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Regularity for nonlinear equations involving square Hörmander operators
Nonlinear
Analysis,
Theory,
Methods
Pergamon
REGULARITY
& Applications, Vol. 23, No. 1, pp. 49-73, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/94 $7.00+ .Ml
FOR NONLINEAR EQUATIONS INVOLVING HijRMANDER OPERATORS
SQUARE
Uco GIANAZZA Dipartimento di Matematica, Universita di Pavia, via Abbiategrasso 209, I-27100 Pavia, Italy (Received 12 July 1992; received for publication
16 June 1993)
Square HBrmander operators, homogeneous spaces, extraintegrability.
Key words and phrases:
1. INTRODUCTION IN 1973 Gehring
[l] showed that an Lq(&‘) function g is actually Lp integrable in a cube Q for some p > q provided a certain inequality relating to the maximal functions holds. In a successive paper Giaquinta and Modica [2] observed that the lemma is true only if g = 0 on a” - Q and proved a correct local version which is the main tool in their proof of an Lp estimate for a nonlinear elliptic system. In this way they gave an extension of a similar result that Meyers had obtained for an elliptic equation in divergence form [3]. At the same time a big interest arose in the development of a theory of singular integrals on homogeneous spaces; therefore, some useful results, such as covering lemmas and, above all, the Calderon-Zygmund lemma, were proved in the new setting (see [4]). Recently the structure of homogeneous space has been of great importance in studying Dirichlet forms as a tool in the description of the variational behaviour of possibly highly nonhomogeneous and nonisotropic bodies (see [5]). In this work we consider the problem - ;
Xi*x((u) + b(x, 24,X;(u)) = 0
in L!
(1.1)
i=l
u=o
on
where Xi satisfies a uniform Hormander condition k span the tangent space), Xi* is the formal adjoint Precisely we have W,
u, Xi(U)) = a,(x)u
asz
(1.2)
(Xi and its commutators up to a fixed order and b(x, s, p) is a first order nonlinear term. + “0% u, Xi(U))
with @o(X)> CY> 0 and f is a Caratheodory function which has at most a quadratic growth in Xi(u). Our aim is to study the regularity properties of weak solutions for equation (1.1). Under a smoothness assumption on L2, we will prove in theorem 5.1 that a weak solution u exists, which actually belongs to H,‘(a) n L”(a). To do that we adapt to our context an idea of Boccardo-Murat-Puel (see [6]), which is essentially based on the weak maximum principle. This result opens the way to finer estimates relative to our nonlinear equation. 49
50
U GIANAZZA
As a matter of fact, it is known that an intrinsic distance d(x, y) can be associated to our Xi in a natural way and this, together with Lebesgue measure, gives to @li” the structure of homogeneous space. Therefore, in theorem 4.1 we prove a homogeneous version of the Giaquinta-Modica lemma (see [2]) that will be a major tool in the proof of theorem 7.1 where we show that an L!’ estimate is still valid, much along the lines of Meyers’ result (see [3]). Finally in theorem 7.2 we demonstrate that our solutions are also locally Holder continuous. 2. FUNCTIONAL
Let us consider C”(al”) functions now define the vector fields
FRAMEWORK
a;,, where i = 1, . . . , m (m fixed) and I = 1, . . ., n. We can
Xi = ~ ai, ~ I=1 I and suppose that they satisfy a uniform (for further details see [7]). By making our vector fields (see [7-91) we denote
i=
l,...,m
Hormander condition as discussed in the Introduction use of the intrinsic distance which can be associated to
B(r, x) = (y: d(x, y) < r). Its main feature is that this ball has different dimensions along the directions defined by the Xi and their commutators. In [7] a Poincare inequality has been proved for intrinsic balls. Precisely we see that given a relatively compact open subset A of a”, there exists a constant C, > 0 and an integer k 2 1 such that for every x E CR”and every r > 0 with B(r, x) C A the following inequality holds
is
”
124 - iiJ2dx 5 C,r2 i where u E C”(A) let us consider
IxiCu)12
dX
(2.1)
.w, x)
B(r/k,x)
and ii is the average
of u on @r/k,
x). Given an open set Q such that Q E 61n 0
E(u, v; Q) = : i=l
Xi(U)Xi(V) dx +
uvdx !
.i Cl
R
which naturally gives rise to the norm [E(u, U; Q)]“2. We can then extend the classical Sobolev spaces to our setting. Therefore, by H’(SZ; E) and Hd(0; E) we denote the closure of C”(Q) and C,“(a) for the norm E. An easy extension of (2.1) to these new spaces is then proved. Let us now consider L = -t
We can associate
the following
bilinear
i=l
x;xi.
form on H’(Q;
(2.2) E) x Hd(sZ; E)
xi (“)xi(v) dX
(2.3)
Square
Hiirmander
operators
51
and the question arises about the existence of the Green function for the Dirichlet problem relative to our operator in CR”. Such a problem has a positive answer and the following estimates have been proved r2
r2 (2.4)
A, J&r, x)1 s GX(y) 5 *l JB(r, x)1 2-s lxj,
9 *..9 x,G”W
5
As ,&,
where G’(Y) is the Green function relative to L in 611”with singularity r I f?, R > 0 suitable and ]B(r, x)1 is the volume of the ball. As in the usual Dirichlet problem, G”(Y) is such that a@n(u, GX) = U(X)
(2.5)
x),
at x and Y E aB(r, x),
v U E C,“(@“).
Since a(u, v; B(r, x)) is coercive on Hd(B(r, x); E) it has been possible to define the Green function relative to L in B(r, x) with singularity at x (see [lo]). The above written estimates (2.4) and (2.5) are easily extended to the new setting. In particular we have
(2.6) for y E dB(qr, x), q E (0, qO], q0 < 1 suitable, A3 and A, constants which do not depend on x. It is also useful to consider the so-called regularized Green function relative to the ball B(r, x) (G,“(y)) as defined in [IO]. We refer to that work for the properties of (or to a”), GpX;BB(r,xj convergence
of G,“;.,,,,(G,“(y))
to G&,,,(GX(~)) 3. HOMOGENEOUS
Let us consider a set X and a function it satisfies the following properties: (i) &,Y) > 0 es x f Y; (ii) P(X, Y) = P(Y, x); (iii) there exists a positive
constant
as P + 0. SPACES
p: X x X --t 6l’. We say that p is a pseudodistance
K such that for every x,y, z E X
P(X, Y) 5 K]P(Y, z) + P(Z, We define
the pseudoball
if
41.
B(r, x) of ray r and centre x as the set B(r,x)
= lY:Y EX,P(X,Y)
< rl.
To simplify the notations we will use “distance” and “ball” instead of “pseudodistance” and “pseudoball”. Let us suppose that the set X considered above is a topological space and that the associated distance is such that: (i) the balls B(r, x) form a base of open neighbourhoods of x; (ii) there exists an N E N such that for every x E X and for every r > 0 there are no more than N points xi belonging to B(r, x) such that p(Xi, Xj) > r/2 (homogeneity property). We say that X is a homogeneous space and the two numbers N and K occurring in the definitions are its constants. In the following the letter C will stand for a generical constant, different in various contexts, depending only on N and K.
U. GIANAZZA
52
It is interesting to observe that the homogeneity property is automatically verified if a positive Borelian measure ,D defined on a o-algebra of subsets of X which contains the balls B(r, x) satisfies the following inequality 0 < p(B(r, x)) < C/@(r/2,
x)) < +=J
(3.1) for every x E X and r > 0. Therefore, we shall say that a topological space X is a homogeneous space if it is endowed with a distance p and a measure ,u satisfying those conditions. We now want to state some lemmas whose proofs have already been given in [4]. They are practically the homogeneous version of the analogous ones given in [ 1l] for the Euclidean case of 6li”. LEMMA 3.1. Let E 5 X be a bounded subset of a homogeneous
space X which is covered by the family (B@(x), x)). From this family we can then select a sequence of disjoint balls B(r(x,), Xi) so that the family (B(kr(Xi), Xi)) covers E, the constant k depending only on Nand K. Moreover, we have CP(E) 5 C P(B(‘?xJ, Xi)). Actually the last part concerning the inequality about the measure is new, but can be proved very easily thanks to (3.1) and the numerable additivity of Borelian measures. LEMMA 3.2. Let 0 c X be a bounded open set. There exists a sequence of balls B(p,, xi) such
that:
(i) Q = Ui &pi,
X;);
(ii) a point x E a cannot belong to more than M balls &pi, Xi); (iii) the balls B(hpi, Xi) intersect the complement of Q, CQ. The two constants M and h depend only on X. Proof. The proof is the same as in [4]. It is important to observe that in the above-mentioned work we have r(x) = & where K is the constant of the homogeneous
dist(x, CQ) space and k is defined in lemma 3.1
1 pi = kri = - dist(x,, CSJ) 2K h = 3K * hpi = 3 dist(xi, CQ) but there is no need to put exactly hp, = t dist(xi, CQ). It is enough that the balls intersect CQ; therefore, any choice of h that brings about this result, will do. Definition 3.1. We define the local maximal function function f as 1
@cx)= suPR
s BR
li;r,(x) of an L’(X, dp) nonnegative
f(t) Wt).
Square Hiirmander operators It
is obvious
53
that a.e.
f(x> 5 n;i,cx>
PROPOSITION3.1. Let us consider f E L’(X, dp), f 2 0. We have p((x E x: &-f(X) > CrJ)5 5o1 x f(t) d/At). I Proof.
The proof is as in [4]. There is no formal change due to the upper limit we now set
for R. PROPOSITION 3.2. Let f be a bounded support function, such that f 2 0. Let us further suppose that f E L’(X, dp) and that
There exists a sequence of balls B(pi, Xi) such that: (i) f(X) I cd a.e. on X - Ui B(pi, Xi) = X - S2; (3 l/IB(Pi, XJI SB(,,,,~ f(t)+(t) (iii) Ci IB(pj, xi)1 5 C/a jxf(t)
5 CW W);
(iv) a point x E X cannot belong to more than M balls B(p,, Xi). Proof. Again we work as in [4]. The fact that we consider the local maximum function instead of the global one requires that hp;
Vi.
This is always possible (see the remark in the proof of lemma 3.2) even if any set X can require a different choice of R, and (or) of h. 4. L!’ INTEGRABILITY
ON HOMOGENEOUS
SPACES
We can now come to theorem 4.1. For the sake of completeness, we will first state a global version of it which is of some interest but cannot be applied to our context. PROPOSITION 4.1. Let us consider a homogeneous space X and a function g with bounded support such that g: X + [0, +oo] and g E L*(X, dp) with q E (1, +a). Suppose that p(X) < +co, b E (1, +a~) and that
M&X) 5 NM, WI4
a.e. in X
(4.1)
where Mf(x) is the global maximal function for f. Then g is Lp integrable in X and satisfies (4.2)
for p E [q, q + c), where c is a positive constant which depends only on q, b and the constants of the homogeneous space.
U. GIANAZZA
54
Remark 4.1. An inequality such as (4.1) is commonly known as reverse Holder. It is interesting to observe that the global nature of this proposition is deeply linked to the use it makes of global maximal functions. Under the point of view of the applications, we are led to consider H{(X) functions, where the boundary value gives no problem, but this can be the biggest limitation. That is why we have to give a local version of the above result and that can be done only by putting an upper limit on R in the maximal function, as done in the preceding paragraph. With the new idea and a finer but rather technical subdivision argument, we can overcome the main difficulties and prove theorem 4.1. Proof. See [12]. Let us finally state the main theorem of this section. space X and two nonnegative functions g and f such that g E Lq(x), 4 > 1 andf E L’(X), r > q. Fix a ball B, = B(R*, x0) in Xand suppose that for every x E B, and R < i dist(x, SS,) the following estimate holds
THEOREM 4.1.Let us consider a homogeneous
(4.3) with b > 1. There exists a constant 8, = &(q, r, K, N) such that if 6 < &,, then g E L$,,(B,) for P E h, q +
~1and (,&,,
{.,gpdp)l’p 5 C/(\B(2;,x),
i,,,gqdp)L’9
+ (,,(,A,.,
.i,,,fp4’p]
where C and E are positive constants depending only on b, 8, q, r, K, N. Proof.
Let us consider the ball B, such that B, = B(3R, x,)
and let us define c, = B(R, x0) C, = (x E X: 2rekR < dist(x, dB,) 5 22-kR) It is clear that B,(3R,
X0) =
u ck. kz0
k = 1,2...
C4e4)
Square
HGrmander
operators
55
Let us now set (Yk
=
(KILL)
K, a proper
g(x)
(g’)B, =
G(x)= (g”);;
constant
j$ i B, gq dp
on ck
f(x)
F(x)= (gQ);( S(x)
=
F<“>
on ck
Olk E(h, t) = (x Fix t E [ 1, +a)
E
B,: h > t).
and choose s = K2 t such that for convenience
Since
we can apply lemma 3.2 on each Gk. If the constant h,p,
and the upper bound
k+5
= k
+
4
diNxi, c&d
R,, in the definition &,k
we obtain
a sequence
that is
k+4R
= -
k+32k
-
in lemma 3.2 is such that
k+5i? 5
k+4
of the local maximal
of balls Bh (pi, Xi) such that:
(i) BL (Pi 3Xi> C ck ; (ii) G 5 (Y,$ a.e. in Gk - UjBi(pi,xi); (iii) l/IBL[ IBj, Gq(t) d&t) I K&/$)‘. If we divide by CY$we obtain
hk which appears
2
function
is given as
U. GIANAZZA
56
Then
IE(s,s)
- uj,kBjkI
= 0 and, hence,
where we take into account
the doubling
property
and we put
K4 = K,C. Let us now recall that !& is such that MGQ(X) > sqcYq k Therefore,
for each x E fik there exists a ball @(r,
vx
E
Qk.
x) such that (4.6)
with r < ROk = (k + 4)/(k + 3)R/2k. Denote by Bi the ball which has the same centre
as i$ and F = $1. Of course
B{ c c, u c, u c,
(4.7)
k+2 Bi
c
u
(4.7’)
cj
i=k-1
and also (4.8) i=O k+4 Lg
c
u
(4.8’)
cj.
i=k-1
Thanks
to (4.7) we have @ = (Bi n ck) u @j, n ck+r) u (@ n &+2).
Therefore,
If we divide
(4.6) by CY~we have
I
(4.9)
51
Square HGrmander operators
Let us now consider
our inequality
+
.i elBIL 4/3R
and take two balls & and B’,. To simplify (4.10) becomes
Dividing
qdp
+
ak
I BnCk
i BnCk+i
z+
+
sQd,z + 2 =+I
If we also take into account
If we reason
Taking
as before
into account
5q
-t(B*l q-1
I
let us simply
write B and B*. Then
side
[BnCk+23p]
Rq&+4[BnCk+2Sq‘b]
+,[BS’q4-;-
s BnCk+l
Bnc,,
LI
(4.10)
dpu.
the notations
by (gq)B, 01: we get on the left-hand
,A -li
gq
the right-hand
we finally
the definition
s
gdp
side, we have
get
of s, we have
+ (B*(‘-
llfl’B*~Qlp)i’q
($i’q([B*SqdP>“q
+
lB*1’-“‘.
B*
It is also easy to see that in this new setting
[B*sdp ([,*CFqdlr)Lb
we have as in [2]
5 i’ris,rlnB*Sdp
5 [jEC5,CloB*5qd,u
5 2t\B*)
+ flB*l + tqiB*~]l’*jB*I’-l’q
t’-’ S CFq dp
+
.E(‘S,t)nB*
lB*ll-l’q (;y’q( I’..6’ dp]‘q
I
2tlB*I
+
;tl-q
6’ dp
j w2.r)
nB*
U.
58
G~ANAZZA
if 0 < b. Therefore, n
5 -t(P q-1
cr
6 dp + t1-q
e + _t’-4 b
5q dp I E(S,f)fIB’
EG,t) oB*
.i EG,1) nfs*
Sq b.
(4.11)
Let us now define Dk = u@. j Since Dk is bounded
we can apply the usual covering
lemma
and obtain (4.12)
where the Bici) are pairwise
disjoint.
Combining
ID,1 5 K, c I@fi’\ I KS? i
(4.11) and (4.12) we get
‘i4 i=k-I I>
x
t-l i
Sdfi
+ t-q
! E(5.t) nci
If k = 0 the sum has obviously to be corrected observe that Dk 2 a,. Therefore, 1%
c
k
Moreover, ; Thanks
5
when we sum over k, the integrals
ID,\
5
;&(q-
1) t-’ i
c k
to the estimates
!
t-l i
Sdp
0 I ws*s)
E(S,t)
.
(4.8). Let us now
at most six times each. Then
!Yqd,u + %f-q
+ t-q i E(3.t)
- 1)
+ trq
Tqdp
.i E(S,t)
1
ID,\.
over Ci are counted
sd,u
Sq&
to (4.5) we have
x
with
+ Bf-q i m,r)nc,
according
‘;K,(q
or
Fqdfi
+
;t-q .i,,,,,
‘”
dpj
Square
Hbrmander
operators
59
On the other hand $j4dp =
&
66’-’ s E(S,1)-E(S,s)
E(s,t)-HS,s)
sq-1
5
6 diu
i E(S,~)-E(S,s)
K 7 btq-’
< -
6 !
dp
E(S,r)
with
Therefore,
if K
(& + K7)b>
=
o
1 -Kg0
8
that is, if K,d<
1
then sqdp
I KS tq-’
.iE(S,t) From
(
here on we can reason 5. EXISTENCE
c E(S,t)
as in [2] and we come to the result.
OF WEAK
SOLUTIONS
QUADRATIC
FOR
Let us now come back to our nonlinear - ~ ~Xj(U)
A QUASILINEAR
GROWTH
IN THE
equation.
+ a,(X)u
+ f(X,
We consider U,
EQUATION
WITH
X, TERM
Xi(U)) = 0
the following in D
i=l
on
u=o
and we suppose
%(4
E J?fi)
we take a Caratheodory if(x,.sp)I
function
5 Co + b(bl)l#
with C,, > 0 and b: CR++ CR+monotone
(5.1) a.e. in Sz.
go(x) z cyo > 0 For f(x. s,p)
an
that
(5.2)
which satisfies a.e. in Q
increasing.
vsE63,vpEw
(5.3)
U. GI.~NAzzA
60
We see that f is bounded relative to the variable x and has at most a quadratic growth in the Xi term. We will prove the existence of at least a solution u E Hd(fiz; E) n L”(Q). We are, therefore, dealing with a weak solution in a sense we will explain in the following. The hypotheses are almost minimal with regard to the nonlinear term, whereas we must assume Sz = B(R, x0) as we rely upon a maximum principle which is valid only on balls: this is a typical situation when working with subelliptic operators as we are doing now. A particular simple case which satisfies the hypotheses given above is - f
x,*X,(u)
+ a&u
+ g(u) i
i=l
lx;(u)12
= h
i=l
with h E L”, g: @ --t CFtcontinuous and a,(x) such as required in (5.1). First of all we require a preliminary lemma, a sort of weak maximum principle equations. LEMMA 5.1.Let u E Hi(Q
E) be a weak solution -i~lx*xitu)
with a = B(R, x0), R suitable,
for nonlinear
of +
aOu
=
-f
f E L”(Q) and a,(x) > o. > 0 a.e. in Q. Then
where M = We now state our theorem
Ilf IL”(n) *
that is just an adapted
version
of the analogous
result given in [6].
THEOREM 5.1.Let Q be a ball B(R, x0) and let us consider a Caratheodory function f(x, S, p) defined in Q x 6l x 61”. Actually this means that it verifies the following hypotheses: (i) v (s, p) E & x @I” the function x + f(x, s, p) is measurable; (ii) v a.e. x E Q the function (s, p) + f(x, S, p) is continuous. We further suppose that f(x, s, p) satisfies (5.3). Finally we consider a coefficient a, defined in 0 such that (5.1) and (5.2) are valid. Under the previous hypotheses we claim that there exists a solution u such that u E Hd(Q; E) n L”(cI) - ~ ~Xi(U)
+ aO(X)u + f(X, 24,Xi(U)) = 0
(5.4) in D’(Q).
(5.5)
i=l
Remark 5.1. It is interesting to observe that as u E Hd n L”, f belongs to L’ thanks hypothesis made above and each term of our equation is therefore a distribution. Remark function
5.2. Generally, when f is a Caratheodory that assigns f(Z, u(2), Xi(U(X))) to Z E Q.
function,
to the
by f(x, u, Xi(U)) we mean
the
61
Square Hiirmander operators
Proof of lemma. As u E H&2; E), given a positive constant C, the function v = (u - C)’ is admissible as a test function. Then Xi(U)Xi((U - C)‘) dX +
iR
i cl
Thanks to the coercivity of the operator,
.r R f,
I&(@ - C>‘)12~+
i
a,(x)u(u
- C)’ dx = -
f(u - C)’ dx.
at least if R is not too big, we have
Qol(U - a+12~
5
1(-f
- a,C)(u
- C)’ dx.
R
R
If we choose C = M/a,
In
we have on the right-hand side
s (-f
- M)(u
- C)’ dx I 0.
n
Therefore,
each of the integrals on the left has to be zero and we conclude (U-c)+=o*U
Proof of theorem. We will proceed in the following proof by steps. of our problem. We consider the function f, defined by
Step 1. Approximation
f fc = 1 + Elf/
(5.6)
where E belongs to a sequence that converges to zero but must be considered fixed for the moment. As If,] < (fl, our new function is still a Caratheodory one and verifies growth condition (5.3). Moreover, we have (5.7) If we consider the approximated
problem U, E H&2; E)
- F Xxi(&) i=l
+ aou, + f,(x, 4,
xi(Uc))
(5.8) =
0
in LO’
(5.8’)
we can reason as in [6] and applying the Schauder fixed point theorem, we can be sure that (5.8), (5.8’) admits at least a solution u,. What is most important, as f, belongs to L”(O) we can use any Hd(!22;E) function as a test function in the approximated problem. Moreover, we can apply lemma 5.1 because of (5.7) and we obtain that any solution of (5.8), (5.8’) belongs to L”(Q) and satisfies the following inequality (5.9)
62
U.
GIANAZZA
Remark 5.3. If we had a maximum
principle valid on a more general class of domain result with weaker regularity assumptions on aQ.
could prove an existence
Step 2. Estimate in L”(Q). We now want to refine estimate does not depend
on E any more.
(5.9) and give a better version we want to show that
Precisely,
Q, we
that
(5.10) As a matter
of fact we will prove that
co
u, < -
a.e. in Q.
(5.11)
010
The symmetric relation U, > -Co/~, is given in the same way and together marized in (5.10). To simplify the notations we define 2, = 2.4,- -
they are sum-
.
010
This function
solves
xi*X.(
- f
I
-fEk,
zE) + aOzE =
XA4J) - a0
i=l
Furthermore,
s
in YB’.
(5.12)
010
if we define (5.13)
where b is the function
that appears
in (5.3), we have
Wl4) 5 c, as b is nondecreasing.
It easily follows
that
-h(Uc,Xi(uc))- a02 5 CO+ b(lu,l) i lXi(Q12 - a02 i=l
s c.5 f Ixi(utd12 i=l = c&E, lXkE)12 Let us now consider
the test function v = exp(t,lz:12)z:
where: (i) z,’ = maxtO, zEl; (ii) t, = C,2/2 and let us define e, =
exp(t,lz:l’).
a.e. in a.
(5.14)
Square
Hiirmander
63
operators
We observe that Z: E L”(a) thanks to the estimate proved at the end of step 1 and Z: E Hd(Q; E); in fact z,’ is zero on &2 as Z, = -Co/o,, a.e. on %2. Moreover, as Xi(U) = e,Xi(Z,‘) u also belongs
+ 2tEe,jZ,‘12Xi($)
to Hd(G; E). We can then use it in equations
Xi(Z,)Xi(Z,+)dX + 2t,
e, i i .n
at least if R is not too big, we can estimate
I ReeiC, lxi(Z:)12dX+ 2tc e,lzi12 ! ,Q
Hfc
69 uc
5-
9
.n
xi Cu.&)+
aO
Ixi(Z:)l k + Qo 46 I2 i .a
f i=l
co 1
e,z,C dx.
by (5.14) we have 7
I eE i
.I fl
&,‘I” f
C,
,,fl
If we apply the Young
jl
dX +
Ixi(Z:)12
C$J
i=l
,I fi
4 5-
n
IXi(Z:)I'dx + 2, I
i=l
, a e, i
1
eelz,f12dx
R
2 IXi(Z,>12e,Z: d-X.
(5.15)
i=l
inequality
we further
lXAz312 do + %
i .a
get
&,+I2 f
&,+I2 b
+ a0 i ,fi
”
1 ~ R e, j, 2 1
Ixi(Z,‘>12dx
i=l
Ixi(Z,‘)12 dx + i
e,ClzE+12 5
IR
IXi(Z,+)J2dX.
i=l
to the choice for t,, 2 t,
we
=
+
have 1” 2 !n
In eE J,
the
QO
0
Thanks
eEuoz,z,+ dx
n
n
,?I
On the other hand,
s
i=l
Thanks to the coercivity of our operator, left-hand term and we obtain
, I
(5.12) and we have
eh,‘12ii xi(Z.E)xi(Z:) dX +
i .R
i=l
also
Ixik:)12 d.Y+ i
cl
As e, 2 1 it must be Z: = 0. Therefore,
e,lz,+I’ E Ixi(zZ>l”~ i= 1 U, I Co/~,
.
+
eE(zE+12
010 s
cl
dx
5
0.
64
U.
GIAN~~ZA
Step 3. Estimate in Hd(CJ; E). If we define
thanks
to the estimate
proved
in the previous
step, we have
b(lu,l) as b is a growing
function.
Let us consider
5 C,
the function +.
u, = exp(tu,2h4, If we define
E, = exp(tuz) then we have + 2t,uz exp{tuz)Xi(u,)
X~(U,) = exp(tu~jXi(u,)
= EEXi(UE) + 2t,u,2E,Xi(u,).
Since u, E H&S; E) fl L”(Q) we can conclude that u, E Hd(Cl; E) and can be used as a test function in our approximated equation (5.8). We have m a0E, uf dx u.ZEe F IXi(U.&12 dx + Ee iC, Ixi(UJ* b + 2t i=l I cl ia .i a
Thanks
to the estimate
on a, we have
m
EciClIXi(u.s)I’ do + 2t
.i cl
=--
R
s
E.zu.Z f Ixi(uc)12~+ QO i=l
E&X
n
fe(~3UC Xi (ucJ)Ec UCdo-
(5.16)
3
i n On the other hand -
fe
txY
u.5 9 xi
(“.A)Ec
ue
dX
cl
i=l
f i=l
IXi(U,)(*dX
(5.17)
Square
HBrmander
65
operators
where IQz(is the measure of Q. If we rewrite (5.16) taking into account estimate (5.17), we have 1 E, : 2 s D i=l
IXi(u,)?~
Ku,2 ,! IXi(u,)l*~
+ f n
+ ~0
i=l
E,U,2 dX n
If we recall that E, 2 1 we have (5.18) and, therefore,
U, E #(a;
E) fl L”(0).
Step 4. Strong convergence in Ho@; E). As the functions u, are bounded in H,‘(Q; E), we can extract a subsequence, which we still denote by uE, such that UE ‘U
weakly in H&Q E)
% + 24
strongly in L*(Q)
u, + u
a.e. in 52.
What we want to show is that actually U, + u strongly in Hd(Q; E). We define ii, = u, - u. This function belongs to Hd(Q; E) n L”(L2) and is a solution for the equation - ~ xi”Xi(ii,) + a(jE, = -f,(X, U,,Xi(U,)) + E xi*Xi(U) - UOU. i=l
i=l
In a similar way as before we define UE= exp(tii,2Jii, where t = 2C,2 and as usual we put E, = exp(t@). It is easy to see that v, E H,‘(Q; E) and can, therefore,
in
E, i
IX&)l*ti
+ 2t
5
Eciiz i .i n
i=l
i -
R
i=l
be used as a test function. We have
IXi(U,)1*dx + (~0 E,tiif ti ia
Et iil xi(~tJK(U)do If,(x, u.s> K(Ue))IEegc ia 2t
n
a, uE, ii, dx
E, ~ Xi(ii,)Xi(U)ii,2 dx i=l
n
(5.19)
66
GIANAZZA
U
If we recall that Ixi(u.5>Iz
=
U + &)I2 = IX;(U) + X;(U,)l2 12JXi(U)12
Ixi(
+ 2lX;(ii,)l2
we get from (5.19) n R
E, 2 ~X;(Z&))~dx + 2t E,U,2 ~ (X;(ii,)l’dx i=l i=l i .Q 1 5
.i[cl
CO + 2C,
f
lX;(~)1~ E,Izi,( dx + ;
E,i$ dX !a
.i a
1
i=l
+ ~yg
E, t
bM%)i2~
i=l
n + i
E,(2C,)2ii,z i ,e
i
IX;(ii,)12dx
-
E, f !
i=l
n
Xi(%)X; (u) dx
i=l
m -
2t
E, C X;(U,)X;(U)~,2 dx R
Taking
into account
a, uE, ii, dx.
(5.20)
n
i=l
the definition
of t we can rewrite
(5.20) in this way
(5.21) As Ei, + 0 strongly in L2(sZ) and also X;(U,) --t 0 at least weakly in L2(n), it is easy to see that all the terms in the right-hand side converge to zero and the same must happen to the other side. As E, 2 1 we have, therefore, that ii, + 0 strongly in H,‘(Qz; E). Step 5. Passage to the limit. As u, converges extracting a new subsequence) Xi(&)
--f
to u strongly
Xi(U)
in HJ(Oz; E) we have (eventually
a.e. in Q.
As in [6] we claim that: (i) f,(X, u,, xi(Uc)) * .I%, U, xi(u)) a.e. in Q (ii) f8(U,, Xi(U,)) + f(U, Xi(u)) strongly in L’(a). Therefore, going to the limit in each term of equation (5.8), we have that the limit function belongs to Hd(Q; E) fl L”(Q) and satisfies equation (5.5). Remark 5.4. It is interesting that we have proved by providing a function which satisfies
the existence
u
of a Hd(Q; E) fl L”(Q) solution
(5.22) In [6] it is proved
that all limited
solutions
satisfy
(5.22).
SquareH(irmanderoperators
67
6. A CACCIOPPOLI INEQUALITY FOR A NONLINEAR PROBLEM We can now prove a Caccioppoli inequality for our problem: it will basically be for the regularity estimates of the next paragraph. Just like the previous paragraph, we have &2= &R, x0). As we work with H,‘(fiz; E) n L”(a) solutions, we can use exponential cut-off functions, so that we can take into account a quadratic growth in the Xi terms. Otherwise a milder condition should be considered.
PROPOSITION6.1. Consider problem (5.4), (5.5) and suppose that (5.1)-(5.3) u E HJ(nz; E) fl L”(a) be a solution. We have
I’ [
+
bi2 G’&,) dx + 1
B(qr,xo)
I ,B(%,),
hold.
Let
su~~(qr,x& - k/*
Iu - kl*dx + C2ra
B(r,xo)-B(w,xo)
(6.1)
where 0 < q 5 i& with Go E (0, l), 0 < r < R, k E 03, C, and C2 are constants which depend on 4, Il&-~~~9k and the quantities that appear in the nonlinear term estimate.
Proof. Firstly we observe that to prove (6.1) it is enough to prove the same inequality with G$2r,x0j replaced by GXo. We use as a test function
u = (u - k) e”“-kt2cp2((G~)-1)Gj z E B(qr, x0) where G,” is the regularized Green function relative to a” and the point Z, t will be fixed in the following and ~1E C,“(B) with
2 @(t) s (C, _ 1y
A and Co suitable. We have
E Xj(u)Xj((u
- k) et’u-k’2cp2GJ dx +
,i Ri=l
+
n
f(x, u, Xj(u))(u
a,u(u - k) er’u-k’2p2G~ dx 0
- k) et~U-k~Zp2G~ dx = 0.
68
U. GIANAZZA
If we put Et = e tiu-k’2 for short as we did in the previous
T lXi(u)12Et~2G;
cl
(U - k)E, i
+ n
+
z
(U - k)‘E,
dx + 2t
fl i=l
we get
(Xi(U)12~‘GpZdX
i=l
Xi(U)Xi(p*)G,ZdX
a, u(u - k)E,p2G; d_x
+
i=l
a
(U - k)Ef f n
paragraph,
Xi(U)Xi(G~)~*dX
i=l
f(xv u, Xi(U>)(U- ~)E~P*G,Z k
=R
and also
Et f IXi(u)~*~*G;dx + 2t .i n
i=l
12)~ IXi(U)12~2G,ZdX i=l
s
Et ~ Xi((U - k)2~2)Xi(G,“)~
n
n
s
(U - k)E,~2G~ dx + ~
-aok
1
i=l
E~(u -
0
--
F
E,(u -
(u - /c)~E,~~~G,Zdx + ;
+ 010
I
i .n
k)* ~
E, ~ Xi((U - k)*)Xi((P2)GpZ dw -
i! .i n
i=l
S[ + ~ Co
Cl
n
i
a
term, putting
f(xv ~9 Xi(U))(U - k)Etp2G;
dx-
+
D zif,
Et E IXi(U)12p2GpZti. i=l
for short and applying
I
i
n
Cl = b(ll~&~,)
IU - klE~~*G,Zdx
IXi(U)12
i=l
C,lu - kjE,p*G;dx
I
dx
n
Recalling the estimate on the nonlinear the Young inequality we have
5
Xi(~")Xi(G,Z)
i=l
cl
IXi
Square
t = CF/2 and apply the Young
If we choose 1 i sa
Hiirmander
E, ~ IXi(U)12~2GpZdX + ~ i=l +
(u
010
1”
k)2E,cp2G;dx
-
(U - k)E,lo’G,Zdx
1
+ ;
+ ~
E,(u - k)2 ~ Xi(lo’)Xi(G~)dx in
(u - k)2E,p2G;
\u - k~E,q~~G;dx
+ Co .i n E,p2G,Z dx
n
1
E,(u - k)’ ~ Xi(~2)Xi(G,Z)dx
2 ia +
i=l
dx + C, k2
0 f-
IX,(~)(~cp~GpZd_x
E1 ~ Xi((U - k)‘~‘)Xi(GpZ)dX i=l
E, ~ Xi((U - k)2)Xi(~2)G~dX i=l
I ?
we find
i=l
iR
2
once more,
E,(u - k12 f
sD
I -aok
--
inequality
.r0
D
.r
operators
Iu
co
- ~
Et E Xi((U - k)2)Xi(p2)G,” dx
i=l
i=l
n
k[Etq2G;dx.
-
fl
Therefore, 1 2 1n
Ed ~ IXi(U)12~2GpZdx + ~ i=I 3 + 4ao 1
(U - k)‘Etp2G;dx
2
i 0
+
T
+ ;
Et i i ,a
Xi(p’)Xi(G,Z)dx
-
Xi((U - k)2~2)Xi(G,“)dw
i=l
- i
Et ~ Xi((U - k)2)Xi(~2)GpZ~
i=l
(u
lXi(U)12v2GpZdx
i=l
i cl
E~(u - k)2 f
I-
E~(u - k)2 t n
sn
k)2Etp2G;dx
+ C2k2
i=l
Etp2G;dx
+ C,
.rn
in
E,cp’G; dx. in
Finally,
sn
Et E [Xi(U)(2p2Gidx i=l + c:
Et f In
5
+
R c4
IXi(U)I’(U - k)2p2G,ZdX +
Et i .i n
i=l
Xi(Cp’)Xi(G,Z) ti
i=l
Et (P’G; dx cl
d.x
sn
E~(u - k)2 f
I
(u - k)2Etp2G;
+ (~0
-
Et i sn
i=l
Xi((U - k)2p2)Xi(GpZ) ti
i=l
Xi((U - k)2)Xi(p2)GidX
U. GIANAZZA
70
having
put c,
If we take into account
and taking
= 2(C#
that u E L”(Q),
into account
-t C,).
putting
CS = inf,(u
- k\
C, = mint1
+ CfCi,
01~)
that 1 5 Et 5 C,
where C, stands
G
for its upper
bound,
we conclude
) i( itlIXi(U>12+ N2‘
p2G; dx +
(u -
R
r~
Xi((U - k)2~2)Xi(G,") ~
, I2i=l
s I c,
(U - k)’ u cl
+ c8
~ Xi(~2)Xi(G~) i=l
m c Xi((u
dx -
- @2)Xi((02)G,” dx
! Cli=l
1
p2G; dx. Ia
If we recall that (U - /Q2 > ;
- !$
we find
c9
f i=l
i( n
I&(u)12
I c,
+
u’)v’C~
(U - k)’ f irtJfi
CIX
+
in
Xi((U - k)2~2)Xi(G,“) dx
Fl
Xi(p2)Xi(GpZ) dx -
i=l
f
Xi((U - k)*)Xi(~“)G,” dx
0 i=l
1
I\ +
cp”G; dx
Cl0
n
where C9 = C6/4 and Cl0 takes into account ’
c9
the estimate
given above.
Then
m
.i( a
I
iCl
2
IxiCu)I
+
u*
P’G;
CLY +
,Bc;,
zj,
B(p,~)
(u
-
W2
CLX
)
c,
n
(U
s
-
‘:’ + kj21p’It Ixi(G,Z)I’(~33 i=l
&)k+
Clo[Qr2G;~.
71
Square Hiirmander operators
Passing
to the limit as p --t O+ we obtain
c9
n
jt,
+
Ixi(u)12
u2
>
i(
I c,
.in
p2Gz dx + (u - /c)~(z)
$$
; Ixi(Gz)/2 i=l
(u - k)‘Iv’I
Choosing A and C, such that: (i) (x: GZ 1 l/A) > B(2qr, z) (ii) (x: Gz 2 l/C,,A) c B((l - q)r,z) (possibly for small r) and taking into account derivatives, we obtain
s(
+ &++
the estimates
j, lxitu)12+ u2 GZ dx
B(v, z)
>
C,,sflGz~.
on the Green
function
and its
+ (u - /c)~(z)
>
s &j
~/ B(r,z)-B(qr,z)(’ - k)2 dx + c12ra
with 01 suitable (we use the duplication property of measures of intrinsic only on the dimension of the space and the Hormander constant k). Finally, taking the supremum for z E B(qr, x0) we get
balls and LYdepends
m C
i=l
Ixi(U)I’
+
s
u2
>
G”‘~-x
(24-
+
SUPB(qr,x,,)(U
-
k12
Jq2dx + C12P.
B(r,q+BWvW
7. REGULARITY
FOR THE
WEAK
We can now come to the estimate
SOLUTION
for the solution
OF A NONLINEAR
to our nonlinear
EQUATION
problem
in Q = B(R, x0).
THEOREM
7.1. Let u be a Hd(Q; E) II L”(Q) solution of (5.5) and suppose that (5.1)-(5.3) are satisfied. Then there exists an exponent p > 2 such that u E yt;p(sZ; E) and the following estimate holds for B, c BZr c Sz 1
(f (
p’2
lU12+ ~ IXi(U)12 )
B,
i=l
&)l”
If we define
we also have the following
theorem.
s
KI[(,~Br(lulz + j,
lxi(u)12) &y”]
+
K2*
U.
12
THEOREM 7.2. Under
the same hypothesis V(r) I
GIANAZZA
as above, r
0
Cl
-
RCI
with r 5 (q/2)Ro < R, 5 (q/2)R,
y
u is locally
V(R,) + CzR,
q as in the previous
Holder
continuous
and
6
paragraph
and y, 6 E (0, 1).
Proof of theorem 7.1. It is easy to see that the metric topology induced by the intrinsic distance defined in Section 1 is equivalent to the natural Euclidean topology of Gl”. Moreover, the Lebesgue measure satisfies the doubling condition for our intrinsic balls. We can then see that the space a” wth the intrinsic distance acquires the structure of homogeneous space, as discussed in Section 2. Let us now come back to our Caccioppoli inequality
s( BG7r,xo)
i
f i=l
k(#
+
Gg2r,xoj
u2
b
+
su~s(~‘.x,,)(u
- Q2
>
,B&,l,
(u -
k)‘dx + C2ra.
~(~,~o)-~(qr,xO)
If we neglect the second term on the left-hand Green function, we have
side and take into account
the estimate
for the
n
i B(V,%)
(
i~llxioi2+~2)~r~~xii,i,,smi,r,,(~~k)2dX+C~’B(~~~‘r~.
If we recall how ra was calculated
we find
Now putting q = a, k the average of u on B(qr, x,), using the Sobolev inequality q* such that l/q* + l/s = 1 where s is the Sobolev exponent, we have
If C, = 0 we could directly apply theorem 4.1 and we have the result. the case, we can reason as in [2] and again we obtain our inequality.
Otherwise,
and assuming
as is usually
Proof of theorem 7.2. We use as a test function u = (u _ 0) er(l~-“‘*G;rp where ii is the average of u on B(qr, x0), p < r, Gj = G&f,,zj,p is the regularized Green function of B(2r, z), rp is the potential of B(2r, z) with respect to B(tr, z) and I is as in proposition 6.1.
Square
Hormander
operators
73
We can reason as in theorem 1 of [lo], to which we refer. The initial difference due to the presence of the exponential term can be adjusted as in the proof of proposition 6.1, thanks to the fact that u E H~(Qz; E) n L”(S2). REFERENCES of the partial derivatives of a quasiconformal 1. GEHRINC F. W., The Lp integrability 265-271 (1973). 2. GIAQUINTA M. & MODICA G., Regularity results for some classes of higher order
mapping, nonlinear
Acta Math. 130, elliptic
systems,
J.
Math. 311, 145-169 (1979). 3. MEYERS N. G., An Lp estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scu. norm. sup. III XVII, 189-205 (1963). 4. COIFMAN R. R. & WEISS G., Analyse harmonique non-commutative sur certaines espaces homogenes, Lecture Notes in Mathematics, Vol. 242. Springer, Berlin (1971). principle for Dirichlet forms on discontinuous media (preprint). 5. BIROLI M. & Mosco U., A Saint-Venant 6. BOCCARDO L., MURAT F. & PUEL J. P., Existence de solutions faibles pour des equations elliptique quasi-lineares a croissance quadratique, Nonlinear Partial Differential Equations and their Applications, College de France Seminar, (Edited by H. BREZIS and J. L. LIONS), Vol. IV, Research Notes in Mathematics. Pitman, London (1983). I. JERISON D. & SANCHEZ-CALLE A., Subelliptic second order differential operators, in Lecture Notes in Mathematics, Vol. 1277, pp. 46-77. Springer, Berlin (1987). 8. NAGEL A., STEIN E. M. & WAINGER S., Balls and metrics defined by vector fields: basic properties, ActaMath. 137,
247-320 (1986). 9. SANCHEZ-CALLE A., Fundamental
solutions
and geometry
of the sum of squares
of vector fields, Invent. Math. 78,
143-160 (1984). 10. BIROLI M., Local
properties
of solutions
to equations
involving
Potential Theory, Nagoya, August (1990). 11. STEIN E. M., Singular Integrals and Differentiability Princeton (1970). 12. GIANAZZA U., The Lp integrability
on homogeneous
square
Hormander
Properties of Functions.
spaces,
operators, Princeton
Proc. Int. Conf. University
Rend. Ist. Lomb. A 126, 82-92 (1992).
Press,
Analysis,
Theory,
Methods
Pergamon
REGULARITY
& Applications, Vol. 23, No. 1, pp. 49-73, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/94 $7.00+ .Ml
FOR NONLINEAR EQUATIONS INVOLVING HijRMANDER OPERATORS
SQUARE
Uco GIANAZZA Dipartimento di Matematica, Universita di Pavia, via Abbiategrasso 209, I-27100 Pavia, Italy (Received 12 July 1992; received for publication
16 June 1993)
Square HBrmander operators, homogeneous spaces, extraintegrability.
Key words and phrases:
1. INTRODUCTION IN 1973 Gehring
[l] showed that an Lq(&‘) function g is actually Lp integrable in a cube Q for some p > q provided a certain inequality relating to the maximal functions holds. In a successive paper Giaquinta and Modica [2] observed that the lemma is true only if g = 0 on a” - Q and proved a correct local version which is the main tool in their proof of an Lp estimate for a nonlinear elliptic system. In this way they gave an extension of a similar result that Meyers had obtained for an elliptic equation in divergence form [3]. At the same time a big interest arose in the development of a theory of singular integrals on homogeneous spaces; therefore, some useful results, such as covering lemmas and, above all, the Calderon-Zygmund lemma, were proved in the new setting (see [4]). Recently the structure of homogeneous space has been of great importance in studying Dirichlet forms as a tool in the description of the variational behaviour of possibly highly nonhomogeneous and nonisotropic bodies (see [5]). In this work we consider the problem - ;
Xi*x((u) + b(x, 24,X;(u)) = 0
in L!
(1.1)
i=l
u=o
on
where Xi satisfies a uniform Hormander condition k span the tangent space), Xi* is the formal adjoint Precisely we have W,
u, Xi(U)) = a,(x)u
asz
(1.2)
(Xi and its commutators up to a fixed order and b(x, s, p) is a first order nonlinear term. + “0% u, Xi(U))
with @o(X)> CY> 0 and f is a Caratheodory function which has at most a quadratic growth in Xi(u). Our aim is to study the regularity properties of weak solutions for equation (1.1). Under a smoothness assumption on L2, we will prove in theorem 5.1 that a weak solution u exists, which actually belongs to H,‘(a) n L”(a). To do that we adapt to our context an idea of Boccardo-Murat-Puel (see [6]), which is essentially based on the weak maximum principle. This result opens the way to finer estimates relative to our nonlinear equation. 49
50
U GIANAZZA
As a matter of fact, it is known that an intrinsic distance d(x, y) can be associated to our Xi in a natural way and this, together with Lebesgue measure, gives to @li” the structure of homogeneous space. Therefore, in theorem 4.1 we prove a homogeneous version of the Giaquinta-Modica lemma (see [2]) that will be a major tool in the proof of theorem 7.1 where we show that an L!’ estimate is still valid, much along the lines of Meyers’ result (see [3]). Finally in theorem 7.2 we demonstrate that our solutions are also locally Holder continuous. 2. FUNCTIONAL
Let us consider C”(al”) functions now define the vector fields
FRAMEWORK
a;,, where i = 1, . . . , m (m fixed) and I = 1, . . ., n. We can
Xi = ~ ai, ~ I=1 I and suppose that they satisfy a uniform (for further details see [7]). By making our vector fields (see [7-91) we denote
i=
l,...,m
Hormander condition as discussed in the Introduction use of the intrinsic distance which can be associated to
B(r, x) = (y: d(x, y) < r). Its main feature is that this ball has different dimensions along the directions defined by the Xi and their commutators. In [7] a Poincare inequality has been proved for intrinsic balls. Precisely we see that given a relatively compact open subset A of a”, there exists a constant C, > 0 and an integer k 2 1 such that for every x E CR”and every r > 0 with B(r, x) C A the following inequality holds
is
”
124 - iiJ2dx 5 C,r2 i where u E C”(A) let us consider
IxiCu)12
dX
(2.1)
.w, x)
B(r/k,x)
and ii is the average
of u on @r/k,
x). Given an open set Q such that Q E 61n 0
E(u, v; Q) = : i=l
Xi(U)Xi(V) dx +
uvdx !
.i Cl
R
which naturally gives rise to the norm [E(u, U; Q)]“2. We can then extend the classical Sobolev spaces to our setting. Therefore, by H’(SZ; E) and Hd(0; E) we denote the closure of C”(Q) and C,“(a) for the norm E. An easy extension of (2.1) to these new spaces is then proved. Let us now consider L = -t
We can associate
the following
bilinear
i=l
x;xi.
form on H’(Q;
(2.2) E) x Hd(sZ; E)
xi (“)xi(v) dX
(2.3)
Square
Hiirmander
operators
51
and the question arises about the existence of the Green function for the Dirichlet problem relative to our operator in CR”. Such a problem has a positive answer and the following estimates have been proved r2
r2 (2.4)
A, J&r, x)1 s GX(y) 5 *l JB(r, x)1 2-s lxj,
9 *..9 x,G”W
5
As ,&,
where G’(Y) is the Green function relative to L in 611”with singularity r I f?, R > 0 suitable and ]B(r, x)1 is the volume of the ball. As in the usual Dirichlet problem, G”(Y) is such that a@n(u, GX) = U(X)
(2.5)
x),
at x and Y E aB(r, x),
v U E C,“(@“).
Since a(u, v; B(r, x)) is coercive on Hd(B(r, x); E) it has been possible to define the Green function relative to L in B(r, x) with singularity at x (see [lo]). The above written estimates (2.4) and (2.5) are easily extended to the new setting. In particular we have
(2.6) for y E dB(qr, x), q E (0, qO], q0 < 1 suitable, A3 and A, constants which do not depend on x. It is also useful to consider the so-called regularized Green function relative to the ball B(r, x) (G,“(y)) as defined in [IO]. We refer to that work for the properties of (or to a”), GpX;BB(r,xj convergence
of G,“;.,,,,(G,“(y))
to G&,,,(GX(~)) 3. HOMOGENEOUS
Let us consider a set X and a function it satisfies the following properties: (i) &,Y) > 0 es x f Y; (ii) P(X, Y) = P(Y, x); (iii) there exists a positive
constant
as P + 0. SPACES
p: X x X --t 6l’. We say that p is a pseudodistance
K such that for every x,y, z E X
P(X, Y) 5 K]P(Y, z) + P(Z, We define
the pseudoball
if
41.
B(r, x) of ray r and centre x as the set B(r,x)
= lY:Y EX,P(X,Y)
< rl.
To simplify the notations we will use “distance” and “ball” instead of “pseudodistance” and “pseudoball”. Let us suppose that the set X considered above is a topological space and that the associated distance is such that: (i) the balls B(r, x) form a base of open neighbourhoods of x; (ii) there exists an N E N such that for every x E X and for every r > 0 there are no more than N points xi belonging to B(r, x) such that p(Xi, Xj) > r/2 (homogeneity property). We say that X is a homogeneous space and the two numbers N and K occurring in the definitions are its constants. In the following the letter C will stand for a generical constant, different in various contexts, depending only on N and K.
U. GIANAZZA
52
It is interesting to observe that the homogeneity property is automatically verified if a positive Borelian measure ,D defined on a o-algebra of subsets of X which contains the balls B(r, x) satisfies the following inequality 0 < p(B(r, x)) < C/@(r/2,
x)) < +=J
(3.1) for every x E X and r > 0. Therefore, we shall say that a topological space X is a homogeneous space if it is endowed with a distance p and a measure ,u satisfying those conditions. We now want to state some lemmas whose proofs have already been given in [4]. They are practically the homogeneous version of the analogous ones given in [ 1l] for the Euclidean case of 6li”. LEMMA 3.1. Let E 5 X be a bounded subset of a homogeneous
space X which is covered by the family (B@(x), x)). From this family we can then select a sequence of disjoint balls B(r(x,), Xi) so that the family (B(kr(Xi), Xi)) covers E, the constant k depending only on Nand K. Moreover, we have CP(E) 5 C P(B(‘?xJ, Xi)). Actually the last part concerning the inequality about the measure is new, but can be proved very easily thanks to (3.1) and the numerable additivity of Borelian measures. LEMMA 3.2. Let 0 c X be a bounded open set. There exists a sequence of balls B(p,, xi) such
that:
(i) Q = Ui &pi,
X;);
(ii) a point x E a cannot belong to more than M balls &pi, Xi); (iii) the balls B(hpi, Xi) intersect the complement of Q, CQ. The two constants M and h depend only on X. Proof. The proof is the same as in [4]. It is important to observe that in the above-mentioned work we have r(x) = & where K is the constant of the homogeneous
dist(x, CQ) space and k is defined in lemma 3.1
1 pi = kri = - dist(x,, CSJ) 2K h = 3K * hpi = 3 dist(xi, CQ) but there is no need to put exactly hp, = t dist(xi, CQ). It is enough that the balls intersect CQ; therefore, any choice of h that brings about this result, will do. Definition 3.1. We define the local maximal function function f as 1
@cx)= suPR
s BR
li;r,(x) of an L’(X, dp) nonnegative
f(t) Wt).
Square Hiirmander operators It
is obvious
53
that a.e.
f(x> 5 n;i,cx>
PROPOSITION3.1. Let us consider f E L’(X, dp), f 2 0. We have p((x E x: &-f(X) > CrJ)5 5o1 x f(t) d/At). I Proof.
The proof is as in [4]. There is no formal change due to the upper limit we now set
for R. PROPOSITION 3.2. Let f be a bounded support function, such that f 2 0. Let us further suppose that f E L’(X, dp) and that
There exists a sequence of balls B(pi, Xi) such that: (i) f(X) I cd a.e. on X - Ui B(pi, Xi) = X - S2; (3 l/IB(Pi, XJI SB(,,,,~ f(t)+(t) (iii) Ci IB(pj, xi)1 5 C/a jxf(t)
5 CW W);
(iv) a point x E X cannot belong to more than M balls B(p,, Xi). Proof. Again we work as in [4]. The fact that we consider the local maximum function instead of the global one requires that hp;
Vi.
This is always possible (see the remark in the proof of lemma 3.2) even if any set X can require a different choice of R, and (or) of h. 4. L!’ INTEGRABILITY
ON HOMOGENEOUS
SPACES
We can now come to theorem 4.1. For the sake of completeness, we will first state a global version of it which is of some interest but cannot be applied to our context. PROPOSITION 4.1. Let us consider a homogeneous space X and a function g with bounded support such that g: X + [0, +oo] and g E L*(X, dp) with q E (1, +a). Suppose that p(X) < +co, b E (1, +a~) and that
M&X) 5 NM, WI4
a.e. in X
(4.1)
where Mf(x) is the global maximal function for f. Then g is Lp integrable in X and satisfies (4.2)
for p E [q, q + c), where c is a positive constant which depends only on q, b and the constants of the homogeneous space.
U. GIANAZZA
54
Remark 4.1. An inequality such as (4.1) is commonly known as reverse Holder. It is interesting to observe that the global nature of this proposition is deeply linked to the use it makes of global maximal functions. Under the point of view of the applications, we are led to consider H{(X) functions, where the boundary value gives no problem, but this can be the biggest limitation. That is why we have to give a local version of the above result and that can be done only by putting an upper limit on R in the maximal function, as done in the preceding paragraph. With the new idea and a finer but rather technical subdivision argument, we can overcome the main difficulties and prove theorem 4.1. Proof. See [12]. Let us finally state the main theorem of this section. space X and two nonnegative functions g and f such that g E Lq(x), 4 > 1 andf E L’(X), r > q. Fix a ball B, = B(R*, x0) in Xand suppose that for every x E B, and R < i dist(x, SS,) the following estimate holds
THEOREM 4.1.Let us consider a homogeneous
(4.3) with b > 1. There exists a constant 8, = &(q, r, K, N) such that if 6 < &,, then g E L$,,(B,) for P E h, q +
~1and (,&,,
{.,gpdp)l’p 5 C/(\B(2;,x),
i,,,gqdp)L’9
+ (,,(,A,.,
.i,,,fp4’p]
where C and E are positive constants depending only on b, 8, q, r, K, N. Proof.
Let us consider the ball B, such that B, = B(3R, x,)
and let us define c, = B(R, x0) C, = (x E X: 2rekR < dist(x, dB,) 5 22-kR) It is clear that B,(3R,
X0) =
u ck. kz0
k = 1,2...
C4e4)
Square
HGrmander
operators
55
Let us now set (Yk
=
(KILL)
K, a proper
g(x)
(g’)B, =
G(x)= (g”);;
constant
j$ i B, gq dp
on ck
f(x)
F(x)= (gQ);( S(x)
=
F<“>
on ck
Olk E(h, t) = (x Fix t E [ 1, +a)
E
B,: h > t).
and choose s = K2 t such that for convenience
Since
we can apply lemma 3.2 on each Gk. If the constant h,p,
and the upper bound
k+5
= k
+
4
diNxi, c&d
R,, in the definition &,k
we obtain
a sequence
that is
k+4R
= -
k+32k
-
in lemma 3.2 is such that
k+5i? 5
k+4
of the local maximal
of balls Bh (pi, Xi) such that:
(i) BL (Pi 3Xi> C ck ; (ii) G 5 (Y,$ a.e. in Gk - UjBi(pi,xi); (iii) l/IBL[ IBj, Gq(t) d&t) I K&/$)‘. If we divide by CY$we obtain
hk which appears
2
function
is given as
U. GIANAZZA
56
Then
IE(s,s)
- uj,kBjkI
= 0 and, hence,
where we take into account
the doubling
property
and we put
K4 = K,C. Let us now recall that !& is such that MGQ(X) > sqcYq k Therefore,
for each x E fik there exists a ball @(r,
vx
E
Qk.
x) such that (4.6)
with r < ROk = (k + 4)/(k + 3)R/2k. Denote by Bi the ball which has the same centre
as i$ and F = $1. Of course
B{ c c, u c, u c,
(4.7)
k+2 Bi
c
u
(4.7’)
cj
i=k-1
and also (4.8) i=O k+4 Lg
c
u
(4.8’)
cj.
i=k-1
Thanks
to (4.7) we have @ = (Bi n ck) u @j, n ck+r) u (@ n &+2).
Therefore,
If we divide
(4.6) by CY~we have
I
(4.9)
51
Square HGrmander operators
Let us now consider
our inequality
+
.i elBIL 4/3R
and take two balls & and B’,. To simplify (4.10) becomes
Dividing
qdp
+
ak
I BnCk
i BnCk+i
z+
+
sQd,z + 2 =+I
If we also take into account
If we reason
Taking
as before
into account
5q
-t(B*l q-1
I
let us simply
write B and B*. Then
side
[BnCk+23p]
Rq&+4[BnCk+2Sq‘b]
+,[BS’q4-;-
s BnCk+l
Bnc,,
LI
(4.10)
dpu.
the notations
by (gq)B, 01: we get on the left-hand
,A -li
gq
the right-hand
we finally
the definition
s
gdp
side, we have
get
of s, we have
+ (B*(‘-
llfl’B*~Qlp)i’q
($i’q([B*SqdP>“q
+
lB*1’-“‘.
B*
It is also easy to see that in this new setting
[B*sdp ([,*CFqdlr)Lb
we have as in [2]
5 i’ris,rlnB*Sdp
5 [jEC5,CloB*5qd,u
5 2t\B*)
+ flB*l + tqiB*~]l’*jB*I’-l’q
t’-’ S CFq dp
+
.E(‘S,t)nB*
lB*ll-l’q (;y’q( I’..6’ dp]‘q
I
2tlB*I
+
;tl-q
6’ dp
j w2.r)
nB*
U.
58
G~ANAZZA
if 0 < b. Therefore, n
5 -t(P q-1
cr
6 dp + t1-q
e + _t’-4 b
5q dp I E(S,f)fIB’
EG,t) oB*
.i EG,1) nfs*
Sq b.
(4.11)
Let us now define Dk = u@. j Since Dk is bounded
we can apply the usual covering
lemma
and obtain (4.12)
where the Bici) are pairwise
disjoint.
Combining
ID,1 5 K, c I@fi’\ I KS? i
(4.11) and (4.12) we get
‘i4 i=k-I I>
x
t-l i
Sdfi
+ t-q
! E(5.t) nci
If k = 0 the sum has obviously to be corrected observe that Dk 2 a,. Therefore, 1%
c
k
Moreover, ; Thanks
5
when we sum over k, the integrals
ID,\
5
;&(q-
1) t-’ i
c k
to the estimates
!
t-l i
Sdp
0 I ws*s)
E(S,t)
.
(4.8). Let us now
at most six times each. Then
!Yqd,u + %f-q
+ t-q i E(3.t)
- 1)
+ trq
Tqdp
.i E(S,t)
1
ID,\.
over Ci are counted
sd,u
Sq&
to (4.5) we have
x
with
+ Bf-q i m,r)nc,
according
‘;K,(q
or
Fqdfi
+
;t-q .i,,,,,
‘”
dpj
Square
Hbrmander
operators
59
On the other hand $j4dp =
&
66’-’ s E(S,1)-E(S,s)
E(s,t)-HS,s)
sq-1
5
6 diu
i E(S,~)-E(S,s)
K 7 btq-’
< -
6 !
dp
E(S,r)
with
Therefore,
if K
(& + K7)b>
=
o
1 -Kg0
8
that is, if K,d<
1
then sqdp
I KS tq-’
.iE(S,t) From
(
here on we can reason 5. EXISTENCE
c E(S,t)
as in [2] and we come to the result.
OF WEAK
SOLUTIONS
QUADRATIC
FOR
Let us now come back to our nonlinear - ~ ~Xj(U)
A QUASILINEAR
GROWTH
IN THE
equation.
+ a,(X)u
+ f(X,
We consider U,
EQUATION
WITH
X, TERM
Xi(U)) = 0
the following in D
i=l
on
u=o
and we suppose
%(4
E J?fi)
we take a Caratheodory if(x,.sp)I
function
5 Co + b(bl)l#
with C,, > 0 and b: CR++ CR+monotone
(5.1) a.e. in Sz.
go(x) z cyo > 0 For f(x. s,p)
an
that
(5.2)
which satisfies a.e. in Q
increasing.
vsE63,vpEw
(5.3)
U. GI.~NAzzA
60
We see that f is bounded relative to the variable x and has at most a quadratic growth in the Xi term. We will prove the existence of at least a solution u E Hd(fiz; E) n L”(Q). We are, therefore, dealing with a weak solution in a sense we will explain in the following. The hypotheses are almost minimal with regard to the nonlinear term, whereas we must assume Sz = B(R, x0) as we rely upon a maximum principle which is valid only on balls: this is a typical situation when working with subelliptic operators as we are doing now. A particular simple case which satisfies the hypotheses given above is - f
x,*X,(u)
+ a&u
+ g(u) i
i=l
lx;(u)12
= h
i=l
with h E L”, g: @ --t CFtcontinuous and a,(x) such as required in (5.1). First of all we require a preliminary lemma, a sort of weak maximum principle equations. LEMMA 5.1.Let u E Hi(Q
E) be a weak solution -i~lx*xitu)
with a = B(R, x0), R suitable,
for nonlinear
of +
aOu
=
-f
f E L”(Q) and a,(x) > o. > 0 a.e. in Q. Then
where M = We now state our theorem
Ilf IL”(n) *
that is just an adapted
version
of the analogous
result given in [6].
THEOREM 5.1.Let Q be a ball B(R, x0) and let us consider a Caratheodory function f(x, S, p) defined in Q x 6l x 61”. Actually this means that it verifies the following hypotheses: (i) v (s, p) E & x @I” the function x + f(x, s, p) is measurable; (ii) v a.e. x E Q the function (s, p) + f(x, S, p) is continuous. We further suppose that f(x, s, p) satisfies (5.3). Finally we consider a coefficient a, defined in 0 such that (5.1) and (5.2) are valid. Under the previous hypotheses we claim that there exists a solution u such that u E Hd(Q; E) n L”(cI) - ~ ~Xi(U)
+ aO(X)u + f(X, 24,Xi(U)) = 0
(5.4) in D’(Q).
(5.5)
i=l
Remark 5.1. It is interesting to observe that as u E Hd n L”, f belongs to L’ thanks hypothesis made above and each term of our equation is therefore a distribution. Remark function
5.2. Generally, when f is a Caratheodory that assigns f(Z, u(2), Xi(U(X))) to Z E Q.
function,
to the
by f(x, u, Xi(U)) we mean
the
61
Square Hiirmander operators
Proof of lemma. As u E H&2; E), given a positive constant C, the function v = (u - C)’ is admissible as a test function. Then Xi(U)Xi((U - C)‘) dX +
iR
i cl
Thanks to the coercivity of the operator,
.r R f,
I&(@ - C>‘)12~+
i
a,(x)u(u
- C)’ dx = -
f(u - C)’ dx.
at least if R is not too big, we have
Qol(U - a+12~
5
1(-f
- a,C)(u
- C)’ dx.
R
R
If we choose C = M/a,
In
we have on the right-hand side
s (-f
- M)(u
- C)’ dx I 0.
n
Therefore,
each of the integrals on the left has to be zero and we conclude (U-c)+=o*U
Proof of theorem. We will proceed in the following proof by steps. of our problem. We consider the function f, defined by
Step 1. Approximation
f fc = 1 + Elf/
(5.6)
where E belongs to a sequence that converges to zero but must be considered fixed for the moment. As If,] < (fl, our new function is still a Caratheodory one and verifies growth condition (5.3). Moreover, we have (5.7) If we consider the approximated
problem U, E H&2; E)
- F Xxi(&) i=l
+ aou, + f,(x, 4,
xi(Uc))
(5.8) =
0
in LO’
(5.8’)
we can reason as in [6] and applying the Schauder fixed point theorem, we can be sure that (5.8), (5.8’) admits at least a solution u,. What is most important, as f, belongs to L”(O) we can use any Hd(!22;E) function as a test function in the approximated problem. Moreover, we can apply lemma 5.1 because of (5.7) and we obtain that any solution of (5.8), (5.8’) belongs to L”(Q) and satisfies the following inequality (5.9)
62
U.
GIANAZZA
Remark 5.3. If we had a maximum
principle valid on a more general class of domain result with weaker regularity assumptions on aQ.
could prove an existence
Step 2. Estimate in L”(Q). We now want to refine estimate does not depend
on E any more.
(5.9) and give a better version we want to show that
Precisely,
Q, we
that
(5.10) As a matter
of fact we will prove that
co
u, < -
a.e. in Q.
(5.11)
010
The symmetric relation U, > -Co/~, is given in the same way and together marized in (5.10). To simplify the notations we define 2, = 2.4,- -
they are sum-
.
010
This function
solves
xi*X.(
- f
I
-fEk,
zE) + aOzE =
XA4J) - a0
i=l
Furthermore,
s
in YB’.
(5.12)
010
if we define (5.13)
where b is the function
that appears
in (5.3), we have
Wl4) 5 c, as b is nondecreasing.
It easily follows
that
-h(Uc,Xi(uc))- a02 5 CO+ b(lu,l) i lXi(Q12 - a02 i=l
s c.5 f Ixi(utd12 i=l = c&E, lXkE)12 Let us now consider
the test function v = exp(t,lz:12)z:
where: (i) z,’ = maxtO, zEl; (ii) t, = C,2/2 and let us define e, =
exp(t,lz:l’).
a.e. in a.
(5.14)
Square
Hiirmander
63
operators
We observe that Z: E L”(a) thanks to the estimate proved at the end of step 1 and Z: E Hd(Q; E); in fact z,’ is zero on &2 as Z, = -Co/o,, a.e. on %2. Moreover, as Xi(U) = e,Xi(Z,‘) u also belongs
+ 2tEe,jZ,‘12Xi($)
to Hd(G; E). We can then use it in equations
Xi(Z,)Xi(Z,+)dX + 2t,
e, i i .n
at least if R is not too big, we can estimate
I ReeiC, lxi(Z:)12dX+ 2tc e,lzi12 ! ,Q
Hfc
69 uc
5-
9
.n
xi Cu.&)+
aO
Ixi(Z:)l k + Qo 46 I2 i .a
f i=l
co 1
e,z,C dx.
by (5.14) we have 7
I eE i
.I fl
&,‘I” f
C,
,,fl
If we apply the Young
jl
dX +
Ixi(Z:)12
C$J
i=l
,I fi
4 5-
n
IXi(Z:)I'dx + 2, I
i=l
, a e, i
1
eelz,f12dx
R
2 IXi(Z,>12e,Z: d-X.
(5.15)
i=l
inequality
we further
lXAz312 do + %
i .a
get
&,+I2 f
&,+I2 b
+ a0 i ,fi
”
1 ~ R e, j, 2 1
Ixi(Z,‘>12dx
i=l
Ixi(Z,‘)12 dx + i
e,ClzE+12 5
IR
IXi(Z,+)J2dX.
i=l
to the choice for t,, 2 t,
we
=
+
have 1” 2 !n
In eE J,
the
QO
0
Thanks
eEuoz,z,+ dx
n
n
,?I
On the other hand,
s
i=l
Thanks to the coercivity of our operator, left-hand term and we obtain
, I
(5.12) and we have
eh,‘12ii xi(Z.E)xi(Z:) dX +
i .R
i=l
also
Ixik:)12 d.Y+ i
cl
As e, 2 1 it must be Z: = 0. Therefore,
e,lz,+I’ E Ixi(zZ>l”~ i= 1 U, I Co/~,
.
+
eE(zE+12
010 s
cl
dx
5
0.
64
U.
GIAN~~ZA
Step 3. Estimate in Hd(CJ; E). If we define
thanks
to the estimate
proved
in the previous
step, we have
b(lu,l) as b is a growing
function.
Let us consider
5 C,
the function +.
u, = exp(tu,2h4, If we define
E, = exp(tuz) then we have + 2t,uz exp{tuz)Xi(u,)
X~(U,) = exp(tu~jXi(u,)
= EEXi(UE) + 2t,u,2E,Xi(u,).
Since u, E H&S; E) fl L”(Q) we can conclude that u, E Hd(Cl; E) and can be used as a test function in our approximated equation (5.8). We have m a0E, uf dx u.ZEe F IXi(U.&12 dx + Ee iC, Ixi(UJ* b + 2t i=l I cl ia .i a
Thanks
to the estimate
on a, we have
m
EciClIXi(u.s)I’ do + 2t
.i cl
=--
R
s
E.zu.Z f Ixi(uc)12~+ QO i=l
E&X
n
fe(~3UC Xi (ucJ)Ec UCdo-
(5.16)
3
i n On the other hand -
fe
txY
u.5 9 xi
(“.A)Ec
ue
dX
cl
i=l
f i=l
IXi(U,)(*dX
(5.17)
Square
HBrmander
65
operators
where IQz(is the measure of Q. If we rewrite (5.16) taking into account estimate (5.17), we have 1 E, : 2 s D i=l
IXi(u,)?~
Ku,2 ,! IXi(u,)l*~
+ f n
+ ~0
i=l
E,U,2 dX n
If we recall that E, 2 1 we have (5.18) and, therefore,
U, E #(a;
E) fl L”(0).
Step 4. Strong convergence in Ho@; E). As the functions u, are bounded in H,‘(Q; E), we can extract a subsequence, which we still denote by uE, such that UE ‘U
weakly in H&Q E)
% + 24
strongly in L*(Q)
u, + u
a.e. in 52.
What we want to show is that actually U, + u strongly in Hd(Q; E). We define ii, = u, - u. This function belongs to Hd(Q; E) n L”(L2) and is a solution for the equation - ~ xi”Xi(ii,) + a(jE, = -f,(X, U,,Xi(U,)) + E xi*Xi(U) - UOU. i=l
i=l
In a similar way as before we define UE= exp(tii,2Jii, where t = 2C,2 and as usual we put E, = exp(t@). It is easy to see that v, E H,‘(Q; E) and can, therefore,
in
E, i
IX&)l*ti
+ 2t
5
Eciiz i .i n
i=l
i -
R
i=l
be used as a test function. We have
IXi(U,)1*dx + (~0 E,tiif ti ia
Et iil xi(~tJK(U)do If,(x, u.s> K(Ue))IEegc ia 2t
n
a, uE, ii, dx
E, ~ Xi(ii,)Xi(U)ii,2 dx i=l
n
(5.19)
66
GIANAZZA
U
If we recall that Ixi(u.5>Iz
=
U + &)I2 = IX;(U) + X;(U,)l2 12JXi(U)12
Ixi(
+ 2lX;(ii,)l2
we get from (5.19) n R
E, 2 ~X;(Z&))~dx + 2t E,U,2 ~ (X;(ii,)l’dx i=l i=l i .Q 1 5
.i[cl
CO + 2C,
f
lX;(~)1~ E,Izi,( dx + ;
E,i$ dX !a
.i a
1
i=l
+ ~yg
E, t
bM%)i2~
i=l
n + i
E,(2C,)2ii,z i ,e
i
IX;(ii,)12dx
-
E, f !
i=l
n
Xi(%)X; (u) dx
i=l
m -
2t
E, C X;(U,)X;(U)~,2 dx R
Taking
into account
a, uE, ii, dx.
(5.20)
n
i=l
the definition
of t we can rewrite
(5.20) in this way
(5.21) As Ei, + 0 strongly in L2(sZ) and also X;(U,) --t 0 at least weakly in L2(n), it is easy to see that all the terms in the right-hand side converge to zero and the same must happen to the other side. As E, 2 1 we have, therefore, that ii, + 0 strongly in H,‘(Qz; E). Step 5. Passage to the limit. As u, converges extracting a new subsequence) Xi(&)
--f
to u strongly
Xi(U)
in HJ(Oz; E) we have (eventually
a.e. in Q.
As in [6] we claim that: (i) f,(X, u,, xi(Uc)) * .I%, U, xi(u)) a.e. in Q (ii) f8(U,, Xi(U,)) + f(U, Xi(u)) strongly in L’(a). Therefore, going to the limit in each term of equation (5.8), we have that the limit function belongs to Hd(Q; E) fl L”(Q) and satisfies equation (5.5). Remark 5.4. It is interesting that we have proved by providing a function which satisfies
the existence
u
of a Hd(Q; E) fl L”(Q) solution
(5.22) In [6] it is proved
that all limited
solutions
satisfy
(5.22).
SquareH(irmanderoperators
67
6. A CACCIOPPOLI INEQUALITY FOR A NONLINEAR PROBLEM We can now prove a Caccioppoli inequality for our problem: it will basically be for the regularity estimates of the next paragraph. Just like the previous paragraph, we have &2= &R, x0). As we work with H,‘(fiz; E) n L”(a) solutions, we can use exponential cut-off functions, so that we can take into account a quadratic growth in the Xi terms. Otherwise a milder condition should be considered.
PROPOSITION6.1. Consider problem (5.4), (5.5) and suppose that (5.1)-(5.3) u E HJ(nz; E) fl L”(a) be a solution. We have
I’ [
+
bi2 G’&,) dx + 1
B(qr,xo)
I ,B(%,),
hold.
Let
su~~(qr,x& - k/*
Iu - kl*dx + C2ra
B(r,xo)-B(w,xo)
(6.1)
where 0 < q 5 i& with Go E (0, l), 0 < r < R, k E 03, C, and C2 are constants which depend on 4, Il&-~~~9k and the quantities that appear in the nonlinear term estimate.
Proof. Firstly we observe that to prove (6.1) it is enough to prove the same inequality with G$2r,x0j replaced by GXo. We use as a test function
u = (u - k) e”“-kt2cp2((G~)-1)Gj z E B(qr, x0) where G,” is the regularized Green function relative to a” and the point Z, t will be fixed in the following and ~1E C,“(B) with
2 @(t) s (C, _ 1y
A and Co suitable. We have
E Xj(u)Xj((u
- k) et’u-k’2cp2GJ dx +
,i Ri=l
+
n
f(x, u, Xj(u))(u
a,u(u - k) er’u-k’2p2G~ dx 0
- k) et~U-k~Zp2G~ dx = 0.
68
U. GIANAZZA
If we put Et = e tiu-k’2 for short as we did in the previous
T lXi(u)12Et~2G;
cl
(U - k)E, i
+ n
+
z
(U - k)‘E,
dx + 2t
fl i=l
we get
(Xi(U)12~‘GpZdX
i=l
Xi(U)Xi(p*)G,ZdX
a, u(u - k)E,p2G; d_x
+
i=l
a
(U - k)Ef f n
paragraph,
Xi(U)Xi(G~)~*dX
i=l
f(xv u, Xi(U>)(U- ~)E~P*G,Z k
=R
and also
Et f IXi(u)~*~*G;dx + 2t .i n
i=l
12)~ IXi(U)12~2G,ZdX i=l
s
Et ~ Xi((U - k)2~2)Xi(G,“)~
n
n
s
(U - k)E,~2G~ dx + ~
-aok
1
i=l
E~(u -
0
--
F
E,(u -
(u - /c)~E,~~~G,Zdx + ;
+ 010
I
i .n
k)* ~
E, ~ Xi((U - k)*)Xi((P2)GpZ dw -
i! .i n
i=l
S[ + ~ Co
Cl
n
i
a
term, putting
f(xv ~9 Xi(U))(U - k)Etp2G;
dx-
+
D zif,
Et E IXi(U)12p2GpZti. i=l
for short and applying
I
i
n
Cl = b(ll~&~,)
IU - klE~~*G,Zdx
IXi(U)12
i=l
C,lu - kjE,p*G;dx
I
dx
n
Recalling the estimate on the nonlinear the Young inequality we have
5
Xi(~")Xi(G,Z)
i=l
cl
IXi
Square
t = CF/2 and apply the Young
If we choose 1 i sa
Hiirmander
E, ~ IXi(U)12~2GpZdX + ~ i=l +
(u
010
1”
k)2E,cp2G;dx
-
(U - k)E,lo’G,Zdx
1
+ ;
+ ~
E,(u - k)2 ~ Xi(lo’)Xi(G~)dx in
(u - k)2E,p2G;
\u - k~E,q~~G;dx
+ Co .i n E,p2G,Z dx
n
1
E,(u - k)’ ~ Xi(~2)Xi(G,Z)dx
2 ia +
i=l
dx + C, k2
0 f-
IX,(~)(~cp~GpZd_x
E1 ~ Xi((U - k)‘~‘)Xi(GpZ)dX i=l
E, ~ Xi((U - k)2)Xi(~2)G~dX i=l
I ?
we find
i=l
iR
2
once more,
E,(u - k12 f
sD
I -aok
--
inequality
.r0
D
.r
operators
Iu
co
- ~
Et E Xi((U - k)2)Xi(p2)G,” dx
i=l
i=l
n
k[Etq2G;dx.
-
fl
Therefore, 1 2 1n
Ed ~ IXi(U)12~2GpZdx + ~ i=I 3 + 4ao 1
(U - k)‘Etp2G;dx
2
i 0
+
T
+ ;
Et i i ,a
Xi(p’)Xi(G,Z)dx
-
Xi((U - k)2~2)Xi(G,“)dw
i=l
- i
Et ~ Xi((U - k)2)Xi(~2)GpZ~
i=l
(u
lXi(U)12v2GpZdx
i=l
i cl
E~(u - k)2 f
I-
E~(u - k)2 t n
sn
k)2Etp2G;dx
+ C2k2
i=l
Etp2G;dx
+ C,
.rn
in
E,cp’G; dx. in
Finally,
sn
Et E [Xi(U)(2p2Gidx i=l + c:
Et f In
5
+
R c4
IXi(U)I’(U - k)2p2G,ZdX +
Et i .i n
i=l
Xi(Cp’)Xi(G,Z) ti
i=l
Et (P’G; dx cl
d.x
sn
E~(u - k)2 f
I
(u - k)2Etp2G;
+ (~0
-
Et i sn
i=l
Xi((U - k)2p2)Xi(GpZ) ti
i=l
Xi((U - k)2)Xi(p2)GidX
U. GIANAZZA
70
having
put c,
If we take into account
and taking
= 2(C#
that u E L”(Q),
into account
-t C,).
putting
CS = inf,(u
- k\
C, = mint1
+ CfCi,
01~)
that 1 5 Et 5 C,
where C, stands
G
for its upper
bound,
we conclude
) i( itlIXi(U>12+ N2‘
p2G; dx +
(u -
R
r~
Xi((U - k)2~2)Xi(G,") ~
, I2i=l
s I c,
(U - k)’ u cl
+ c8
~ Xi(~2)Xi(G~) i=l
m c Xi((u
dx -
- @2)Xi((02)G,” dx
! Cli=l
1
p2G; dx. Ia
If we recall that (U - /Q2 > ;
- !$
we find
c9
f i=l
i( n
I&(u)12
I c,
+
u’)v’C~
(U - k)’ f irtJfi
CIX
+
in
Xi((U - k)2~2)Xi(G,“) dx
Fl
Xi(p2)Xi(GpZ) dx -
i=l
f
Xi((U - k)*)Xi(~“)G,” dx
0 i=l
1
I\ +
cp”G; dx
Cl0
n
where C9 = C6/4 and Cl0 takes into account ’
c9
the estimate
given above.
Then
m
.i( a
I
iCl
2
IxiCu)I
+
u*
P’G;
CLY +
,Bc;,
zj,
B(p,~)
(u
-
W2
CLX
)
c,
n
(U
s
-
‘:’ + kj21p’It Ixi(G,Z)I’(~33 i=l
&)k+
Clo[Qr2G;~.
71
Square Hiirmander operators
Passing
to the limit as p --t O+ we obtain
c9
n
jt,
+
Ixi(u)12
u2
>
i(
I c,
.in
p2Gz dx + (u - /c)~(z)
$$
; Ixi(Gz)/2 i=l
(u - k)‘Iv’I
Choosing A and C, such that: (i) (x: GZ 1 l/A) > B(2qr, z) (ii) (x: Gz 2 l/C,,A) c B((l - q)r,z) (possibly for small r) and taking into account derivatives, we obtain
s(
+ &++
the estimates
j, lxitu)12+ u2 GZ dx
B(v, z)
>
C,,sflGz~.
on the Green
function
and its
+ (u - /c)~(z)
>
s &j
~/ B(r,z)-B(qr,z)(’ - k)2 dx + c12ra
with 01 suitable (we use the duplication property of measures of intrinsic only on the dimension of the space and the Hormander constant k). Finally, taking the supremum for z E B(qr, x0) we get
balls and LYdepends
m C
i=l
Ixi(U)I’
+
s
u2
>
G”‘~-x
(24-
+
SUPB(qr,x,,)(U
-
k12
Jq2dx + C12P.
B(r,q+BWvW
7. REGULARITY
FOR THE
WEAK
We can now come to the estimate
SOLUTION
for the solution
OF A NONLINEAR
to our nonlinear
EQUATION
problem
in Q = B(R, x0).
THEOREM
7.1. Let u be a Hd(Q; E) II L”(Q) solution of (5.5) and suppose that (5.1)-(5.3) are satisfied. Then there exists an exponent p > 2 such that u E yt;p(sZ; E) and the following estimate holds for B, c BZr c Sz 1
(f (
p’2
lU12+ ~ IXi(U)12 )
B,
i=l
&)l”
If we define
we also have the following
theorem.
s
KI[(,~Br(lulz + j,
lxi(u)12) &y”]
+
K2*
U.
12
THEOREM 7.2. Under
the same hypothesis V(r) I
GIANAZZA
as above, r
0
Cl
-
RCI
with r 5 (q/2)Ro < R, 5 (q/2)R,
y
u is locally
V(R,) + CzR,
q as in the previous
Holder
continuous
and
6
paragraph
and y, 6 E (0, 1).
Proof of theorem 7.1. It is easy to see that the metric topology induced by the intrinsic distance defined in Section 1 is equivalent to the natural Euclidean topology of Gl”. Moreover, the Lebesgue measure satisfies the doubling condition for our intrinsic balls. We can then see that the space a” wth the intrinsic distance acquires the structure of homogeneous space, as discussed in Section 2. Let us now come back to our Caccioppoli inequality
s( BG7r,xo)
i
f i=l
k(#
+
Gg2r,xoj
u2
b
+
su~s(~‘.x,,)(u
- Q2
>
,B&,l,
(u -
k)‘dx + C2ra.
~(~,~o)-~(qr,xO)
If we neglect the second term on the left-hand Green function, we have
side and take into account
the estimate
for the
n
i B(V,%)
(
i~llxioi2+~2)~r~~xii,i,,smi,r,,(~~k)2dX+C~’B(~~~‘r~.
If we recall how ra was calculated
we find
Now putting q = a, k the average of u on B(qr, x,), using the Sobolev inequality q* such that l/q* + l/s = 1 where s is the Sobolev exponent, we have
If C, = 0 we could directly apply theorem 4.1 and we have the result. the case, we can reason as in [2] and again we obtain our inequality.
Otherwise,
and assuming
as is usually
Proof of theorem 7.2. We use as a test function u = (u _ 0) er(l~-“‘*G;rp where ii is the average of u on B(qr, x0), p < r, Gj = G&f,,zj,p is the regularized Green function of B(2r, z), rp is the potential of B(2r, z) with respect to B(tr, z) and I is as in proposition 6.1.
Square
Hormander
operators
73
We can reason as in theorem 1 of [lo], to which we refer. The initial difference due to the presence of the exponential term can be adjusted as in the proof of proposition 6.1, thanks to the fact that u E H~(Qz; E) n L”(S2). REFERENCES of the partial derivatives of a quasiconformal 1. GEHRINC F. W., The Lp integrability 265-271 (1973). 2. GIAQUINTA M. & MODICA G., Regularity results for some classes of higher order
mapping, nonlinear
Acta Math. 130, elliptic
systems,
J.
Math. 311, 145-169 (1979). 3. MEYERS N. G., An Lp estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scu. norm. sup. III XVII, 189-205 (1963). 4. COIFMAN R. R. & WEISS G., Analyse harmonique non-commutative sur certaines espaces homogenes, Lecture Notes in Mathematics, Vol. 242. Springer, Berlin (1971). principle for Dirichlet forms on discontinuous media (preprint). 5. BIROLI M. & Mosco U., A Saint-Venant 6. BOCCARDO L., MURAT F. & PUEL J. P., Existence de solutions faibles pour des equations elliptique quasi-lineares a croissance quadratique, Nonlinear Partial Differential Equations and their Applications, College de France Seminar, (Edited by H. BREZIS and J. L. LIONS), Vol. IV, Research Notes in Mathematics. Pitman, London (1983). I. JERISON D. & SANCHEZ-CALLE A., Subelliptic second order differential operators, in Lecture Notes in Mathematics, Vol. 1277, pp. 46-77. Springer, Berlin (1987). 8. NAGEL A., STEIN E. M. & WAINGER S., Balls and metrics defined by vector fields: basic properties, ActaMath. 137,
247-320 (1986). 9. SANCHEZ-CALLE A., Fundamental
solutions
and geometry
of the sum of squares
of vector fields, Invent. Math. 78,
143-160 (1984). 10. BIROLI M., Local
properties
of solutions
to equations
involving
Potential Theory, Nagoya, August (1990). 11. STEIN E. M., Singular Integrals and Differentiability Princeton (1970). 12. GIANAZZA U., The Lp integrability
on homogeneous
square
Hormander
Properties of Functions.
spaces,
operators, Princeton
Proc. Int. Conf. University
Rend. Ist. Lomb. A 126, 82-92 (1992).
Press,