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- sub-elliptic equations
- interior regularities
- semilinear sub-elliptic
- see front matter
- small enough
- hormander's condition
- isotropic holder's space
- chao-jiang xu
- jx ?
Nonlinear Analysis 37 (1999) 1039{1049
The dirichlet problems for a class of semilinear
1sub-elliptic equations
Chao-Jiang Xu
Institute of Mathematics, Wuhan University, Wuhan 430072, China
Received 16 June 1995; accepted 14 May 1996
Keywords:Semilinearsub-ellipticequation;Vector elds;Non-isotropicHolder’sspace;Dirichlet
problems
1. Introduction
In this work, we study the following semilinear Dirichlet problem:
(Pm X X u+cu=f(x;u); in
;jj=1 j
(1)
u=’; on @
;
where X=fX ;:::;X gisasystemofrealsmoothvector eldsde nedinanopendo-1 m
nmain MR;n2;
isaboundedopensubdomainof M with @
smooth, c(x)c0
>0. X denotetheadjointof X.Weassumethatthesystemofvector elds X=fX ;j 1j
:::;X g satis es the following Hormander’s condition:m
X ;:::;X together with their commutators X =[X ;:::[X ;X ]:::]1 m 1 s−1 s
up to some xed length r span the tangent space at each point of M.
We have the following theorem:
Theorem 1. Assume that the system of vector elds X=fX ;:::;X g satis es the1 m
Hormander ’scondition, @
issmoothandnon-characteristicforthesystem X ;:::;X ;1 m
1 1 1f2C (
R);@ f(x;u)0;’2C (@ ) . Then there exists a solution u2C ( )u
of Dirichlet problem (1).
1This work was supported by National Natural Science Foundation of P.R. China.
0362-546X/99/$ { see front matter? 1999 Elsevier Science Ltd. All rights reserved.
PII:S0362-546X(97)00722-01040 C.-J. Xu / Nonlinear Analysis 37 (1999) 1039{1049
SinceEq.(1)issubelliptic,wecallEq.(1)semilinearsubelliptic.Hormander’s con-
Pm ditionpermitsustogetsomepropertiesofHormander’s operators H= X X +cjj=1 j
similar to those of the Laplacian (see [1,3,4,9,11]). Using these properties, we have
proved the interior regularities for quasilinear second-order subelliptic equation of the
Pm
form A (x;u;Xu)X X u+B(x;u;XU)=0, and the existence of weak solution forij i jij=1
variational problems (see [13,15]). The results of this work is about the existence
1and C regularitiesuptotheboundaryforthesemilineardegenerateellipticDirichlet
problems.
2. Function spaces and preliminary lemmas
We de ne now the metric on M associated with X as in [9,15].
De nition 1. Let C() be a class of absolutely continuous mappings :[0;1]!M
which almost everywhere satisfy the di erential equation
X
0(t)= a (t)X ((t)) (2)J J
jJj r
jJjwith ja (t)j< ; then we de neJ
(x;y)=inff > 0j92C() with (0)=x; (1)=yg: (3)
Then, is a local metric on M, and for any small compact subset KM; there
exists a constant C>0 such that
−1 1=rC jx− yj (x;y)Cjx− yj
for any x;y2K. We can de ne a family of balls by this metric.
B(x;)=fy2M;(x;y)< g
for x2M,and > 0smallenough.Denoteby B (x;)theEuclideanball.Then,forallE
compact KM, there exists constants C >0;C>0, and >0 such that1 2 0
r
B (x;C )B(x;)B (x;C )E 1 E 2
for all x2K and 0< .0
We introduce now a class of \non-isotropic" Holder continuous functions. For 1>
0 0 1>0, we de ne ( S ( )= C ( ) \L ( ))
( )
jf(x)− f(y)j 0 XS ( )= f2S ( );[ f] = sup <+1 (4) ;
(x;y)x;y2
and for k2N;1> 0; we de ne
k; J
S ( )= fu2S ( ); X u2S ( ) ;8jJj kg: (5)C.-J. Xu / Nonlinear Analysis 37 (1999) 1039{1049 1041
Set
X J[u] =supsupjX u(x)jk;0;
x2
jJj=k
and
X J X[u] =sup[X u(x)] :k; ;
k; ;
jJj=k
k;The norms on S ( ) are given by
kX
X Xkuk = [u] +[u] : (6)k;S ( ) j;0;
k; ;
j=0
k;Then the space S ( ) is a Banach space (see [15]).
As for the classical Holder space, we also have the interpolation inequalities in the
k; k;space S ( ). For j+ <k + ; j;k 2N;0 ; 1;u2S ( ) ; and any > 0, we
have
kuk kuk k; +C( ;j;k;
;r)kuk 1 : (7)j; S ( ) L ( )S ( )
This implies the following compactness results.
k;Lemma 1. Let K be a bounded subset of S ( ) ;k+ > 0: If j+ <k + ; then K
j;is precompact in S ( ) .
3. Schauder estimates for the Hormander operators
We study in this section the following linear Dirichlet problem:
Hu=f in ; u=’ on @
: (8)
with c(x)c >0. From the subellipticity of Hormander’s operators H, we have (see0
[1,3,6])
Lemma 2. Assume that the system of vector elds X ;:::;X satis es the1 m
Hormander ’scondition;and @
isnon-characteristicforoperators H.Thenthelinear
1Dirichlet problem (8) possess a unique solution u2C ( ) .
By [1], there exists Green’s kernel G(x;y) for H. From [11,15] we have
Lemma 3. For n2;K
; and (x;y)2KK; we have
J 2−jJj −1jX G(x;y)j C (x;y) jB(x;(x;y))j ; (9)J
wherethedi erentialistakenin x or y.Andforany > 0;thereexistsaconstant C
such that
Z
−1 (x;y) jB(x;(x;y))j C :
B(x;)
1042 C.-J. Xu / Nonlinear Analysis 37 (1999) 1039{1049
We shall use the inequality (9) to prove the Schauder estimate of Hormander oper-
k;ators in the \non-isotropic" Holder spaces S . Firstly, we have the weak maximum
principle
2 0Lemma 4. If u2S ( ) \C ( ) is a solution of Dirichlet problem (9), c(x)c >0.0
Then we have
−1
1 1kuk c kfk : (10)L ( ) L ( )0
This is the theorem of Bony [1]. We also have the strong maximum principle of
Bony.
2Lemma 5. Assume that X ;:::;X satis es the H ormander’s condition, u2S ( ) \1 m
0C ( ) veri es Hu0 in
; and u0 on @
. Then u0 in
.
Wenowprovetheestimateof kXuk 1,andSchauder-typeestimateforoperators HL
in the interior of .
2 0Lemma 6. Let u2S ( ) \C ( ) ;uj =0; then for all K
; there exists a con-@
stant C such that; for 1km;
maxjX uj CsupjHuj: (11)k
K
2; 0And if u2S ( ) \C ( ) ;uj =0; >0; then@
kuk 2; CkHuk ; (12)S (K) eS (K)
ewhere K K
.
Proof. For > 0 small enough, we denote by K =fy2 ; (x;y)< ;x 2Kg. Take
1’2C ( ) ;’(x)=1; for x2K. Using the Green’s kernel of the Dirichlet problem,0
0we have H(’u)2C ( ), and for x2K0
Z
u(x)= G(x;y)H(’u)(y)dy:
Since
mX
H(’u)= X X(’u)+c(’u)jj
j=1
m mX X
=’H(u)+ (X ’X u+X ’X u)+u X X(’);j j jj j j
j=1 j=1C.-J. Xu / Nonlinear Analysis 37 (1999) 1039{1049 1043
we have, for x2K,
Z Z mX
u(x)= ’(y)G(x;y)H(u)(y)dy+ u(y)G(x;y) X X(’)(y)dyjj
j=1
Z mX
+ G(x;y) (X ’X u+X ’X u)(y)dy:j jj j
j=1
Since supp’ , integrating by parts, we have
Z
xX u(x)= X (’(y)G(x;y))H(u)(y)dyk k
Z mX
x + u(y)X G(x;y) X X(’)(y)dyjk j
j=1
Z mX
yx+ [X X (G(x;y)X ’(y))jk j
j=1
yx +X X (G(x;y)X ’(y))]u(y)dyk j j
=I+II+III:
Using Lemma 3, we have for x2K,
Z
−1djIj CsupjH(u)j (x;y)jB(x;(x;y))j dy
eCsupjH(u)j;
Z
−1jIIj Cmaxjuj (x;y)jB(x;(x;y))j dy
eCsupjuj;
Z
−1
jIIIj Cmaxjuj jB(x;(x;y))j dy
nK
Z
−1 eCmaxjuj jB(x;)j dyCsupjuj:
nK
which give the estimates
maxjX uj C supjH(u)j+maxjuj :k
K
By Lemma 4, we obtain the rst part of lemma. The second part is just the results
in [15].1044 C.-J. Xu / Nonlinear Analysis 37 (1999) 1039{1049
4. Existence of solutions
We now prove the Theorem 1 in two steps:
(a) Assume that f is bounded
jf(x;u)j N; (13)
we prove that for the semilinear Dirichlet problem (1) there exists a solution.
Wecanassumethat ’=0,sinceif wisthesolutionofthefollowinglinearDirichlet
problems,
Hw=0 in ; w=’ on @
;
1then Lemma 2 gives that w2C ( ). Set u=v+w, then the Dirichlet problem (1) is
equivalent to the following homogeneous semilinear Dirichlet problem:
Hv=f(x;v+ w)in;
v=0 on@
: (14)
Let v be a solution of following linear problem:
Hv=N in ; v=0 on@
;
1then, by Lemmas 2 and 5, we have v0in and v2C ( ). Take
@f(x;u)
k= inf ;
−maxvumaxv @u
then k0, and for all −maxvuwmaxv we have
f(x;u)− f(x;w)− k(u− w)0:
The solution of problem (14) will be the limit of u, where u =v, and u is thel 0 l
solutions of following problems:
L[u] H(u)− ku =f(x;u )− ku ; uj =0: (15)l l l l−1 l−1 l @
1FromLemma2,thereexistsasolutionofEq.(15)and u 2C ( )forall l2N.Notel
that
L[u]=f(x;v)− kvN − kv=L[v]:1
Using Lemma 5, we have u v. And1
Hu =k(u − v)+f(x;v)f(x;u ) − N=H(−v);1 1 1
which give u − v, so that1
−vu v:1
We will prove the following estimates by induction:
−vu u v (l=1;2;:::): (16)l l−1C.-J. Xu / Nonlinear Analysis 37 (1999) 1039{1049 1045
Assume that Eq. (16) is true for l, then
L[u − u]=f(x;u)− f(x;u )− k(u − u )0:l+1 l l l−1 l l−1
By Lemma 5, we have u u. On the other hand,l+1 l
H(u )=k(u − u)+f(x;u)f(x;u ) − N=H(−v);l+1 l+1 l l l+1
whichgives−vu .WehaveprovedthatEq.(16)isalsotruefor l+1.SoEq.(16)l+1
is true for all l2N.
We have proved that max juj max jvj=M<+1; then fjujg is bounded in
l
l
0C ( ), and
ejH(u)j=jk(u − u )+f(x;u )j 2jkjM+ N=M:l l l−1 l−1
eFrom Lemma 7, for K K , we have
emaxjX uj CM;j l
Ke
ewhere C;M are independent of l. On the other hand,
