Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 191, pp. 1–7.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
CORDES NONLINEAR OPERATORS IN CARNOT GROUPS
GIUSEPPE DI FAZIO, MARIA STELLA FANCIULLO
Abstract. Our aim is to obtain Lp estimates for the second-order horizontal
derivatives of the solutions for a nondivergence form nonlinear equation in
Carnot groups.
1. Introduction
The W 2,p estimates for elliptic differential equations and systems is a very interesting problem and many Authors have given several contributions to this problem
from several different points of view (see [5, 4, 7]) using different approaches. There
are essentially two main approaches to the problem: assuming on the coefficients of
the equation the Cordes condition or the VMO condition. The first one consists of
a geometric condition on the eigenvalues of linear operators. Cordes condition was
introduced in [6] and studied by many authors (in the cases of nonlinear nonvariational equations and systems we quote [3, 4]). The second technique consists in
assuming the coefficients of the operator to be in VMO-type classes (see [5, 7, 9, 10],
and for more general setting see [1, 2]).
Here we obtain W 2,p estimates for the following nonlinear nondivergence form
equation
a(x, u, Xu, X 2 u) = f,
where X = (X1 , X2 , . . . , Xl ) is a system of Hörmander’s vector fields on a Carnot
group, and we assume a condition that in the particular case of a linear equation
gives back the Cordes condition (see [8] for the case of the Heisenberg group).
Namely, we show that there exists a critical exponent p0 > 2 such that if the
datum f belongs to Lp , with 2 < p < p0 , then the second derivatives X 2 u of the
solutions u have the same integrability as f .
2. Preliminaries
Let G be a finite-dimensional, stratified, nilpotent Lie algebra. We assume G =
⊕si=1 Vi , where [Vi , Vj ] ⊂ Vi+j for i + j ≤ s and [Vi , Vj ] = 0 for i + j > s. Let
X1 , . . . , Xl be a basis for V1 and suppose that X1 , . . . , Xl generate G as a Lie
algebra. Then for 2 ≤ j ≤ s we choose a basis {Xij }, 1 ≤ i ≤ kj , for Vj consisting of
commutators of length j. We set Xi1 = Xi , i = 1, . . . , l and k1 = l, and we call Xi1 a
commutator of length 1. If G is the simply connected Lie group associated to G then
2010 Mathematics Subject Classification. 35H20.
Key words and phrases. Cordes condition; Carnot groups; nonlinear equations.
c
2015
Texas State University - San Marcos.
Submitted April 8, 2015. Published July 20, 2015.
1
2
G. DI FAZIO, M. S. FANCIULLO
EJDE-2015/191
G is called Carnot group. It is well known that the exponential mapping is a global
n
diffeomorphism from G to P
G and then for any g ∈ G there
P exists x = (xij ) ∈ R ,
s
1 ≤ i ≤ kj , 1 ≤ j ≤ s, n = j=1 kj , such that g = exp( xij Xij ).
We now recall the definition of polynomials on the Carnot group G given by
Folland and Stein in [15].
A function P on G is said to be a polynomial on G if P ◦ exp is a polynomial on
the Lie algebra G.
Let X1 , X2 , . . . , Xn be a basis of G and ξ1 , ξ2 , . . . , ξn be the dual basis for G ∗ we
set ηi = ξi ◦ exp−1 . Each ηi is a polynomial on G, and η1 , η2 , . . . , ηn form a system
of global coordinates on G. Then every polynomial on G can be written uniquely
as
X
P (x) =
aI η I (x), η I = η1i1 · · · ηnin , aI ∈ R
I
I
where all but finitely
Pnmany of the coefficients aI vanish. Clearly η is homogeneous
of degree d(I) = j=1 ij dj , where dj is the length of Xj as a commutator. We
define the homogeneous degree of the polynomial P as max{d(I) : aI 6= 0}.
Here we recall the definition of the Carnot-Carathéodory metric. An absolutely
continuous curve γ : [0, τ ] → Rn is called subunitary if there exists a measurable
Pl
function c = (c1 , c2 , . . . , cl ) : [0, τ ] → Rl such that γ 0 (t) =
j=1 cj (t)Xj (γ(t)),
for a.e. t ∈ [0, τ ], and kck∞ ≤ 1. The Carnot-Carathéodory distance d(x0 , x00 ) is
defined as the infimum of those τ > 0 for which there exists a subunitary curve
γ : [0, τ ] → Rn with γ(0) = x0 and γ(τ ) = x00 .
We set Br (x0 ) = {x ∈ Rn : d(x, x0 ) < r}. When it is clear from the setting we
will omit x0 or r. It is well known that the Carnot-Carathéodory balls satisfy a
doubling condition, that is
|B2r (x0 )| ≤ 2Q |Br (x0 )|
for all r > 0 and x0 ∈ Rn . The constant Q is the homogeneous dimension of G .
We define the intrinsic Sobolev spaces for a bounded domain Ω in Rn . Let k ∈ N
and p ≥ 1 we set
W k,p (Ω) = u : Ω → R : u, Xi1 . . . Xij u ∈ Lp (Ω), 1 ≤ j ≤ k
endowed with the norm
kukW k,p (Ω) = kuk
Lp (Ω)
+
k X
l
X
kXi1 Xi2 . . . Xih ukLp (Ω) .
h=1 ij =1
We define W0k,p (Ω) as the closure of C0∞ (Ω) in W k,p (Ω) with respect to the above
norm.
For I = (i1 , i2 , . . . , in ) we denote the differential operator X1i1 X2i2 . . . Xnin by
I
X and d(I) the homogeneous degree of X I . We denote by Xu the gradient of u
(X1 u, X2 u, . . . , Xl ) and by X 2 u the hessian matrix {Xij u}i,j=1,...,l .
In [17] the existence of approximation polynomials of Sobolev functions in Carnot
groups and related Poincaré-type inequalities have been obtained. Here we state
some results [17, Theorems 2.7 and 5.1], that we will use in the sequel.
Theorem 2.1. Let k be a positive integer and u a function
in W k,1 (Ω). Then there
R
I
exists a polynomial P of degree less than k such that Ω X (u − P )dx = 0 for any
0 ≤ d(I) < k.
EJDE-2015/191
CORDES NONLINEAR OPERATORS
3
2Q
Choosing first q10 = p = 2 and q21 = 2, p = Q+2
in [17, Theorem 5.1], we obtain
the following two inequalities that we collect in the same statement.
2Q
Theorem 2.2. Let Br be a ball of Rn and u ∈ W 2, Q+2 (B r ). Then there exists a
polynomial of degree ≤ 1 such that
Z
Z
2
2
|u − P | dx ≤ cr
|X(u − P )|2 dx,
(2.1)
Br
Z
Br
Z
|X(u − P )|2 dx ≤ c
Br
2Q
|X 2 u| Q+2 dx
Q+2
Q
,
(2.2)
Br
where the constant c is independent of Br and u. (The polynomial P is the same
as in Theorem 2.1).
The following Theorem has been proved in [14], (for different cases see [13, 11,
12]).
Theorem 2.3. There exists a constant CG ≥ 1 such that for every u ∈ W02,2 (Ω)
the following inequality holds
Z
Z
|X 2 u|2 dx ≤ CG
|∆u|2 dx ,
(2.3)
Ω
where ∆u =
Pl
i=1
Ω
Xi Xi u.
3. Caccioppoli-type inequality and W 2,p estimates
2
Let Ω be a bounded domain in Rn . Let a(x, u, p, m) : Ω × R × Rl × Rl → R be
a Carathéodory function satisfying the condition
(A) there exist three positive constants, α, γ and δ such that CG γ + δ < 1, for
all M = {mij }i,j=1,...,l ∈ Rl × Rl , u ∈ R, p ∈ Rl ,
l
l
2
X
2
X
mii ,
mii − α[a(x, u, p, m)] ≤ γ|M |2 + δ a.e. x ∈ Ω .
i=1
i=1
We consider the nonlinear nonvariational elliptic equation
a(x, u, X, X 2 u) = f ,
(3.1)
where f ∈ L2 (Ω).
Definition 3.1. A function u ∈ W 2,2 (Ω) is called a solution of (3.1) if u satisfies
(3.1) for a.e. x in Ω.
Remark 3.2. In the case of linear equation, i.e.
l
X
aij (x)Xi Xj u(x) = f
i,j=1
condition (A) is stronger than the following Cordes condition (see [6, 18] for a
comparison between condition (A) and Cordes condition in Euclidean setting).
4
G. DI FAZIO, M. S. FANCIULLO
EJDE-2015/191
Definition 3.3. The linear operator L ≡ aij (x)Xi Xj satisfies the Cordes condition K,σ,θ if there exist ∈ (0, 1], σ > 0 and θ > 0 such that for a.e. x ∈ Ω,
Pl
i=1 aii (x) > 0 and
0<
l
l
X
2
X
1
1
θ2
≤
aii (x) ≤
.
a2ij (x) ≤
σ i,j=1
l − 1 + i=1
l−1+
Now we prove a Caccioppoli type inequality for solutions of (3.1).
Theorem 3.4. Let condition (A) hold true and f ∈ L2 (Ω). Then for any u ∈
W 2,2 (Ω) solution of (3.1), for any r >R0 such that B2r b Ω, there exists a polynomial P of degree less than 2 such that B2r X I (u − P )dx = 0 for any 0 ≤ d(I) < 2,
and
Z
Z
Z
f 2 dx .
(3.2)
|X(u − P )|2 dx + c
|X 2 u|2 dx ≤ cr−2
Br
B2r
B2r
Proof. Let B2r b Ω.R From Theorem 2.1 there exists a polynomial P of degree less
than two such that B2r X I (u − P )dx = 0, for I with d(I) < 2.
Let η be a C0∞ (Rn ) with the properties 0 ≤ η ≤ 1, η = 1 in Br , η = 0 in Rn \ B2r
and |Xη| ≤ rc .
If we set U = η(u − P ) ∈ W02,2 (B2r ), since X 2 P = 0 (see the proof of [17,
Theorem 2.7]), we have X 2 (u − P ) = X 2 u and X 2 U = X 2 u in Br . We have
η∆u = η(∆u − αa(x, u, Xu, X 2 u)) + ηαf ,
which implies
|η∆u| ≤ η|∆u − αa(x, u, Xu, X 2 u)| + |ηαf |
≤ η[γ|X 2 u|2 + δ|∆u|2 ]1/2 + ηα|f |.
Note that
∆U = η∆u + A(u − P ),
(3.3)
ηX 2 u = X 2 U − B(u − P ),
(3.4)
where
A(u − P ) = (u − P )∆η + 2
X
Xi ηXi (u − P )
(3.5)
and
B(u − P ) = {(u − P )Xi Xj η + Xi ηXj (u − P ) + Xj ηXi (u − P )}ij .
Then for x ∈ B2r ,
|∆U| ≤ |η∆u| + |A(u − P )| ≤ η(γ|X 2 u|2 + δ|∆u|2 )1/2 + ηα|f | + |A(u − P )|,
from which it follows that for all > 0,
|∆U|2 ≤ η 2 (γ|X 2 u|2 + δ|∆u|2 ) + (ηα|f | + |A(u − P )|)2
+ 2η(γ|X 2 u|2 + δ|∆u|2 )1/2 (ηα|f | + |A(u − P )|)
≤ η 2 (γ|X 2 u|2 + δ|∆u|2 ) + (ηα|f | + |A(u − P )|)2
1
+ η 2 (γ|X 2 u|2 + δ|∆u|2 ) + (ηα|f | + |A(u − P )|)2
1
2
2 2
2
= (1 + )η (γ|X u| + δ|∆u| ) + 1 + (ηα|f | + |A(u − P )|)2
EJDE-2015/191
CORDES NONLINEAR OPERATORS
1 2 2 2
(η α |f | + |A(u − P )|2 ).
≤ (1 + )η 2 (γ|X 2 u|2 + δ|∆u|2 ) + 2 1 +
Then from (3.3) and (3.4),
|∆U|2 ≤ (1 + )γ|X 2 (U) − B(u − P )|2 + (1 + )δ|∆U − A(u − P )|2
1
1
+ 2 1 + η 2 α2 |f |2 + 2 1 + |A(u − P )|2
≤ (1 + )γ(|X 2 (U)|2 + |B(u − P )|2 + 2|X 2 (U)||B(u − P )|)
+ (1 + )δ(|∆U|2 + |A(u − P )|2 + 2|∆U||A(u − P )|)
1
1
+ 2 1 + η 2 α2 |f |2 + 2 1 + |A(u − P )|2
1
2
2
≤ (1 + )γ (1 + )|X (U)| + 1 + |B(u − P )|2
1
2
+ (1 + )δ (1 + )|∆U| + 1 + |A(u − P )|2
1 2 2 2
1
+ 2 1 + η α |f | + 2 1 + |A(u − P )|2
≤ (1 + )2 γ|X 2 (U)|2 + (1 + )2 δ|∆U|2
+ c(, α, γ, δ)[|A(u − P )|2 + |B(u − P )|2 + |f |2 ].
We integrate on B2r and apply (2.3) in Theorem 2.3 to obtain
Z
Z
Z
|∆U|2 dx ≤ (1 + )2 γ
|X 2 (U)|2 dx + (1 + )2 δ
|∆U|2 dx
B2r
B2r
B2r
Z
2
2
+c
(|f | + |A(u − P )| + |B(u − P )|2 )dx
B2r
Z
≤ (1 + )2 (γCG + δ)
|∆U|2 dx
B2r
Z
2
+c
(|f | + |A(u − P )|2 + |B(u − P )|2 )dx.
B2r
It follows that
Z
[1 − (1 + )2 (γCG + δ)]
|∆U|2 dx
B2r
Z
≤c
(|f |2 + |A(u − P )|2 + |B(u − P )|2 )dx,
B2r
and then
Z
Z
|∆U|2 dx ≤ c
B2r
(|f |2 + |A(u − P )|2 + |B(u − P )|2 )dx.
B2r
Finally, we get that
Z
Z
2 2
|X u| dx ≤
Br
2
2
Z
|X U| dx ≤ CG
B2r
Z
≤c
B2r
|∆U|2 dx
B2r
(|f |2 + |A(u − P )|2 + |B(u − P )|2 )dx.
5
6
G. DI FAZIO, M. S. FANCIULLO
EJDE-2015/191
Now we observe that from (3.5) and Poincaré inequality (2.1) we obtain
Z
|A(u − P )|2 dx
B2r
Z
Z
X
2
2
≤c
|∆η| |u − P | dx + c
|Xi η|2 |Xi (u − P )|2 dx
B2r
B2r
Z
Z
n
o
|u − P |2 dx +
|X(u − P )|2 dx
≤ cr−2 r−2
B2r
B2r
Z
−2
2
≤ cr
|X(u − P )| dx.
B2r
In the same way, we find that
Z
Z
2
−2
|B(u − P )| dx ≤ cr
B2r
|X(u − P )|2 dx,
B2r
from which the Caccioppoli type inequality follows.
Next we state [19, Theorem 3.3] which is a generalization of the Gehring lemma
[16].
Lemma 3.5. Let U and G be non-negative functions in Ω such that
U ∈ Ltloc (Ω),
G ∈ Lsloc (Ω),
1 < t < s.
If there exists c > 1 such that for every B2r b Ω, r < 1,
Z
Z
t
Z
t
− U dx ≤ c − U dx + c − Gt dx,
Br
B2r
B2r
Lploc (Ω),
then there exists ∈ (0, s − t] such that U ∈
for all p ∈ [t, t + ) and, for
every B2r b Ω, with r < 1, we have
Z
1/p
hZ
1/t Z
1/p i
p
t
− U dx
≤ K − U dx
+ − Gp dx
,
Br
B2r
B2r
where the constant K depends on c, t and Q.
Our main Theorem is now an easy consequence of Caccioppoli-type inequality
(3.2) and Lemma 3.5.
Theorem 3.6. Let u ∈ W 2,2 (Ω) be a solution of (3.1) then there exists p0 > 2
2,p
such that, if f ∈ Lp (Ω), with 2 ≤ p < p0 , then u ∈ Wloc
(Ω) and for all B2r ⊂⊂ Ω
we have
Z
1/p
Z
1/2 Z
1/p
−Br |X 2 u|p dx
≤ c −B2r |X 2 u|2 dx
+ −B2r |f |p dx
.
Proof. Let B2r ⊂⊂ Ω, from the Caccioppoli-type inequality (3.2) and Poincaré
inequality (2.2) it follows
Z
Z
Q+2
Z
2Q
Q
+
f 2 dx ,
|X 2 u|2 dx ≤ cr−2
|X 2 u| Q+2 dx
Br
B2r
B2r
from which
Z
Z
Z
Q+2
2Q
Q
2 2
2 Q+2
−Br |X u| dx ≤ c −B2r |X u|
dx
+ −B2r f 2 dx.
(3.6)
EJDE-2015/191
CORDES NONLINEAR OPERATORS
2Q
Now we can apply Lemma 3.5 with U = |X 2 u| Q+2 , t =
s=
p(Q+2)
2Q ,
to obtain the thesis.
Q+2
Q ,
7
2Q
G = |f | Q+2 and
References
[1] M. Bramanti, L. Brandolini; Lp -estimates for nonvariational hypoelliptic operators with
VMO coefficients. Trans. Amer. Math. Soc., 352, n. 2 (2000), 781–822.
[2] M. Bramanti, M. S. Fanciullo; BM O estimates for nonvariational operators with discontinuous coefficients structured on Hormander’s vector fields on Carnot groups, Advances in Diff.
Eq., 18, 9-10, (2013), 955-1004.
[3] S. Campanato; Sistemi ellitticiin forma di divergenza. Regolaritá all’interno, Quaderni SNS
di Pisa, (1980).
[4] S. Campanato; L2,λ theory for nonlinear nonvariational differential systems, Rend. Mat.
Appl., 10 (1990), 531-549.
[5] F. Chiarenza, M. Frasca, P. Longo; Interior W 2,p estimates for non divergence elliptic equations with discontinuous coefficients, Ric. di Mat., 60, (1991), 149-168.
[6] H. O. Cordes; Zero order a priori estimates for solutions of elliptic differential equations,
Proceedings of Symposia in Pure Mathematics IV (1961).
[7] G. Di Fazio; Lp estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Un. Mat. Ital. A (7) 10, n.2, (1996) 409–420.
2,p
[8] G. Di Fazio, M. S. Fanciullo; Wloc
estimates for Cordes nonlinear operators in the Heisenberg
group, J. of Math. Analysis and Appl., 411, 947-052, (2014).
[9] G. Di Fazio, M. S. Fanciullo, P. Zamboni; Interior Lp estimates for degenerate elliptic equations in divergence form with VMO coefficients, Differential and Integral Equations, 25, 7-8
(2012), 619-628.
[10] G. Di Fazio, M. S. Fanciullo, P. Zamboni; Lp estimates for degenerate elliptic systems with
VMO coefficients , St. Petersburg Math. J., 25, 6 (2014).
[11] G. Di Fazio, A. Domokos, M. S. Fanciullo, J. J. Manfredi; Subelliptic Cordes estimates in the
Grushin plane, Manuscripta Mathematica, 120, 419-433 (2006).
[12] A. Domokos, M. S. Fanciullo; On the best constant for the Friedrichs-Knapp-Stein inequality
in free nilpotent Lie groups of step two and applications to subelliptic PDE, J. of Geometric
Analysis, 17,2, (2007), 245-252.
[13] A. Domokos, J. J. Manfredi; Subelliptic Cordes estimates, Proc. AMS, 133, no. 4, 1047-1056
(2005).
[14] G. B. Folland; Applications of analysis on nilpotent groups to partial differential equations,
Bull. Amer. Math. Soc. 83 (1977) 912-930.
[15] G. B. Folland, E. M. Stein; Hardy spaces on homogeneous groups, Mathematical Notes 28,
Princeton Univ. Press and Univ. of Tokyo Press 1982.
[16] M. Giaquinta; Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Math. Studies No. 105, Princeton Univ. Press, Princeton 1983.
[17] G. Lu; Polynomials, Higher Order Sobolev Extension Theorems and Interpolation Inequalities
on Weighted Folland-Stein Spaces on Stratified group, Acta Mathematica Sinica 16, n. 3,
(2000), 405-444.
[18] A. Tarsia; On Cordes and Campanato conditions, Arch. of Inequalities and Applications, 2,
(2004), 25-40.
[19] A. Zatorska-Goldstein; Very weak solutions of nonlinear subelliptic equations, Annales Acc.
Scientiarum Fennicae, 30, (2005), 407-436.
Giuseppe Di Fazio
Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6,
95125, Catania, Italy
E-mail address: difazio@dmi.unict.it
Maria Stella Fanciullo
Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6,
95125, Catania, Italy
E-mail address: fanciullo@dmi.unict.it