具有低阶项的散度型椭圆方程的解

502Vol.50,No.2 20073ACTA MATHEMATICA SINICA,Chinese Series Mar.,2007 :0583-1431(2007)02-0299-12:A

Morrey

454003

E-mail:wangys1962@;heyxiang63@

−(a ij u x

i )x

j

+b i u x

i

−(d j u)x

j

+cu=

(f j)x

j

,a.e.x∈ΩMorrey,a ij∈VMO∩L∞(Ω),

Morrey.

;Morrey;

MR(2000)35R05,46E35,42B20

O175.24

The Local Regularity of Solutions in Morrey Space to the Elliptic Equation

in Divergence Form with Lower Order Terms

Yue Shan WANG Yue Xiang HE

Department of Basic Science,Jiaozuo University,Jiaozuo454003,P.R.China

E-mail:wangys1962@;heyxiang63@

Abstract The aim of this note is to study the local regularity of solutions in Morrey

spaces to the elliptic equation in divergence form−(a ij u x

i )x

j

+b i u x

i

−(d j u)x

j

+cu=

(f j)x

j ,a.e.x∈Ω.Where a ij∈VMO∩L∞(Ω),and the lower order terms belong to

suitable Morrey spaces.

Keywords elliptic equation;Morrey space;regularity

MR(2000)Subject Classification35R05,46E35,42B20

Chinese Library Classification O175.24

1

,.,Ω⊂R n L,L u[1]L uΩP,Ω ⊂⊂Ω,u D m uΩ P?

Ω⊂R n(n≥3)

−(a ij u x

i )x

j

+b i u x

i

−(d j u)x

j

+cu=(f j)x

j

, a.e.x∈Ω.(1.1)

:2004-09-20;:2006-03-01

:(10426029)(2004189)

30050

a ij Sarason VMO([2]),

b i,c,d i Morrey

.,H:

a ij∈VMO∩L∞(Ω),a ij(x)=a ji(x),∀i,j=1,...,n;

∃Λ>0:Λ−1|ξ|2≤a ij(x)ξiξj≤Λ|ξ|2.

(1.2)

F:

b i,d i∈Lα(p),α(p)μn(Ω)(∀i=1,...,n),c∈Lα(p)2,α(p)μ2n(Ω),(1.3)

α(p)μ

n

n;p>n,α(p)=p.

1991Chiarenza,Frasca Longo BMO ,L p VMO

[3],,VMO

Morrey[4].

[5,6].(1.1)0,[7,8]Morrey

.H F(1.1)Morrey

L p,λ(Ω),(1.1)Morrey L p,λ(Ω) .

1.1u∈W1,p(Ω)(1.1),

Ω((a i,j u x

i

+f j+d j u)φx

j

+b i u x

i

φ+cuφ)dx=0,∀φ∈C∞0(Ω).(1.4)

1.1H F,f=(f1,f2,...,f n)∈[L p,λ(Ω)]n,u∈W1,p(Ω) (1.1),Du∈L p,λ

loc

(Ω),C=C(n,λ,p,Λ,M,dist(Ω ,∂Ω )),

Du L p,λ(Ω )≤C

u L p,λ(Ω )+ f L p,λ(Ω )+ Du L2(Ω )

,∀Ω ⊂⊂Ω ⊂⊂Ω.(1.5)

,p≥2,

Du L p,λ(Ω )≤C

u L p,λ(Ω )+ f L p,λ(Ω )

.(1.6)

2

2.1R n f∈BMO(R n),sup B1|B|

B

|f(x)−f B|dx=

f ∗<∞.B R n,f B=1|B|

B

f(x)dx.f∈BMO(R n),η(r)=

supρ≤r1

|Bρ|

Bρ

|f(x)−f B

ρ

|dx.lim r→0+η(r)=0,f∈VMO(R n).

2.1B∩ΩBρ∩ΩB Bρ,BMO(Ω)

VMO(Ω).

2.21

f p L p,λ(Ω)=sup

x∈Ω,ρ>0

1

ρλ

Bρ(x)∩Ω

|f(y)|p dy<∞.