\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 191, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/191\hfil Cordes nonlinear operators] {Cordes nonlinear operators in Carnot groups} \author[G. Di Fazio, M. S. Fanciullo \hfil EJDE-2015/191\hfilneg] {Giuseppe Di Fazio, Maria Stella Fanciullo} \address{Giuseppe Di Fazio \newline Dipartimento di Matematica e Informatica, Universit\a di Catania, Viale A. Doria 6, 95125, Catania, Italy} \email{difazio@dmi.unict.it} \address{Maria Stella Fanciullo \newline Dipartimento di Matematica e Informatica, Universit\a di Catania, Viale A. Doria 6, 95125, Catania, Italy} \email{fanciullo@dmi.unict.it} \thanks{Submitted April 8, 2015. Published July 20, 2015.} \subjclass{35H20} \keywords{Cordes condition; Carnot groups; nonlinear equations} \begin{abstract} Our aim is to obtain $L^p$ estimates for the second-order horizontal derivatives of the solutions for a nondivergence form nonlinear equation in Carnot groups. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{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 \cite{cfl,cam,d}) 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 \cite{c} and studied by many authors (in the cases of nonlinear nonvariational equations and systems we quote \cite{cam1,cam}). The second technique consists in assuming the coefficients of the operator to be in $\operatorname{VMO}$-type classes (see \cite{cfl,d,dfz1,dfz2}, and for more general setting see \cite{bb2,bf}). Here we obtain $W^{2,p}$ estimates for the following nonlinear nondivergence form equation $$a(x,u,Xu,X^2u)=f,$$ where $X=(X_1,X_2,\dots,X_l)$ is a system of H\"ormander'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 \cite{df1} for the case of the Heisenberg group). Namely, we show that there exists a critical exponent $p_0 > 2$ such that if the datum $f$ belongs to $L^p$, with $2s$. Let $X_1, \dots,X_l$ be a basis for $V_1$ and suppose that $X_1,\dots,X_l$ generate $\mathcal{G}$ as a Lie algebra. Then for $2\le j\le s$ we choose a basis $\{X_{ij}\}$, $1\le i\le k_j$, for $V_j$ consisting of commutators of length $j$. We set $X_{i1}=X_i$, $i=1,\dots,l$ and $k_1=l$, and we call $X_{i1}$ a commutator of length $1$. If $\mathbb G$ is the simply connected Lie group associated to $\mathcal{G}$ then $\mathbb G$ is called Carnot group. It is well known that the exponential mapping is a global diffeomorphism from $\mathcal{G}$ to $\mathbb G$ and then for any $g\in \mathbb G$ there exists $x=(x_{ij})\in \mathbb{R}^n$, $1\le i\le k_j$, $1\le j\le s$, $n=\sum_{j=1}^sk_j$, such that $g=\exp (\sum x_{ij}X_{ij})$. We now recall the definition of polynomials on the Carnot group $\mathbb G$ given by Folland and Stein in \cite{fs}. A function $P$ on $\mathbb G$ is said to be a polynomial on $\mathbb G$ if $P\circ$ exp is a polynomial on the Lie algebra $\mathcal{G}$. Let $X_1,X_2,\dots, X_{n}$ be a basis of $\mathbb G$ and $\xi_1,\xi_2,\dots,\xi_{n}$ be the dual basis for $\mathcal{G}^*$ we set $\eta_i=\xi_i\circ {\rm exp}^{-1}$. Each $\eta_i$ is a polynomial on $\mathbb G$, and $\eta_1,\eta_2,\dots,\eta_{n}$ form a system of global coordinates on $\mathbb G$. Then every polynomial on $\mathbb G$ can be written uniquely as $$P(x)=\sum_Ia_I\eta^I(x),\quad \eta^I=\eta_1^{i_1}\cdots\eta_{n}^{i_{n}},\; a_I\in \mathbb{R}$$ where all but finitely many of the coefficients $a_I$ vanish. Clearly $\eta^I$ is homogeneous of degree $d(I)=\sum_{j=1}^{n}i_jd_j$, where $d_j$ is the length of $X_j$ as a commutator. We define the homogeneous degree of the polynomial $P$ as max$\{d(I):a_I \neq 0 \}$. Here we recall the definition of the Carnot-Carath\'eodory metric. An absolutely continuous curve $\gamma:[0,\tau]\to \mathbb{R}^{n}$ is called subunitary if there exists a measurable function $c=(c_1,c_2,\dots ,c_{l}):[0,\tau]\to \mathbb{R}^{l}$ such that $\gamma'(t)=\sum_{j=1}^{l} c_j(t)X_j(\gamma(t))$, for a.e. $t\in [0,\tau]$, and $\|c\|_\infty\le 1$. The Carnot-Carath\'eodory distance $d(x',x'')$ is defined as the infimum of those $\tau >0$ for which there exists a subunitary curve $\gamma :[0,\tau]\to \mathbb{R}^n$ with $\gamma(0)=x'$ and $\gamma(\tau)=x''$. We set $B_r(x_0)=\{x\in \mathbb{R}^{n}: d(x,x_0)0$ and $x_0\in \mathbb{R}^{n}$. The constant $Q$ is the homogeneous dimension of $\mathbb G$ . We define the intrinsic Sobolev spaces for a bounded domain $\Omega$ in $\mathbb{R}^n$. Let $k\in \mathbb{N}$ and $p\ge 1$ we set $$W^{k,p} (\Omega ) = \big\{ u : \Omega \to {\mathbb R} : u , X_{i_1} \dots X_{i_j} u \in L^p (\Omega ), \; 1 \leq j \leq k \bigr\}$$ endowed with the norm $$\|u\|_{W^{k,p}(\Omega)}=\|u\|_{L^p(\Omega)} +\sum_{h=1}^k\sum_{i_j=1}^{l} \|X_{i_1}X_{i_2}\dots X_{i_h}u\|_ {L^p(\Omega)} \,.$$ We define $W^{k,p}_0 (\Omega )$ as the closure of $C_0^{\infty} (\Omega )$ in $W^{k,p} (\Omega )$ with respect to the above norm. For $I=(i_1,i_2,\dots ,i_{n})$ we denote the differential operator $X^{i_1}_{1}X^{i_2}_{2}\dots X^{i_{n}}_{n}$ by $X^I$ and $d(I)$ the homogeneous degree of $X^I$. We denote by $Xu$ the gradient of $u$ $(X_1u,X_2u,\dots,X_{l})$ and by $X^2u$ the hessian matrix $\{X_{ij}u\}_{i,j=1,\dots,l}$. In \cite{lu} the existence of approximation polynomials of Sobolev functions in Carnot groups and related Poincar\'{e}-type inequalities have been obtained. Here we state some results \cite[Theorems 2.7 and 5.1]{lu}, that we will use in the sequel. \begin{theorem}\label{existencepolynomials} Let $k$ be a positive integer and $u$ a function in $W^{k,1}(\Omega)$. Then there exists a polynomial P of degree less than $k$ such that $\int_\Omega X^I(u-P)dx=0$ for any $0\le d(I)0$ and $\theta > 0$ such that for a.e. $x\in\Omega$, $\sum_{i=1}^{l} a_{ii} (x) > 0$ and $$0< \frac{1}{\sigma} \leq \sum_{i,j=1}^{l}a^2_{ij}(x)\le \frac{1}{l-1+\epsilon}\Big(\sum_{i=1}^{l}a_{ii} (x) \Big)^2 \leq \frac{\theta^2}{l-1+\epsilon} \, .$$ \end{definition} Now we prove a Caccioppoli type inequality for solutions of \eqref{equation}. \begin{theorem} \label{thm3.1} Let condition $(A)$ hold true and $f\in L^2(\Omega)$. Then for any $u\in W^{2,2}(\Omega)$ solution of \eqref{equation}, for any $r>0$ such that $B_{2r}\Subset \Omega$, there exists a polynomial $P$ of degree less than $2$ such that $\int_{B_{2r}} X^I(u-P)dx=0$ for any $0\le d(I)<2$, and \begin{equation}\label{cacci} \int_{B_r}|X^2u|^2dx\le c r^{-2}\int_{B_{2r}}|X(u-P)|^2dx + c\int_{B_{2r}}f^2dx\,. \end{equation} \end{theorem} \begin{proof} Let $B_{2r}\Subset \Omega$. From Theorem \ref{existencepolynomials} there exists a polynomial $P$ of degree less than two such that $\int_{B_{2r}}X^I (u-P)dx=0$, for $I$ with $d(I)<2$. Let $\eta$ be a $C^\infty_0(\mathbb{R}^{n})$ with the properties $0\le \eta\le 1$, $\eta=1$ in $B_r$, $\eta=0$ in $\mathbb{R}^{n}\setminus B_{2r}$ and $|X \eta|\le \frac{c}{ r}$. If we set $\mathcal U= \eta(u-P) \in W^{2,2}_0(B_{2r})$, since $X^2P=0$ (see the proof of \cite[Theorem 2.7]{lu}), we have $X^2(u-P)=X^2u$ and $X^2\mathcal U=X^2 u$ in $B_r$. We have $$\eta \Delta u =\eta (\Delta u - \alpha a(x,u,Xu,X^2u))+\eta\alpha f \, ,$$ which implies \begin{align*} |\eta\Delta u| &\le \eta |\Delta u -\alpha a(x,u,Xu,X^2u)|+|\eta\alpha f| \\ &\le \eta[\gamma |X^2u|^2+\delta |\Delta u|^2]^{1/2}+\eta \alpha |f|. \end{align*} Note that \begin{gather}\label{Delta} \Delta \mathcal U =\eta \Delta u + A(u-P),\\ \label{X} \eta X^2u=X^2\mathcal U-B(u-P), \end{gather} where \begin{equation}\label{AA} A(u-P)= (u-P)\Delta \eta+2\sum X_i\eta X_i(u-P) \end{equation} and $$B(u-P)=\{(u-P)X_i X_j \eta+X_i\eta X_j(u-P)+X_j\eta X_i(u-P)\}_{ij} \, .$$ Then for $x\in B_{2r}$, $$|\Delta\mathcal U |\le |\eta \Delta u|+|A(u-P)| \le \eta (\gamma |X^2u|^2+\delta|\Delta u|^2)^{1/2}+\eta \alpha |f|+|A(u-P)|,$$ from which it follows that for all $\epsilon>0$, \begin{align*} |\Delta\mathcal U |^2 &\le \eta^2(\gamma |X^2u|^2 +\delta|\Delta u|^2)+ ( \eta \alpha |f|+|A(u-P)| )^2\\ &\quad +2\eta (\gamma |X^2u|^2+\delta|\Delta u|^2)^{1/2} ( \eta \alpha |f|+|A(u-P)| ) \\ &\le \eta^2(\gamma |X^2u|^2+\delta|\Delta u|^2)+( \eta \alpha |f|+|A(u-P)| )^2\\ &\quad +\epsilon \eta^2 (\gamma |X^2u|^2+\delta|\Delta u|^2) +\frac{1}{\epsilon} ( \eta \alpha |f|+|A(u-P)| )^2\\ &=(1+\epsilon)\eta^2 (\gamma |X^2u|^2+\delta|\Delta u|^2) + \big(1+\frac{1}{\epsilon}\big) ( \eta \alpha |f|+|A(u-P)| )^2\\ &\le (1+\epsilon)\eta^2 (\gamma |X^2u|^2+\delta|\Delta u|^2) +2\big(1+\frac{1}{\epsilon}\big) ( \eta^2 \alpha^2 |f|^2+|A(u-P)|^2 ). \end{align*} Then from \eqref{Delta} and \eqref{X}, \begin{align*} |\Delta\mathcal U |^2 &\le (1+\epsilon)\gamma |X^2(\mathcal U)-B(u-P)|^2+(1+\epsilon)\delta |\Delta\mathcal U -A(u-P)|^2\\ &\quad +2\big(1+\frac{1}{\epsilon}\big)\eta^2\alpha^2 |f|^2+2\big(1+\frac{1}{\epsilon}\big)|A(u-P)|^2 \\ &\le (1+\epsilon)\gamma (|X^2(\mathcal U)|^2+|B(u-P)|^2+2|X^2(\mathcal U)||B(u-P)|)\\ &\quad +(1+\epsilon)\delta (|\Delta\mathcal U|^2+|A(u-P)|^2+2|\Delta\mathcal U||A(u-P)|)\\ &\quad +2\big(1+\frac{1}{\epsilon}\big)\eta^2\alpha^2 |f|^2+2\big(1+\frac{1}{\epsilon}\big) |A(u-P)|^2 \\ &\le (1+\epsilon)\gamma \big[(1+\epsilon)|X^2(\mathcal U)|^2 +\big(1+\frac{1}{\epsilon}\big)|B(u-P)|^2\big]\\ &\quad +(1+\epsilon)\delta \big[(1+\epsilon)|\Delta\mathcal U|^2 +\big(1+\frac{1}{\epsilon}\big)|A(u-P)|^2\big]\\ &\quad +2\big(1+\frac{1}{\epsilon}\big)\eta^2\alpha^2 |f|^2 +2\big(1+\frac{1}{\epsilon}\big)|A(u-P)|^2 \\ &\le (1+\epsilon)^2\gamma |X^2(\mathcal U)|^2+(1+\epsilon)^2\delta |\Delta\mathcal U|^2\\ &\quad +c(\epsilon, \alpha,\gamma,\delta)[ |A(u-P)|^2+ |B(u-P)|^2+|f|^2]. \end{align*} We integrate on $B_{2r}$ and apply \eqref{eqcz} in Theorem \ref{cz} to obtain \begin{align*} \int_{B_{2r}}|\Delta\mathcal U|^2dx &\le (1+\epsilon)^2\gamma \int_{B_{2r}}|X^2(\mathcal U)|^2dx+(1+\epsilon)^2\delta \int_{B_{2r}}|\Delta\mathcal U|^2dx \\ &\quad +c \int_{B_{2r}}(|f|^2+|A(u-P)|^2+ |B(u-P)|^2)dx \\ &\le (1+\epsilon)^2 (\gamma C_G+\delta)\int_{B_{2r}}|\Delta\mathcal U|^2dx\\ &\quad +c\int_{B_{2r}}(|f|^2+|A(u-P)|^2+ |B(u-P)|^2)dx. \end{align*} It follows that \begin{align*} &[1-(1+\epsilon)^2(\gamma C_G+\delta)]\int_{B_{2r}}|\Delta\mathcal U|^2dx \\ &\le c\int_{B_{2r}}(|f|^2+|A(u-P)|^2+ |B(u-P)|^2)dx, \end{align*} and then $$\int_{B_{2r}}|\Delta\mathcal U|^2dx \le c\int_{B_{2r}}(|f|^2+|A(u-P)|^2 + |B(u-P)|^2)dx .$$ Finally, we get that \begin{align*} \int_{B_{r}}|X^2u|^2dx &\le \int_{B_{2r}}|X^2\mathcal U|^2dx\le C_G\int_{B_{2r}}|\Delta\mathcal U|^2dx \\ &\le c\int_{B_{2r}}(|f|^2+|A(u-P)|^2+ |B(u-P)|^2)dx. \end{align*} Now we observe that from \eqref{AA} and Poincar\'e inequality \eqref{poincareI} we obtain \begin{align*} &\int_{B_{2r}}|A(u-P)|^2dx \\ &\le c \int_{B_{2r}}|\Delta \eta|^2|u-P|^2dx +c \int_{B_{2r}}\sum |X_i\eta|^2|X_i(u-P)|^2dx \\ &\le c r^{-2}\Big\{r^{-2}\int_{B_{2r}}|u-P|^2dx+\int_{B_{2r}}|X(u-P)|^2dx \Big\}\\ &\le c r^{-2}\int_{B_{2r}}|X(u-P)|^2 dx. \end{align*} In the same way, we find that $$\int_{B_{2r}}|B(u-P)|^2dx \le c r^{-2}\int_{B_{2r}}|X(u-P)|^2dx,$$ from which the Caccioppoli type inequality follows. \end{proof} Next we state \cite[Theorem 3.3]{z} which is a generalization of the Gehring lemma \cite{g}. \begin{lemma}\label{ggm} Let $U$ and $G$ be non-negative functions in $\Omega$ such that $$U\in L^t_{\rm loc}(\Omega), \quad G\in L^s_{\rm loc}(\Omega), \quad 11 such that for every B_{2r}\Subset \Omega, r <1,$$ {\int\hspace{-10pt}-}_{B_{r}}U^tdx\le c \Big({\int\hspace{-10pt}-}_{B_{2r}}Udx\Big)^t +c{\int\hspace{-10pt}-}_{B_{2r}}G^tdx, $$then there exists \epsilon\in (0,s-t] such that U\in L^p_{\rm loc}(\Omega), for all p \in [t,t+\epsilon) and, for every B_{2r}\Subset \Omega, with r <1, we have$$ \Big({\int\hspace{-10pt}-}_{B_{r}}U^pdx\Big)^{1/p} \le K\Big[\Big({\int\hspace{-10pt}-}_{B_{2r}}U^tdx\Big)^{1/t} +\Big({\int\hspace{-10pt}-}_{B_{2r}}G^pdx\Big)^{1/p}\Big],  where the constant $K$ depends on $c, t$ and $Q$. \end{lemma} Our main Theorem is now an easy consequence of Caccioppoli-type inequality \eqref{cacci} and Lemma \ref{ggm}. \begin{theorem} \label{thm3.2} Let $u\in W^{2,2}(\Omega)$ be a solution of \eqref{equation} then there exists $p_0>2$ such that, if $f\in L^p(\Omega)$, with \$2\le p