In this section we will prove by the modified A-harmonic approximation technique a Morrey type estimate for the subelliptic system
$$ {X^{\ast }} \bigl( {A(x)Xu} \bigr) = {X^{\ast }} \bigl( {A(x)X\psi } \bigr) . $$
(3.1)
Let us first recall that a function \(h \in S_{X}^{1,2}(\Omega , \mathbb{R}^{N})\) is called A-harmonic for \(A \in {\mathrm{Bil}}(\mathbb{R}^{mN})\) if h satisfies
$$ \int _{\Omega }{A(Xh,X\varphi )\,dx} = 0,\quad \forall \varphi \in C_{0}^{1}\bigl(\Omega ,\mathbb{R}^{N}\bigr). $$
We cite the A-harmonic approximation lemma for vector fields as follows ([24, 31]).
Lemma 3.1
Consider fixed positive
λ
and Λ, and
\(m,N\in \mathbb{N}\)
with
\(m\geq 2\). Then for any given
\(\varepsilon >0\)
there exists
\(\delta =\delta (m,N,\lambda ,\Lambda ,\varepsilon )\)
with the following property: for any
\(A\in {\mathrm{{Bil}}}(\mathbb{R}^{mN})\)
satisfying
$$ A(\xi ,\xi ) \geq \lambda \vert \xi { \vert ^{2}}, \quad \textit{for all } \xi \in \mathbb{R}^{mN}, $$
(3.2)
and
$$ A(\xi ,\tilde{\xi })\leq \Lambda \vert \xi \vert \vert \tilde{\xi } \vert ,\quad \textit{for all } \xi ,\tilde{\xi }\in\mathbb{R}^{mN}, $$
(3.3)
for any
\(g\in S_{X}^{1,2}({B_{\rho }}(x_{0}),\mathbb{R}^{N})\) (for some
\(\rho >0\), \({x_{0}}\in \mathbb{R}^{n}\)) satisfying
(3.4)
and
(3.5)
there exists an
A-harmonic function
\(h \in S_{X}^{1,2}(B_{\rho }(x_{0}),\mathbb{R}^{N})\)
such that
Similarly to [32] and [33], we can prove the following modification of the A-harmonic approximation lemma.
Lemma 3.2
Let
\(0 < \lambda <\Lambda \)
and
\(m\in \mathbb{N}\)
with
\(m \geq 2\)
be fixed. Then, for any
\(\varepsilon >0\), there exists a constant
\(k= k(m,N,\lambda ,\Lambda ,\varepsilon )\)
such that the following holds: for any
\(A \in {\mathrm{{Bil}}}(\mathbb{R}^{mN})\)
satisfying conditions (3.2), (3.3) and for any
\(u \in S_{X}^{1,2}(B_{\rho }(x_{0}),\mathbb{R}^{N})\), there exists an
A-harmonic function
\(h \in S_{X}^{1,2}(B_{\rho }(x_{0}),\mathbb{R}^{N})\)
such that
$$ \int _{B_{\rho }(x_{0})} \vert Xh \vert ^{2}\,dx \leq \int _{B_{\rho }(x_{0})}\vert Xu \vert ^{2}\,dx $$
(3.6)
and, moreover, there exists
\(\varphi \in C_{0}^{\infty }(B_{\rho }(x_{0}),\mathbb{R}^{N})\)
such that
$$ \Vert X\varphi \Vert _{L^{\infty }(B_{\rho }(x_{0}),\mathbb{R}^{N})} \leq \frac{1}{\rho } $$
(3.7)
and
$$\begin{aligned} &\int _{B_{\rho }(x_{0})} {\vert u - h \vert ^{2}}\,dx \\ &\quad \leq \varepsilon {\rho ^{2}} \int _{B_{\rho }(x_{0})} \vert Xu \vert ^{2}\,dx + k(\varepsilon ) \biggl[ \frac{\rho ^{4}}{\vert B_{\rho }(x_{0}) \vert } \biggl( \int _{B_{\rho }(x_{0})} A(Xu,X\varphi )\,dx \biggr) ^{2} \biggr] . \end{aligned}$$
(3.8)
Proof
For any given \(\varepsilon > 0\) and \(u \in S_{X}^{1,2}({B_{\rho }}({x_{0}}),\mathbb{R}^{N})\), we take \(\delta (\varepsilon )\) as in the above Lemma 3.1 and set
Then (3.4) holds. Assume that for g the inequality (3.5) is true. From Lemma 3.1, there is an A-harmonic function w satisfying
and thus the function satisfies (3.6). Moreover, we have
which implies
(3.9)
If, vice versa, there is a nonconstant function \(\tilde{\varphi }\in C_{0}^{\infty }(B_{\rho }(x_{0}), \mathbb{R}^{N})\) such that
Setting \(\varphi = \frac{\tilde{\varphi }}{\rho \sup _{B_{\rho }(x_{0})} \vert X\tilde{\varphi } \vert }\), it follows that
We now take \(h = {u_{\rho }}\). Using the Poincaré inequality and the fact that we deduce
(3.10)
Combining (3.9) and (3.10) and taking \(k(\varepsilon ) = \frac{c}{\delta ^{2} (\varepsilon ) }\) complete the proof. □
Now we are in a position to establish the Morrey type estimate for gradient of weak solution to (3.1) based on Lemma 3.2.
Lemma 3.3
Suppose that
\(A(x)\)
satisfies (H1) and
\(u \in S_{X,\mathrm{loc}}^{1,2}(\Omega , \mathbb{R}^{N})\)
is a weak solution to the system (3.1), i.e.,
$$ \int _{\Omega }{A(x)} Xu \cdot X\varphi\,dx = \int _{\Omega }{A(x)} X\psi \cdot X\varphi \,dx,\quad \forall \varphi \in C_{0}^{\infty }\bigl(\Omega , \mathbb{R}^{N}\bigr). $$
Then for any
\({x_{0}} \in \Omega \)
there exists a constant
\(c > 0\)
such that, for all
\(B_{\rho }(x_{0}) \subset B_{R} (x_{0}) \subset \Omega \), \(R < {R_{d}}\),
$$ \int _{ B_{\rho }(x_{0})} \vert Xu \vert ^{2}\,dx \leq c \biggl[ \biggl( \frac{\rho }{R} \biggr) ^{Q} + \varepsilon + {\eta _{R}}(A) \biggr] \int _{ B_{R} (x_{0})} \vert Xu \vert ^{2}\,dx+ c \int _{ B_{R} (x_{0}) } \vert X\psi \vert ^{2}\,dx. $$
(3.11)
Proof
For fixed \({x_{0}}\in \Omega \) and \(0 < R < {R_{d}}\), denote \({B_{R}}: = {B_{R}}({x_{0}})\). Let η be a cut-off function on \({B_{R}}\) relative to \({B_{\rho }}\), i.e. \(\eta \in C_{0}^{\infty }(B_{R}, \mathbb{R}^{N})\) and satisfies
$$ 0 \leq \eta (x) \leq 1, \qquad \eta (x) = 1 \quad \text{in } {B_{\rho }}, \qquad \bigl\vert X\eta (x) \bigr\vert \leq \frac{c}{R-\rho }. $$
Taking the function \(\varphi ={\eta ^{2}}(u-u_{R})\) as a test function, it follows that
$$\begin{aligned} &\int _{B_{R}} {\eta ^{2}A(x)Xu \cdot Xu\,dx} \\ &\quad = - 2 \int _{B_{R}} {A(x)\eta (u - {u_{R}})Xu \cdot X\eta \,dx}+ \int _{B_{R}} \eta ^{2} A(x)X\psi \cdot Xu\,dx \\ &\quad \quad {}+ 2 \int _{B_{R}} {A(x)\eta (u - {u_{R}})X\psi \cdot X\eta \,dx}. \end{aligned}$$
From (H1) and Young’s inequality
$$\begin{aligned} \lambda \int _{B_{R}}\eta ^{2}\vert Xu \vert ^{2}\,dx&\leq 2\Lambda \int _{B_{R}}\vert \eta Xu \vert \vert u - {u_{R}} \vert \vert X\eta \vert \,dx \\ &\quad +\Lambda \int _{B_{R}} \vert \eta X\psi \vert \vert \eta Xu \vert \,dx + 2\Lambda \int _{B_{R}} {\vert \eta X\psi \vert \vert u - {u_{R}} \vert \vert X\eta \vert \,dx} \\ &\leq \varepsilon \int _{B_{R}} \eta ^{2}\vert Xu \vert ^{2}\,dx + \frac{c_{\varepsilon }}{(R-\rho )^{2}} \int _{B_{R}} \vert u - {u_{R}} \vert ^{2}\,dx + {c_{\varepsilon }} \int _{B_{R}} {\vert X\psi \vert ^{2}\,dx}. \end{aligned}$$
Choosing \(\varepsilon <\lambda \), it follows that
$$ \int _{{B_{\rho }}} {\vert Xu{ \vert ^{2}}\,dx} \le \frac{c}{{{{(R - \rho )}^{2}}}} \int _{{B_{R}}} {\vert u - {u_{R}} { \vert ^{2}}\,dx} + c \int _{{B_{R}}} {\vert X\psi { \vert ^{2}}\,dx}. $$
(3.12)
Next we define . By Lemma 3.2, there exists an \({A_{R}}\)-harmonic function \(h \in S_{X}^{1,2}(B_{R}, \mathbb{R}^{N})\) such that (3.6)–(3.8) hold. Moreover, by standard results of the subelliptic system with constant coefficients (see for example [34]), we have
$$ \int _{B_{\rho }} {\vert Xh \vert ^{2}\,dx} \leq c \biggl( \frac{\rho }{R} \biggr) ^{Q} \int _{B_{R}} \vert Xh \vert ^{2}\,dx, \quad \forall 0 < \rho \leq R. $$
Therefore, from (3.12) and (3.6) it follows that for any \(0 < \rho < R/2\)
$$\begin{aligned} &\int _{B_{\rho }} {\vert Xu \vert ^{2}\,dx} \\ &\quad \leq \frac{c}{\rho ^{2}} \int _{B_{2\rho }} {\vert u - {u_{2\rho }} \vert ^{2}\,dx} + c \int _{B_{2\rho }} {\vert X\psi \vert ^{2}\,dx} \\ & \quad \leq \frac{c}{\rho ^{2}} \biggl( \int _{B_{2\rho }} \bigl\vert u-u_{2\rho }-(h- h_{2\rho }) \bigr\vert ^{2}\,dx + \int _{B_{2\rho }} \vert h - h_{2\rho } \vert ^{2}\,dx \biggr) + c \int _{B_{2\rho }} \vert X\psi \vert ^{2}\,dx \\ &\quad \leq \frac{c}{\rho ^{2}} \int _{B_{2\rho }} \vert u - h \vert ^{2}\,dx + c \int _{B_{2\rho }} \vert Xh \vert ^{2}\,dx + c \int _{B_{2\rho }} \vert X\psi \vert ^{2}\,dx \\ &\quad \leq \frac{c}{\rho ^{2}} \int _{B_{2\rho }}\vert u-h \vert ^{2}\,dx+ c \biggl( \frac{\rho }{R} \biggr) ^{Q} \int _{B_{R}}\vert Xh \vert ^{2}\,dx+c \int _{B_{R}}\vert X\psi \vert ^{2}\,dx \\ &\quad \leq \frac{c}{\rho ^{2}} \int _{B_{2\rho }} \vert u - h \vert ^{2}\,dx + c \biggl( \frac{\rho }{R} \biggr) ^{Q} \int _{B_{R}} {\vert Xu \vert ^{2}\,dx} + c \int _{B_{R}} {\vert X\psi \vert ^{2}\,dx}. \end{aligned}$$
(3.13)
For the first term in the right-hand side, we have from Lemma 3.2
$$\begin{aligned} \frac{c}{\rho ^{2}} \int _{B_{2\rho }}\vert u - h \vert ^{2}\,dx & \leq c \varepsilon \int _{B_{2\rho }}\vert Xu \vert ^{2}\,dx + ck( \varepsilon )\frac{\rho ^{2}}{\vert B_{2\rho } \vert } \biggl( \int _{B_{2\rho }} {A_{R}}Xu \cdot X\varphi\,dx \biggr) ^{2} \\ &\leq c\varepsilon \int _{B_{R}} \vert Xu \vert ^{2}\,dx + {c_{\varepsilon }}\frac{\rho ^{2}}{\vert B_{R} \vert } \biggl( \int _{B_{R}} {A_{R}}Xu \cdot X\varphi\,dx \biggr) ^{2}, \end{aligned}$$
(3.14)
where \(\varphi \in C_{0}^{\infty }(B_{2\rho },\mathbb{R}^{N})\) satisfies \(\Vert X\varphi \Vert _{L^{\infty }(B_{2\rho },\mathbb{R}^{N})} \leq \frac{1}{R}<\frac{1}{2\rho }\). Since u is a weak solution to (3.1), it follows that
$$\begin{aligned} \biggl(\int _{B_{R}} {A_{R}}Xu \cdot X\varphi\,dx \biggr)^{2}&= \biggl( \int _{B_{R}} ({A_{R}} - A)Xu \cdot X\varphi\,dx + \int _{B_{R}} AXu \cdot X\varphi\,dx \biggr) ^{2} \\ & \leq 2{ \biggl( { \int _{{B_{R}}} {({A_{R}} - A)Xu \cdot X\varphi \,dx} } \biggr) ^{2}} + 2{ \biggl( { \int _{{B_{R}}} {AXu \cdot X\varphi \,dx} } \biggr) ^{2}} \\ &: = {I_{1}} + {I_{2}}. \end{aligned}$$
From Hölder’s inequality, using (H1), we have
$$\begin{aligned} {I_{1}} &\leq \frac{1}{2\rho ^{2}} \int _{B_{R}} {\vert Xu \vert ^{2}\,dx} \cdot \int _{{B_{R}}} {\vert {A_{R}} - A \vert ^{2}\,dx} \\ & \leq \frac{\Lambda \vert B_{R} \vert }{\rho ^{2}}\frac{1}{\vert B_{R} \vert } \int _{B_{R}} {\vert {A_{R}} - A \vert \,dx} \cdot \int _{B_{R}} {\vert Xu \vert ^{2}\,dx} \\ &\leq \frac{\Lambda \vert B_{R} \vert }{\rho ^{2}}{\eta _{R}}(A) \int _{B_{R}} {\vert Xu \vert ^{2}\,dx} \end{aligned}$$
and
$$ {I_{2}} \leq \frac{\Lambda ^{2}}{2\rho ^{2}} \biggl( \int _{B_{R}} {\vert X\psi \vert \,dx} \biggr) ^{2} \leq \frac{\Lambda ^{2}\vert B_{R} \vert }{2\rho ^{2}} \int _{B_{R}} {\vert X\psi \vert ^{2}\,dx} . $$
Hence
$$ \biggl( \int _{B_{R}}{A_{R}}Xu \cdot X\varphi\,dx \biggr) ^{2} \leq \frac{c\vert {B_{R}} \vert }{\rho ^{2}} \biggl[ \eta _{R} (A) \int _{B_{R}} \vert Xu \vert ^{2}\,dx + \int _{B_{R}} \vert X\psi \vert ^{2}\,dx \biggr] . $$
(3.15)
Combining (3.15), (3.14) and (3.13), we have, for any \(0 < \rho < R/2\),
$$ \int _{B_{\rho }} \vert Xu \vert ^{2}\,dx\leq c \biggl[ \biggl( \frac{\rho }{R} \biggr) ^{Q} + \varepsilon + {\eta _{R}}(A) \biggr] \int _{B_{R}}\vert Xu \vert ^{2}\,dx + c \int _{B_{R}} \vert X\psi \vert ^{2}\,dx . $$
For \(R/2\leq \rho \leq R\), obviously
$$ \int _{B_{\rho }} {\vert Xu \vert ^{2}\,dx} \leq \int _{B_{R}} {\vert Xu \vert ^{2}\,dx} \leq {2^{Q}} { \biggl( \frac{\rho }{R} \biggr) ^{Q}} \int _{B_{R}} {\vert Xu \vert ^{2}\,dx}. $$
A combination of these two cases leads to (3.11) for \(0 < \rho \le R\). □
We end this section with a comparison principle for system (3.1).
Lemma 3.4
Suppose that
\(u,\psi \in S_{X}^{1,2} (B_{R}, \mathbb{R}^{N})\)
satisfy
$$ {X^{\ast }} \bigl( {A(x)Xu} \bigr) = {X^{\ast }} \bigl( {A(x)X\psi } \bigr) $$
where
\(A(x)\)
satisfies (H1). If
\(\psi \leq u\)
on
\(\partial {B_{R}}\), then
\(\psi \leq u\)
a.e. in
\({B_{R}}\).
Proof
For any \(\varphi \in C_{0}^{\infty }(B_{R}, \mathbb{R}^{N})\) we have
$$ \int _{B_{R}} {A(x)Xu \cdot X\varphi \,dx} = \int _{B_{R}}A(x)X\psi \cdot X\varphi \,dx. $$
(3.16)
Set \({u_{+} }= \max \{ u,0\}\). Since \(\psi \le u\) on \(\partial {B_{R}}\), we conclude that (see [40, Lemma 6]) \((\psi -u)_{+} \in S_{X,0}^{1,2}({B_{R}}, \mathbb{R} ^{N})\) and
$$ X{(\psi - u)_{+}}= \textstyle\begin{cases} X(\psi - u),& \psi > u, \\ 0,& \psi \le u. \end{cases} $$
Choosing \(\varphi = {(\psi - u)_{+} }\) in (3.16) gives
$$ \int _{B_{R}} {A(x)X(\psi - u) \cdot X{(\psi - u)}_{+}}\,dx = 0, $$
which implies
$$ \int _{B_{R}\cap \{ \psi > u\} } {A(x)X(\psi - u) \cdot X(\psi - u)\,dx} = 0. $$
From (H1) we have
$$\begin{aligned} \int _{B_{R}} \bigl\vert X{(\psi - u)_{+}} \bigr\vert ^{2}\,dx& = \int _{B_{R} \cap \{ \psi > u\} }\bigl\vert X(\psi - u) \bigr\vert ^{2}\,dx \\ &\leq \frac{1}{\lambda } \int _{B_{R} \cap \{\psi > u\}} A(x)X(\psi - u) \cdot X(\psi - u)\,dx=0. \end{aligned}$$
Thus from Poincaré inequality, we obtain
$$ \int _{B_{R}}\bigl\vert (\psi -u)_{+} \bigr\vert ^{2}\,dx \leq c{R^{2}} \int _{B_{R}}\bigl\vert X(\psi - u)_{+} \bigr\vert ^{2}\,dx = 0, $$
which implies \({(\psi - u)_{+} }=0\), or \(\psi \leq u\) a.e. in \({B_{R}}\). The proof is complete. □