V i et nam J our nal of M at hem at i cs 33:2 (2005) 135–147
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On t he Sm o ot hness of Solut ions of t he F irst
Init ial B oundary Value P roblem for Schr¨odinger
Sy st em s in D om ains w it h C onical P oint s
N g uyen M an h H u ng and Cun g T h e A nh
D epart m en t of M athem at ics, H an oi U n iversity of E du cation ,
136 X uan T hu y R oad, H an oi, V iet n am
Received March 12, 2004
Revised March 14, 2005
A b st ract . Som e result s on t he smoot h ness of generalized solut ions of t he first init ial
b oun dary value problem for st rongly Schr¨odinger syst em s in domains wit h conical
p oint s on b oundary are given .
1. Int ro d uct io n
Boundary value problems for Schr¨od inger equat ion s and Schr ¨od inger syst em s in
a finit e cylin der Ω T= Ω ×(0,T) have b een st udied by many a ut hors [4,8,9].
T h e unique solvabilit y of t he fir st init ia l b ou ndary valu e pr oblem for st rongly
Schr¨odinger syst ems in an in finit e cylinder Ω ∞= Ω ×(0,∞) was given in [5]. T he
aim of t his pap er is t o est ab lish som e t heorem s on t he sm oot hness of genera lized
solut ions of t he prob lem in dom ain s wit h conical p oint s on b ou nda ry.
Let Ω be a bounded dom ain in Rn. It s b oundar y ∂Ω is assu med t o b e a n
infin it ely different ia ble surface ever ywhere, except for t he coordin at e or igin, in
a n eighb orhood of which Ω coincides wit h t he cone K=
x:x / |x|∈G
,where
Gis a sm oot h d om ain on t he unit sph ere Sn−1. We int rodu ce some not at ions:
ΩT= Ω ×(0,T),S
T=∂Ω×(0,T),Ω∞= Ω ×(0,∞),S
∞=∂Ω×(0,∞),x =
(x1,... ,xn)∈Ω,u(x , t )= (u1(x , t ),... ,us(x , t )) is a vect or com plex funct ion,
|Dαu|2=
s
i= 1
|Dαui|2,u
tj=
∂ju1/ ∂ t j,... ,∂jus/ ∂ t j
,|utj|2=
s
i= 1
∂jui/ ∂ t j
2,
dx =dx 1...dxn,r=|x|=
x2
1+···+x2
n.
In t his pap er we use frequ ent ly t he following funct iona l spa ces:
136 N guyen M an h H un g and C ung T he A n h
•Hl
β(Ω ) - t he spa ce of all funct ions u(x)= (u1(x),... ,us(x)) which have gen-
eralized derivat ives Dαui,|α|≤ l,1≤i≤s, sat isfying
u2
Hl
β( Ω ) =
l
|α|= 0
Ω
r2 ( β+|α|− l)|Dαu|2dx < +∞.
•Hl , k (e−γ t ,Ω∞) - t he sp ace of all fun ct ions u(x , t ) which have generalized deriva-
t ives Dαui,∂jui
∂ tj,|α|≤ l,1≤j≤k,1≤i≤s, sa t isfyin g
u2
Hl , k (e−γ t ,Ω∞)=
Ω∞
l
|α|= 0
|Dαu|2+
k
j= 1
|utj|2
e−2γ t dx dt < +∞.
In pa rt icular
u2
Hl , 0(e−γ t ,Ω∞)=
l
|α|= 0
Ω∞
|Dαu|2e−2γ t dx dt .
•
◦
Hl , k (e−γ t ,Ω∞) - t he closure in Hl , k (e−γ t ,Ω∞) of t he set of all infinit ely dif-
ferent iable in Ω ∞funct ions wh ich b elong t o Hl , k (e−γ t ,Ω∞) and vanish near
S∞.
•Hl , k
β(e−γ t ,Ω∞) - t he sp ace of all funct ions u(x , t ) which have generalized deriva-
t ives Dαui,∂jui
∂ tj,|α|≤ l,1≤j≤k,1≤i≤s, sat isfying
u2
Hl , k
β(e−γ t ,Ω∞)=
Ω∞
l
|α|= 0
r2 ( β+|α|− l)|Dαu|2+
k
j= 1
|utj|2
e−2γ t dx dt < +∞.
•Hl
β(e−γ t ,Ω∞) - t he space of all funct ions u(x , t ) which have generalized deriva-
t ives Dα(ui)tj,|α|+j≤l,1≤i≤s, sa t isfying
u2
Hl
β(e−γ t ,Ω∞)=
l
|α|+j= 0
Ω∞
r2 ( β+|α|+j−l)|Dαutj|2e−2γ t dx dt < +∞.
•Let Xb e a Banach space. Denot e by L∞(0,∞;X) t h e spa ce of all m easurab le
funct ions u:(0,∞)− → X
t− → u(t)
, sat isfying
uL∞( 0 ,∞;X)=ess sup
t > 0
u(t)
X<+∞.
Con sider t he differ ent ial op er at or of ord er 2m
L(x , t , D )=
m
|p|,|q|= 0
Dp
ap q (x , t )Dq
,
where ap q are s×s-m at rices of m ea sur able, b ou nded in Ω∞, com plex funct ions,
ap q = (−1)|p|+|q|a∗
qp . Supp ose t hat ap q a re cont inu ous in x∈Ω un iformly wit h
O n t he Sm oot hn ess of Sol ut i on s 137
resp ect t o t∈[0,∞)if|p|=|q|=m, and t ha t for ea ch t∈[0,∞) t he op er at or
L(x , t , D ) is uniform ly ellipt ic in Ω wit h ellip t icit y const ant a0indep endent of
t im e t, i.e., we have
|p|=|q|=m
ap q (x , t )ξpξqηη≥a0|ξ|2m|η|2,(1.1)
for all ξ∈Rn\ {0},η∈Cs\ {0}and (x , t )∈Ω∞.
P ut
B(u, u)(t)=
m
|p|,|q|= 0
(−1)|p|
Ω
ap q DquDpudx , u(x , t )∈
◦
Hm , 0(e−γ t ,Ω∞).
F o r a . e . t∈[0,∞), t he funct ion x→ u(x , t ) belongs t o
◦
Hm(Ω ). On t he ot her
ha nd, since t he p rincipa l coefficient s ap q a re cont inu ous in x∈Ω uniformly
wit h resp ect t o t∈[0,∞) a nd t he const ant a0in (1.1) is independent of t,by
rep ea t ing t he pr oof of G ar ding’s inequalit y [2, p .44], we have
Lem m a 1.1. T here exist two con st an t s µ0an d λ0(µ0>0,λ0≥0) su ch that
(−1)mB(u , u )(t)≥µ0
u(x , t )
2
Hm( Ω ) −λ0
u(x , t )
2
L2( Ω ) (1.2)
for all u(x , t )∈
◦
Hm , 0(e−γ t ,Ω∞).
T h erefore, using t he t ra nsfor mat ion u=ei λ 0tvif necessary, we can assume
t ha t t he op era t or L(x , t , D )satisfies
(−1)mB(u, u )(t)≥µ0u2
Hm( Ω ) (1.3)
for all u(x , t )∈
◦
Hm , 0(e−γ t ,Ω∞). T his in equa lit y is a b asic t ool for pr ovin g t h e
exist ence and uniqu eness of solut ions of a b oun dar y value problem .
2. M ain R esults
In t his p ap er we con sider t h e following problem : F ind a funct ion u(x , t )such
t ha t
(−1)m−1i L (x , t , D )u−ut=f(x , t )inΩ ∞,(2.1)
u|t= 0 = 0,(2.2)
∂ju
∂ νj
S∞
= 0,j= 0,... ,m −1,(2.3)
where νis t he out er u nit norma l t o S∞.
A funct ion u(x , t ) is ca lled a gen era lized solut ion of t he problem (2.1) - (2.3)
in t he space
◦
Hm , 0(e−γ t ,Ω∞) if and only if u(x , t ) b elongs t o
◦
Hm , 0(e−γ t ,Ω∞)
and for each T> 0 t he followin g equ alit y hold s
138 N guyen M an h H un g and C ung T he A n h
(−1)m−1i
m
|p|,|q|= 0
(−1)|p|
ΩT
ap q DquDpηdxdt +
ΩT
uηtdx dt =
ΩT
fηdx dt
(2.4)
for all t est funct ion η∈
◦
Hm , 1(Ω T) sat isfying η(x , T )= 0.
Denot e by m∗t h e numb er of mult i-in dices which h ave order not exceeding
m,µ0is t h e const a nt in (1.3). From T heor ems 3.1, 3.2 in [5] and by using
induct ion we obt ain t he following result .
T he orem 2.1. L e t
i) sup
∂ a p q
∂ t
:(x , t )∈Ω∞,0≤ |p|,|q|≤ m
=µ < +∞;
∂kap q
∂ tk
≤µ1,µ
1=co n s t >0,for 2≤k≤h+ 1;
ii) ftk∈L∞(0,∞;L2(Ω )), fork≤h+ 1;
iii) ftk(x , 0) = 0,for k≤h.
T hen for every γ> γ
0=m∗
µ
2µ0
, the problem (2.1) -(2.3) has exactly on e
gen eralized solu tion u(x , t )in the space
◦
Hm , 0(e−γ t ,Ω∞). M oreover, u(x , t )has
derivat ives wit h respect t o tup to order hbe lon gin g t o
◦
Hm , 0(e−( 2 h+ 1 ) γ t ,Ω∞)
an d t he followin g est im at e holds
uth2
Hm , 0(e−( 2 h+ 1 ) γ t ,Ω∞)≤C
h+ 1
k= 0
ftk2
L∞( 0 ,∞;L2( Ω ) ) ,
where Cis a posit ive con stan t in depen den t of uan d f.
From now on for t he sake of brevit y, we will writ e γhinst ead of (2h+ 1)γ
(h= 1,2, ..., ).
In or der t o st udy t he sm oot hness wit h resp ect t o (x , t ) of generalized solu-
t ions of t h e problem (2.1) - (2.3), we assume t h at coefficient s ap q (x , t )ofthe
op erat or L(x , t , D ) are infin it ely d iffer ent iable in Ω∞. In addit ion, we also as-
su me t hat ap q and it s all derivat ives a re b oun ded in Ω∞.
F irst , we pr ove t h e following lemma.
Lem m a 2.1. L e t f , f t,f t t ∈L∞(0,∞;L2(K)) an d f(x , 0) = ft(x , 0) = 0.If
u(x , t )∈
◦
Hm , 0(e−γ t ,Ω∞)is a gen eralized solu tion of t he problem (2.1) -(2.3) in
the space
◦
Hm , 0(e−γ t ,Ω∞)such that u≡0when ever |x|> R,R=con st, t hen
u∈H2m , 1
m(e−γ1t,K ∞)an d t he followin g est im at e holds
u2
H2m , 1
m(e−γ1t, K ∞)≤C
f2
L∞( 0 ,∞;L2(K) ) +ft2
L∞( 0 ,∞;L2(K) ) +ft t 2
L∞( 0 ,∞;L2(K) )
,
where C=co n s t .
P roof. R ewr it e t he syst em (2.1) in t he following form
O n t he Sm oot hn ess of Sol ut i on s 139
(−1)m
m
|p|,|q|= 0
Dp
ap q (x , t )Dqu
=F , (2.5)
where F=i(ut+f). From T heor em 2.1 it follows t hat F∈L2(K) for a.e.
t∈[0,∞).
Con sider t h e sequ ence of dom ains
Ωk=
x∈K:2
−k≤ |x|≤ 2−k+ 1
,k= 1,2, ...
Ch oosing a sm oot h domain Ω 2,0su ch t ha t Ω 2⊂Ω2,0⊂
Ω1∪Ω2∪Ω3
.By t he
t heorem on t he sm oot hness of solut ions of ellipt ic pr oblems in a sm oot h d om ain
[3, T h. 17.2, p. 67], we obt ain
Ω2,0
|Dαu(x , t )|2dx ≤C
Ω2,0
F(x , t )
2+
u(x , t )
2
dx , |α|≤ 2m , C = const.
Hence
Ω2
|Dαu(x , t )|2dx ≤C
Ω1∪Ω2∪Ω3
F(x , t )
2+
u(x , t )
2
dx , |α|≤ 2m , C = const.
(2.6)
By subst it u t ing x=4
2k1x′(k1>2) in (2.5) and applying t he est imat e (2.6), we
have
Ω2
|Dα
x′u(x′,t)|2dx ≤C1
Ω1∪Ω2∪Ω3
F(x′,t)
2
4
2k1
4m
+
u(x′,t)
2
dx ′,C1= const .
R et ur ning t o va riables x1,... ,xn,weobtain
Ωk1
|Dαu(x , t )|2r2 ( |α|− m)dx
≤C2
Ωk1−1∪Ωk1∪Ωk1+ 1
F(x , t )
2r2m+r– 2 m
u(x , t )
2
dx , C 2= const.
Sum ming t hese inequalit ies for a ll k1>2weobtain
k > 2Ωk
Dαu(x , t )
2r2 ( |α|− m)dx
≤C3
k > 1Ωk
F(x , t )
2r2m+r−2m
u(x , t )
2
dx , C3= const.
(2.7)