Tahara,H., FuchsiantypeequationsandFuchsianhyperbolicequations, μ沁n. J. M出。 N{?加 Seγ
,. , 5
(1979), 245-347.
[6] ---…, Singular hyperbolicsystems. I , Eχistence, uniquenessanddifferentiability. H, Pseudo-differentiaI operators w ith a param eter and their applications to singular hyperbolic system s. Ⅲ , 0 n the Cauchy
problem for Fuchsian hyperbolic partial differential equations. Ⅳ , Remarks on the Cauchy problem for
singular hyperbolic partial differential equations. J
F ac. & f. U戒仇 To胎 o, Sed . I A ,M d h. , 2 6(1979), 213
-238, 391-412, 27 (1980), 465-507, μ帥 n. J. Md 1., 8 (1982), 297-308.

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