\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 48, pp. 1--24.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/48\hfil Complex Monge-Amp\{e}re equation] {Dirichlet problem for degenerate elliptic complex Monge-Amp\{e}re equation} \author[Saoussen Kallel-Jallouli\hfil EJDE-2004/48\hfilneg] {Saoussen Kallel-Jallouli} \address{Saoussen Kallel-Jallouli\hfill\break Facult\'{e} des Sciences, Campus Universitaire, 1060 Tunis, Tunisie} \email{Saoussen.Kallel@fst.rnu.tn} \date{} \thanks{Submitted May 15, 2003. Published April 6, 2004.} \subjclass[2000]{35J70, 32W20, 32W05} \keywords{Degenerate elliptic, omplex Monge-Amp\ere, \hfill\break\indent Plurisubharmonic function} \begin{abstract} We consider the Dirichlet problem $$\det \big({\frac{\partial^2u}{\partial z_i\partial \overline{z_j}}} \big)=g(z,u)\quad\mbox{in }\Omega\,, \quad u\big|_{ \partial \Omega }=\varphi\,,$$ where $\Omega$ is a bounded open set of $\mathbb{C}^{n}$ with regular boundary, $g$ and $\varphi$ are sufficiently smooth functions, and $g$ is non-negative. We prove that, under additional hypotheses on $g$ and $\varphi$, if $|\det \varphi _{i\overline{j}}-g|_{C^{s_{\ast }}}$ is sufficiently small the problem has a plurisubharmonic solution. \end{abstract} \maketitle \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Propositon} \numberwithin{equation}{section} \allowdisplaybreaks \section{Introduction} Let $\Omega$ be a bounded domain in $\mathbb{R}^{2n}$ with smooth boundary and let $z_i=x_i+ix_{i+n}(1\leq i\leq n)$. We shall also denote by $\Omega$ the set of $z=(z_1,z_{2},\dots ,z_n)$ satisfying $(\mathop{\rm Re}z,\mathop{\rm Im}z)\in \Omega$.\ We study the problem of finding a sufficiently smooth plurisubharmonic solution to the degenerate problem $$\begin{gathered} \det \big({\frac{\partial ^2\phi }{\partial z_i\partial \overline{z_j} }}\big)=g(z,\phi )\quad\mbox{in }\Omega \,, \\ \phi \big|_{ \partial \Omega }=\varphi \,. \end{gathered} \label{I.1}$$ In \cite{k1,k2}, the author studies local solutions, while, here we consider global solutions. This problem has received considerable attention both in the non-degenerate case ($g>0$) and in the degenerate case ($g\geq 0$). In particular, Caffarelli, Kohn, Nirenberg and Spruck \cite{c1} established some existence results in strongly pseudoconvex domains based on the construction of a subsolution. The recent work of Guan \cite{g2}, extends some of these results to arbitrary smooth bounded domains. Guan proved for the nondegenerate case that a sufficient condition for the classical solvability is the existence of a subsolution. Here we are concerned with degenerate problems in an arbitrary smooth bounded domain, which need not be Pseudoconvex. Counterexamples due to Bedford and Fornaes \cite{b1} show that the Dirichlet problem, in general, does not have a regular solution. This implies that we should place some restrictions on $g$ and $\varphi$. Let us assume that $\varphi$ is a real function defined in $\overline{\Omega }$, $\Sigma$ is a finite set of points in $\Omega$, and $g(z,\phi )=K(z)f(\mathop{\rm Re}z,\mathop{\rm Im}z,\phi )$. We further assume the following hypotheses. \begin{itemize} \item[(A1)] $K\geq 0$ in $\overline{\Omega }$, and $K^{-1}(0)=\Sigma$ % \label{I.2} \item[(A2)] $f(x,u)>0$ in $\overline{\Omega }\times \mathbb{R}$, and $\frac{\partial f}{\partial u}\geq -\varrho$ in $\overline{\Omega }\times \mathbb{R}$, with $0\leq \varrho <<1$ % \label{I.3} \item[(A3)] $\varphi \big|_{ \overline{\Omega }\backslash \Sigma }$ is strictly plurisubharmonic, $(\varphi _{i\overline{j}})\big|_{\Sigma }$ is of rank $(n-1)$, and the eigenvalues of $(\varphi _{i\overline{j}})$ on $\Sigma$ are distinct. %\label{I.4} \end{itemize} Our main results are the following theorems: \begin{theorem} \label{thmA} Let $s_{\ast }\geq 7+2n$ be an integer, $\alpha \in ] 0,1[$, and $\Gamma >1$. If $\varphi \in C^{s_{\ast}+2,\alpha }(\overline{\Omega })$ satisfies the condition (A3), {\it then one can find a constant $\varepsilon _0>0$ (depending on $s_{\ast }$, $\alpha$, $\Gamma$, $\Omega$ and $\varphi$) such that for any $g=Kf\in C^{s_{\ast }}$ satisfying} (A1), (A2), $$|\det \varphi _{i\overline{j}}-g(\varphi )| _{C^{s_{\ast }}}\leq \varepsilon _0 \label{I.5}$$ and $|\frac{\partial g}{\partial u}|_{C^{s_{\ast }}}\leq \Gamma$, then problem \eqref{I.1} has a plurisubharmonic (real valued) solution $\phi \in C^{s_{\ast }-3-n}(\overline{\Omega })$, which is unique when $\rho =0$. \end{theorem} Let $l_{\alpha }(x)$ denote $\alpha$-th row the matrix of cofactors of $(\varphi _{i\overline{j}})$, and $$D^{k}K(x)(l_{\alpha }(x),l_{\beta }(x))^{(k)} =D^{k}K(x)\big(l_{\alpha}(x),l_{\beta }(x);\dots ;l_{\alpha }(x) ,l_{\beta }(x)\big).$$ \begin{theorem} \label{thmB} Under the assumptions in Theorem \ref{thmA}, suppose that $\varphi \in C^{\infty }(\overline{\Omega })$ and for any point $x_0\in \Sigma$ one can find an integer $k$ such that $D^{j}K(x_0)=0$ for all $j\leq k-1$ and there exists $\alpha \neq \beta \in \{1,\dots ,n\}$ such that $D^{k}K(x_0)(l_{\alpha }(x_0),l_{\beta }(x_0))^{(k)}\neq 0$. Then there exists an integer $s_{\ast }>0$ and a constant $\varepsilon_0>0$ such that for any function $g\in C^{\infty }$ satisfying (A2), (A3) and \eqref{I.5}, the plurisubharmonic solution $\phi$ to the problem \eqref{I.1} is in $C^{\infty }(\overline{\Omega })$. \end{theorem} In Theorem \ref{thmA}, the assumption concerning $\Sigma$ leads to a-priori estimates and the assumption on $g$ and $\varphi$ ensures the convergence of an iteration scheme of Nash-Moser type. It is to be noted that we do not require demonstrating that a subsolution exists as in \cite{c1} and [6]. Under some additional conditions on $g$, we can prove the smoothness of the solution, using the works of Xu \cite{x1} and Xu and Zuily \cite{x2}. This paper is organized as follows. In Section 2 we state some preliminary results. In Section 3, we state fundamental global a-priori estimates for degenerate linearized operators that are crucial to establish an iteration scheme of Nash-Moser type. We then prove Theorem \ref{thmA} in Section 4. We prove Theorem \ref{thmB} in Section 5. Finally, we prove the a-priori estimates stated in Section 3. \section{Preliminary results} We shall use the norms \begin{equation*} |\cdot|_{k}=\|\cdot\|_{C^{k}(\overline{\Omega })}, \quad \|\cdot \|_{k}=\|\cdot\|_{H^{k}(\Omega )}, \quad |\cdot|_{k,\tau }=\|\cdot \|_{C^{k,\tau }(\overline{\Omega })}\, \end{equation*} where $k\in \mathbb{N}$ and $\tau \in ] 0,\alpha [$. In this work, we need some technical lemmas which play important roles in the proof of convergence of our iteration scheme. \begin{lemma} \label{lmII.1} Let $s_{\ast }$ be an integer, $s_{\ast }\geq 7+2n$. We can find a constant $\beta \geq 2$ such that for any $0\leq i,j,k\leq s_{\ast }+2$, $n_{\ast }=n+\tau$ and $u\in C^{s_{\ast }+2,\alpha }(\overline{\Omega })$ we have: The Sobolev inequality $$|u|_{i,\tau }\leq \beta \| u\|_{i+n_{\ast }}\label{II.1}$$ The Gagliardo-Nirenberg inequality \| u\|_j\leq \beta \| u\|_i^{\frac{k-j}{k-i} }\| u\|_{k}^{\frac{j-i}{k-i}},\quad i0$; if$u,v\in L^{\infty }\cap H^{t}$, then$uv\in L^{\infty }\cap H^{t}$and $$\| uv\|_{t}\leq K_1(|u|_0\|v\|_{t}+\| u\|_{t}|v|_0),\label{II.7}$$ where,$K_1$is a constant$\geq 1$independent of$u$and$v$. \noindent (2) Let$H:\mathbb{R}^{m}\to \mathbb{C}$be a function$C^{\infty }$of its arguments.\\ For$s>0$, if$\omega \in (L^{\infty }\cap H^{s})^{m}$and$|\omega |_0\leq M$, then $$\| H(\omega )\|_s\leq K_{2}(s,H,M) (\| \omega \|_s+1), \label{II.8}$$ where$K_{2}\geq 1$and is a constant independent of$\omega $. \\ If$\omega \in (C^{i,\mu })^{m}$,$\mu \in ] 0,1[ $and$i\in \mathbb{N}$, then$H(\omega )\in C^{i,\mu }$. If we suppose that$|\omega |_0\leq M$, then we can find a constant$K_{3}=K_{3}(i,\mu ,H,M)\geq 1$such that $$|H(\omega )|_{i,\mu }\leq K_{3}(|\omega |_{i,\mu }+1). \label{II.9}$$ \end{lemma} We shall also need the following technical lemma. \begin{lemma}[{\cite[Lemma]{k1}}] \label{lmII.3} Let$F(u_{z_i\overline{z_j}})=\det (u_{z_i\overline{z_j}})$. For$1\leq i,j,a,b\leq n$, we have $$F\frac{\partial ^2F}{\partial u_{z_{a}\overline{z_{b}}}\partial u_{z_i \overline{z_j}}}=\frac{\partial F}{\partial u_{z_{a}\overline{z_{b}}}} \frac{\partial F}{\partial u_{z_i\overline{z_j}}}-\frac{\partial F}{ \partial u_{z_i\overline{z_{b}}}}\frac{\partial F}{\partial u_{z_{a} \overline{z_j}}}. \label{II.10}$$ \end{lemma} \section{A priori estimates for the linearized operator} Defining$\phi =\varphi +\varepsilon w$, \eqref{I.1} becomes $$\det (\phi _{z_i\overline{z_j}})=\det (\varphi _{z_i \overline{z_j}}+\varepsilon w_{z_i\overline{z_j}})=g. \label{III.1}$$ Let $$G(w)=\frac{1}{\varepsilon }[\det \Phi -g] .\label{III.2}$$ Then the linearization of$G$at$w$is $$L_{G}(w)=\sum_{i,j=1}^n \phi ^{ij}\partial _{z_i}\partial _{\overline{z_j}}+b, \label{III.3}$$ where$\widetilde{\Phi }=(\phi ^{ij})$is the matrix of cofactors of$\Phi =(\phi _{z_i\overline{z_j}}(z,\varepsilon ,w))$and$b={\frac{\partial g}{\partial u}}$. Now we construct linear elliptic operators, maybe degenerate, related to linearized operators. For any smooth real valued function$w$, the matrix$(\phi _{i\overline{j}})$is Hermitian and we can find a unitary matrix$T(z,\varepsilon )$satisfying $$T(z,\varepsilon )(\phi _{z_i\overline{z_j}}) ^{t}\overline{T}(z,\varepsilon )=\mathop{\rm diag}(\lambda_1,\dots ,\lambda _n). \label{III.4}$$ Without loss of generality we may assume that$\Sigma $is reduced to one point, the origin. By means of change of variables we may assume, using (A3), that $$\varphi _{z_i\overline{z_j}}(0)=\sigma _i\delta _i^{j} \quad i,j=1,\dots ,n, \label{III.5}$$ where$\sigma _i>0$for$i=1,\dots ,n-1$,$\sigma _n=0$and$\sigma_i\neq \sigma _j$for$i\neq j$. %\label{III.6} Let$0<\tau \leq \frac{\alpha }{4}$. \begin{lemma} \label{lmIII.1} There exist constants$\varepsilon _1>0$,$\delta _1>0$and$M>0$depending only on$\varphi $,$n$,$\Omega $such that when $$V_0=\{(z,\varepsilon ,w)/|z|\leq \delta_1,\, 0\leq \varepsilon \leq \varepsilon _1,\, w\in C^{3,\tau}(\overline{\Omega }),\,|w|_{3,\tau }\leq 1\},$$ we have: (i) The eigenvalues$\lambda _i$,$i=1,\dots ,n$of$\Phi $are distinct on$V_0$and of class$C^{1}$in \r{V}$_0$. Moreover,$\lambda _i>0$in$V_0$, for$i=1,\dots ,n-1$.\\ (ii) For$(z,\varepsilon ,w)\in $V$_0$, $$\sum_{i=1}^n |\sigma _i-\lambda _i(z,\varepsilon ,w)| +|\Phi ^{nn}(z,\varepsilon,w)-\prod_{i=1}^{n-1} \sigma _i| \leq M(\varepsilon +|z|). \label{III.7}$$ (iii) For$(z,\varepsilon ,w)\in V_0$and$i=1,\dots ,n-1$, $$\lambda _i\geq \inf_{1\leq i\leq n-1} \sigma _i -M\delta_1-(M+1)\varepsilon _1>0\text{ and }\Phi ^{nn}\geq \prod_{i=1}^{n-1}\sigma _i-M\delta _1-M\varepsilon _1>0. \label{III.8}$$ \end{lemma} \begin{proof} Let us consider the function$H(z,\varepsilon ,w,\lambda )=\det (\varphi _{z_i\overline{z_j}} +\varepsilon w_{z_i\overline{z_j}}-\lambda \delta_i^{j})$. Then$H\in C^{1}$and by \eqref{III.5}, we have $$H(0,0,0,\sigma _i)=0\mbox{ and } \frac{\partial H}{\partial\lambda }(0,0,0,\sigma _i)\neq 0, \quad \forall i\in \{1,\dots ,n\}.$$ By the implicit function theorem, one can find two constants$\varepsilon _1>0$and$\delta _1>0$such that (i) holds. Moreover by \eqref{III.5} we have $${\frac{\partial F}{\partial u_{n\overline{n}}}}(\varphi _{i\overline{j} })(0)=\Phi ^{nn}(0,0,w) =\prod_{i=1}^{n-1} \sigma _i>0,$$ which gives (ii) and (iii). \end{proof} \begin{lemma} \label{lmIII.2} There exists a positive constant$\varepsilon _{2}$such that for any$0<\varepsilon <\varepsilon _{2}$, any real valued function$w\in C^{3,\tau }(\overline{\Omega })$satisfying$|w|_{3,\tau }\leq 1$and$\theta =\max_{z\in \overline{\Omega }}|G(w)|$, the operator $$L=-L_{G}(w)-\theta \triangle \label{III.9}$$ is elliptic, maybe degenerate. (Here$\triangle =\sum_{i=1}^n (\frac{\partial^2}{\partial x_i^2} +\frac{\partial ^2}{\partial y_i^2})$) \end{lemma} \begin{proof} Let $$A=\theta |\xi |^2+\sum_{i,j=1}^n \phi^{ij}\xi _i\overline{\xi _j}\geq 0,\quad \forall (z,\xi ) \in \overline{\Omega }\times \mathbb{C}^{n}. \label{III.10}$$ If$z\in \overline{\Omega }\backslash \{0\}$, as$\varphi $is strictly plurisubharmonic, then$A>0$for all$\xi \in \mathbb{C}^{n}\backslash \{0\}$. If$z=0$, for$\xi \in \mathbb{C}^{n}$, we let$\xi =^{t}T(\tau ,\varepsilon )\widetilde{\xi }$. Then we have $$A=\theta |\xi |^2+^{t}\xi \widetilde{\Phi }\overline{\xi } =\theta |\xi |^2+^{t}\widetilde{\xi }T\widetilde{\Phi }^{t} \overline{T}\overline{\widetilde{\xi }}.$$ Since$\Phi \widetilde{\Phi }=\det \Phi \mathop{\rm \mathop{\rm Id}}, by \eqref{III.4}, \begin{gather*} \det \Phi \mathop{\rm \mathop{\rm Id}}=T\Phi ^{t}\overline{T}T\widetilde{\Phi }^{t}\overline{T} =\mathop{\rm diag}(\lambda _i)T\widetilde{\Phi }^{t}\overline{T}, \\ T\widetilde{\Phi }^{t}\overline{T}=\det \Phi \mathop{\rm diag} (\frac{1}{\lambda _i})=\prod_{i=1}^n \lambda_i \mathop{\rm diag} (\frac{1}{\lambda _i}) =(\varepsilon G+g)\mathop{\rm diag}(\frac{1}{\lambda _i}). \end{gather*} Thus, \begin{align*} A&=\theta |\widetilde{\xi }|^2+\det \Phi \sum^n_{i=1} \frac{|\widetilde{\xi }_i|^2}{\lambda_i} \\ & =\theta |\widetilde{\xi }|^2+\sum^{n-1}_{i=1} \det \Phi \frac{|\widetilde{\xi }_i|^2}{\lambda _i} +\prod_{i=1}^{n-1} \lambda _i|\widetilde{\xi }_n|^2 \\ &=(\theta +\prod_{i=1}^{n-1} \lambda _i)|\widetilde{\xi }_n|^2 +\sum^{n-1}_{i=1} \frac{\varepsilon G+g+\theta \lambda _i}{\lambda _i}| \widetilde{\xi }_i|^2. \end{align*} By \eqref{III.8}, fori=1,\dots ,n-1$,$\varepsilon \leq \varepsilon _1$and$|w|_{3,\tau }\leq 1$, we have $$\varepsilon G+\theta \lambda _i\geq \theta (\sigma _i-M\delta _1-(M+1)\varepsilon _1)\geq 0.$$ Therefore,$A\geq 0$, which proves the lemma. \end{proof} Now we study a boundary-value problem for the degenrate elliptic operator $$L=-L_{G}(w)-\theta \triangle =\sum^n_{i,j=1} b^{ij}\partial _{z_i} \partial _{\overline{z_j}}+b,$$ where $$b^{ij}=-{\frac{\partial F}{\partial u_{i\overline{j}}}}(\varphi _{i \overline{j}}+\varepsilon w_{i\overline{j}})-\theta \delta _i^{j}=-\Phi ^{ij}-\theta \delta _i^{j}$$ and$b=K{\frac{\partial f}{\partial u}}$. For$k,s\in \mathbb{N}$we let $$\begin{gathered} A(k)=\max (1,\max_{1\leq i,j\leq n} |b^{ij}|_{k},|b|_{k}) \\ \Lambda _s=\{(i,j): 0\leq i,j\leq s,\; i+j\leq s,\mbox{ and } i+2\leq \max (s,2)\} \end{gathered}\label{III.11}$$ Now from Lemma \ref{lmIII.2} we have the following statement. \begin{theorem} \label{thmIII.3} Suppose that$\theta \leq 1$and$A(2)\leq M_0$, for some constant$M_0>0$. One can find$\varepsilon _{3}>0$such that for any$\varepsilon \in ] 0,\varepsilon _{3}] $, any real valued function$w\in C^{s_{\ast }+2,\tau }(\overline{\Omega })$satisfying the inequality$|w|_{3,\tau }\leq 1$and any real valued function$h\in H^{s_{\ast }}$, the problem $$\begin{gathered} Lu=h\quad\mbox{in }\Omega \\ u\big|_{\partial \Omega }=0 \end{gathered} \label{III.12}$$ has a unique solution$u\in H^{s_{\ast }}$. Moreover for$0\leq s\leq s_{\ast }$, \begin{gather} \| u\|_0\leq C_0\| h\|_0 \label{III.13}\\ \| u\|_1\leq C_1(\| h\|_1+\|u\|_0) \label{III.14}\\ \| u\|_s\leq C_s\{\| h\|_s+\sum_{j\leq s-1,\, (i,j)\in \Lambda _s} (1+|\varphi +\varepsilon w|_{i+4,\tau })\| u\|_j\},\quad s\geq 2 \label{III.15} \end{gather} for some constant$C_s=C_s(\varphi ,s,\Omega ,M_0,\varepsilon _{3})$independent of$w$and$\varepsilon $. \end{theorem} For$\nu \in ] 0,1[ $, we denote$L_{\nu }=L-\nu \triangle $. To solve the Dirichlet problem \eqref{III.12}, we first establish the following proposition. \begin{proposition} \label{propIII.4} Let$\theta \leq 1$and, for some constant$M_0>0$,$A(2)\leq M_0$. Then there exists$\varepsilon _{3}>0$such that for any$\varepsilon \in ] 0,\varepsilon_{3}] $, any real valued function$w\in C^{s_{\ast }+2,\tau }(\overline{\Omega })$satisfying the inequality$|w|_{3,\tau }\leq 1$and any real valued function$h\in H^{s_{\ast}}(\Omega )$, the regularized problem $$\begin{gathered} L_{\nu }u=h \quad\mbox{in }\Omega , \\ u\big|_{ \partial \Omega }=0, \end{gathered} \label{III.12'}$$ has a unique (real valued) solution$u\in H^{s_{\ast }+1}(\Omega)$. \end{proposition} \begin{proof} Since$L_{G}(w)$is a second order operator with real coefficients, from Lemma \ref{lmIII.2},$L_{\nu }$is uniformly elliptic with coefficients in$C^{s_{\ast },\tau }(\overline{\Omega })$. Thus by \cite[Theorems 6.14 and 8.13]{b2} we see that \eqref{III.12'} has a real valued solution. If \eqref{III.13}--\eqref{III.15} hold for the regularized problem \eqref{III.12'} with an uniform constant$C_s$independent of$\nu \in ] 0,1]$, then by letting$\nu $tend to zero we get a solution$u\in H^{s_{\ast }}(\Omega )$to the original problem which of course satisfies \eqref{III.13}--\eqref{III.15}. \end{proof} Using Theorem \ref{thmIII.3}, we prove Theorem \ref{thmA} by constructing a sequence of approximating solutions and a priori estimates for linearized operators. The hypothesis \eqref{I.5} will play an important role in the proof of the convergence of our iteration scheme of Nash-Moser type. \section{Proof of Theorem \ref{thmA}} \subsection*{Part 1: An iteration scheme of Nash-Moser type} In this section, we use the Nash-Moser procedure \cite{h1,m1} and the results of Section 3 to prove Theorem \ref{thmA}. We construct a sequence which converges to a solution to our problem. We define $$M_0=1+\underset{H\in \mathcal{F}}{\max }K_{3}(2,\tau ,H,( 1+|\varphi |_{2}))(1+|\varphi | _{4,\tau }), \label{IV.1}$$ where$\mathcal{F=}\{\frac{\partial F}{\partial u_{i\overline{j}}}, \frac{\partial g}{\partial u}/1\leq i,j\leq n\}$and$K_{3}$is the constant introduced in \eqref{II.9}. (i.e:$|H(u)|_{j,\mu }\leq K_{3}(j,\mu ,H,M)|u|_{j,\mu }$). We also define $$D=\max \big(\max_{0\leq s\leq s_{\ast }} C_s,\,1\big). \label{IV.2}$$ Here$C_s$is the constant (depending only on$s,\varphi ,\Omega ,M_0$) given by Theorem \ref{thmIII.3}. We let $$\mu =\max (\beta ,3Ds_{\ast }^2(1+|\varphi | _{s_{\ast }+2,\tau }),n,2^{\frac{1}{\tau }})\quad\text{and}\quad \widetilde{\mu }=\beta ^2\mu ^{s_{\ast }}, \label{IV.3}$$ $$a_1=9K_0\mu ^{5}, \quad a_{2}=5a_1\mu ^{s_{\ast }+1}, \quad a_{3}=7K_0\mu ^{5}, \label{IV.4}$$ were$K_0$is the constant given by Proposition \ref{propVI.1}. Also, we fix$\widetilde{\varepsilon }$satisfying $$\widetilde{\varepsilon }\leq \min [1,(\varepsilon _i) _{1\leq i\leq 4},(3D^2a_{2}+6\widetilde{\mu }D^2)^{-2}], \label{IV.5}$$ were$\varepsilon _i$are given in Lemma \ref{lmIII.2}, Theorem \ref{thmIII.3}, the proof of Theorem \ref{thmIII.3} and the proof of \eqref{III.14}. As a consequence of these inequalities, we have$6\widetilde{\varepsilon }\mu ^{s_{\ast }}\leq 1/4$. Let$g\in C^{s_{\ast }}$satisfy $$|\det \varphi _{i\overline{j}}-g(\varphi )| _{s_{\ast }}\leq \widetilde{\varepsilon }^2,$$ with$\varepsilon _0$in Theorem \ref{thmA} equal to$\widetilde{\varepsilon }^2$. Let$S_n=S_{\mu _n}$the family of operators given by Lemma \ref{lmII.1}, with$\mu _n=\mu ^{n}$($\mu $is given by \eqref{IV.3}). Using Theorem \ref{thmIII.3}, we construct$w_n$,$n=0,1,\dots $, by induction on$n$as follows. We let$u_0$,$w_0=0$, and assume$w_0,w_1,\dots,w_n$have been chosen and define$w_{n+1}$by $$w_{n+1}=w_n+u_{n+1}, \label{IV.6}$$ where$u_{n+1}$is the solution to the Dirichlet problem $$\begin{gathered} L_{G}(\widetilde{w}_n)u_{n+1}+\theta _n\triangle u_{n+1}=g_n, \quad \mbox{in }\Omega \\ u_{n+1}\big|_{\partial \Omega }=0, \end{gathered}\label{IV.7}$$ given by Theorem \ref{thmIII.3}. Here \begin{gather} \widetilde{w}_n=S_nw_n, \label{IV.8} \\ \theta _n=|G(\widetilde{w}_n)|_0, \label{IV.9} \\ g_0=-S_0G(0),g_n=S_{n-1}R_{n-1}-S_nR_n+S_{n-1}G( 0)-S_nG(0), \label{IV.10} \\ R_0=0,\quad R_n=\sum^n_{j=1} r_j, \label{IV.11} \\ r_0=0,\quad r_j=[L_{G}(w_{j-1})-L_{G}(\widetilde{w}_{j-1})] u_j+Q_j -\theta _{j-1}\triangle u_j,\quad 1\leq j\leq n, \label{IV.12} \\ Q_j=G(w_j)-G(w_{j-1})-L_{G}(w_{j-1})u_j,\quad 1\leq j\leq n. \label{IV.13} \end{gather} To ensure that the$w_n$'s are well defined, we prove the following proposition. \begin{proposition} \label{propIV.1} Let$s\in \mathbb{N}$. If$s_{\ast }\geq 7+2n$and$4+2n+2\tau \leq \sigma k$, $$\| w_i-w_{k}\|_s\leq \sum^i_{j=k+1} \| u_j\|_s\leq \beta \sqrt{\widetilde{\varepsilon }} \sum^i_{j=k+1} \mu _{j-1}^{-\tau }=\beta \sqrt{\widetilde{\varepsilon }} \sum^i_{j=k+1} (\mu ^{-\tau})^{j-1}.$$ Since$\mu \geq 2$and$\tau >0$, then$\| w_i-w_{k}\|_s\to 0$as$i,k\to \infty $. Hence, there is a function$w\in H^{s_{\ast }-2-2\tau }(\Omega )$satisfying$w_n\to w$in$H^{s_{\ast }-2-2\tau }(\Omega)$. Since$H^{s_{\ast }-2-2\tau }(\Omega )\subset C^{s_{\ast }-2-n-3\tau }(\overline{\Omega })$, it follows that$w\in C^{s_{\ast}-3-n}(\overline{\Omega })$. On the other hand, combining \eqref{IV.7}, \eqref{IV.12} and \eqref{IV.13}, we obtain \begin{equation*} r_j=G(w_j)-G(w_{j-1})-g_{j-1} \end{equation*} Taking the sum between$j=1$and$j=n$, using \eqref{IV.10} and \eqref{IV.11}, we get $$G(w_n)=(I-S_{n-1})R_{n-1}+(I-S_{n-1})G(0)+r_n. \label{IV.22}$$ For$n\geq 2, using \eqref{II.2} and \eqref{IV.18}, we have \begin{equation*} \| r_n\|_{s_{\ast }-2-2\tau }\leq a_1\beta \widetilde{ \varepsilon }\mu _{n-1}^{s_{\ast }-2-2\tau -\sigma }=a_1\beta \widetilde{ \varepsilon }\mu _{n-1}^{-\tau }. \end{equation*} Combining \eqref{II.3} with \eqref{II.6} and \eqref{I.5}, we get \begin{equation*} \| (I-S_{n-1})G(0)\|_{s_{\ast }-2-2\tau }\leq \beta \mu _{n-1}^{-2-2\tau }\| G(0) \|_{s_{\ast }}\leq \beta ^2\mu _{n-1}^{-2-2\tau }\widetilde{ \varepsilon }. \end{equation*} Combining \eqref{II.6}, \eqref{IV.11} and \eqref{IV.18}, we can write \begin{align*} \| (I-S_{n-1})R_{n-1}\|_{s_{\ast }-2-2\tau } &\leq \beta \mu _{n-1}^{-2\tau }\| R_{n-1}\|_{s_{\ast}-2} \leq \beta \mu _{n-1}^{-2\tau }\sum^{n-1}_{j=1}\| r_j\|_{s_{\ast }-2} \\ &\leq \beta \mu _{n-1}^{-2\tau }\widetilde{\varepsilon }a_1 Big(\mu^{s_{\ast }-2-\sigma }+\sum^{n-1}_{j=2} \mu_{j-1}^{s_{\ast }-2-\sigma } \big) \\ &\leq \widetilde{\varepsilon }\beta a_1\mu _{n-1}^{-2\tau }\mu _{n-1}^{s_{\ast }-2-\sigma } \leq \beta a_1\widetilde{\varepsilon }\mu _{n-1}^{-\tau }. \end{align*} These inequalities implyG(w_n)\to 0$in$H^{s_{\ast}-2-2\tau }(\Omega )$as$n\to \infty $. Since$H^{s_{\ast }-2-2\tau }(\Omega )\subset C^2(\overline{\Omega })$and$w_{n\mid \partial \Omega }=0$, we conclude that$G(w)=0$and$w\big|_{ \partial \Omega }=0$. That is$u=\varphi+\varepsilon w$is a solution to the original Monge-Amp\{e}re equation which is by Lemma \ref{lmIII.1} plurisubharmonic since$g$is nonnegative. If we suppose that$\rho =0$, in (A2), then\ the uniqueness of the solution follows immediately from \cite{c1}. \section{Proof of Theorem \ref{thmB}} We shall use the result of Xu and Zuily \cite{x1,x2} that we recall briefly. Let us consider a non linear partial differential equation $$F(x,y,u,\nabla u,D^2u)=0\,,$$ where$F$is$C^{\infty }$. To any solution$u$\ we can associate the vector fields$X_j=\sum_{k} \frac{\partial F}{\partial u_{jk}}\partial_{k}$. Then \begin{theorem}[{[12]}] \label{thmV.1} Suppose$u\in C_{\rm loc}^{\rho }(\Omega )$with$\rho >Max(4,r+2)$for some constant$r\geq 0$and that the brackets of the$X_j$, up to the order$r$, span the tangent space at each point of$\Omega $, then$u$belongs to$C^{\infty }(\Omega )$. \end{theorem} To prove this theorem, it is sufficient to prove that the solution of Theorem \ref{thmA} satisfies Theorem \ref{thmV.1} at any point in$\Sigma $. Suppose$\Sigma =\{0\}$. For$i=1\dots n$; \begin{gather} X_i=\phi ^{ii}\frac{\partial }{\partial x_i} +\sum_{j\neq i,\, j=1} ^n \frac{\phi ^{ij}+\overline{\phi ^{ij}}}{2} \frac{\partial }{\partial x_j} +\sum_{j\neq i,\, j=1}^n \frac{i\phi ^{ij}-i\overline{\phi ^{ij}}}{2} \frac{\partial }{\partial x_{j+n}}, \label{V.1} \\ X_{i+n}=\phi ^{ii}\frac{\partial }{\partial x_{i+n}} +\sum_{j\neq i,\, j=1} ^n \frac{\phi ^{ij}+\overline{\phi ^{ij}}}{2} \frac{\partial }{\partial x_{j+n}} -\sum_{j\neq i,\, j=1} ^n \frac{i\phi ^{ij}-i\overline{\phi ^{ij}}}{2} \frac{\partial }{\partial x_j}. \label{V.2} \end{gather} For computing the Lie algebra generated by the$X_i$, we need the following result. \begin{lemma} \label{lmV.2} For any integer$1\leq m\leq k, \begin{aligned} &(adX_n)^{m-1}[X_n-iX_{2n},X_i-iX_{i+n}] \\ &=\sum^{2n}_{l=1} \sum_{|\beta |\leq m,\, i\neq j} \big[(C_{i\beta p})\partial _x^{\beta }g+\varepsilon d_{pij}\big] \partial _{x_{l}} \\ &\quad + [A_n(\varphi _{i\overline{j}})]^{m-1}A_i(\varphi _{i\overline{j}}) \big[(\partial_{x_n}^{m}g+i\partial _{x_n}^{m-1}\partial _{x_{2n}}g)( \partial _{x_i}+i\partial _{x_i+n})\big] , \end{aligned} \label{V.3} whereC_{i\beta p}$and$d_{pij}$are$C^{s_{\ast }-m,\tau }(\Omega )$(depending on$w$and$\varphi $bounded for$\varepsilon $small enough) satisfying for$|\beta |=m$,$C_{i\beta p}( 0)=0,p=1,\dots ,n$if$n\geq 3$and$C_{i\beta 1}(0)=0$if$n=2$.$A_n=\frac{\partial F}{\partial u_{n\overline{n}}}$and$A_i=\frac{\partial ^2F}{\partial u_{n\overline{n}}\partial u_{i\overline{i}}}$. \end{lemma} \begin{proof} We use induction on the size of the brackets. First we calculate$D_{in}=[X_n+iX_{2n},X_i+iX_{i+n}]$, for$i\leq n-1. \begin{align*} D_{in} & =\Big[\sum_{j=1}^n \Phi ^{nj}\partial _{x_j}+i\sum_{j=1}^n \Phi ^{nj}\partial _{x_{j+n}}, \sum_{l=1}^n \Phi ^{il}\partial _{x_{l}}+i\sum_{l=1}^n \Phi ^{il}\partial _{x_{l+n}}\Big] \\ & =\sum_{l=1}^n \underset{(1)}{ \underbrace{\sum_{j=1}^n \{\Phi ^{nj}\partial _{x_j}(\Phi ^{il})-\Phi ^{ij}\partial _{x_j}(\Phi ^{nl})\}}}\partial _{x_{l}} \\ & \quad +i\underset{(2)}{\sum_{l=1}^n \underbrace{\sum_{j=1}^n \{\Phi ^{nj}\partial _{x_{j+n}}(\Phi ^{il})-\Phi ^{ij}\partial _{x_{j+n}}(\Phi ^{nl})\}}}\partial _{x_{l}} \\ & \quad -\sum_{l=1}^n \underset{(2)}{ \underbrace{\sum_{j=1}^n \{\Phi ^{nj}\partial _{x_{j+n}}(\Phi ^{il})-\Phi ^{ij}\partial _{x_{j+n}}(\Phi ^{nl})\}}}\partial _{x_{l+n}} \\ & \quad +i\underset{(1)}{\sum_{l=1}^n \underbrace{\sum_{j=1}^n \{\Phi ^{nj}\partial _{x_j}(\Phi ^{il})-\Phi ^{ij}\partial _{x_j}(\Phi^{nl})\}}} \partial _{x_{l}+n}, \end{align*} where $$(1)=\sum_{j=1}^n \sum_{p,q=1}^n \{\frac{\partial F}{\partial u_{n\overline{j}}}\frac{ \partial ^2F}{\partial u_{i\overline{l}}\partial u_{p\overline{q}}}-\frac{ \partial F}{\partial u_{i\overline{j}}}\frac{\partial ^2F}{\partial u_{n \overline{l}}\partial u_{p\overline{q}}}\}\partial _{x_j}u_{p \overline{q}}.$$ Using \eqref{II.10}, we get \begin{align*} F.(1) & =\sum_{j=1}^n \sum_{p,q=1}^n {\frac{\partial F}{\partial u_{n\overline{j}}}( \frac{\partial F}{\partial u_{i\overline{l}}}\frac{\partial F}{\partial u_{p \overline{q}}}-\frac{\partial F}{\partial u_{i\overline{q}}}\frac{\partial F }{\partial u_{p\overline{l}}})\partial _{x_j}u_{p\overline{q}}} \\ &\quad -\sum_{j=1}^n \sum_{p,q=1}^n \frac{\partial F}{\partial u_{i\overline{j}}}(\frac{\partial F}{ \partial u_{n\overline{l}}}\frac{\partial F}{\partial u_{p\overline{q}}}- \frac{\partial F}{\partial u_{n\overline{q}}}\frac{\partial F}{\partial u_{p \overline{l}}})\partial _{x_j}u_{p\overline{q}} \\ & =\sum_{j=1}^n \underset{(5)}{ \underbrace{\sum_{p,q=1}^n \frac{\partial F}{\partial u_{p\overline{q}}}\partial _{x_j}u_{p\overline{q}}(\frac{\partial F}{ \partial u_{n\overline{j}}}\frac{\partial F}{\partial u_{i\overline{l}}}- \frac{\partial F}{\partial u_{i\overline{j}}}\frac{\partial F}{\partial u_{n \overline{l}}})}} \\ &\quad +\underset{(6)}{\underbrace{\sum_{j,p,q=1}^n \frac{\partial F}{\partial u_{p\overline{l}}}(\frac{\partial F}{ \partial u_{i\overline{j}}}\frac{\partial F}{\partial u_{n\overline{q}}}- \frac{\partial F}{\partial u_{n\overline{j}}}\frac{\partial F}{\partial u_{i \overline{q}}})\partial _{x_j}u_{p\overline{q}}}}. \end{align*} Using \eqref{II.10}, we have $$(5) = \partial _{x_j}(F)F\frac{\partial ^2F} {\partial u_{n\overline{j}}\partial u_{i\overline{l}}}.$$ Similarly, we prove that \begin{align*} F.(2)& =\sum_{j=1}^n \partial _{x_{j+n}}(F)F\frac{\partial ^2F}{\partial u_{n\overline{j} }\partial u_{i\overline{l}}} \\ &\quad +\underset{(7)}{\underbrace{\sum_{j,p,q=1}^n \frac{\partial F}{\partial u_{p\overline{l}}}(\frac{\partial F}{ \partial u_{i\overline{j}}}\frac{\partial F}{\partial u_{n\overline{q}}}- \frac{\partial F}{\partial u_{n\overline{j}}}\frac{\partial F}{\partial u_{i \overline{q}}})\partial _{x_{j+n}}u_{p\overline{q}}}}. \end{align*} We can easily see that(6)+i(7)=0$, so, $$(1)+i(2)= \sum_{j=1}^n (\partial _{x_j}(F)+i\partial _{x_{j+n}}(F) )\frac{\partial ^2F}{\partial u_{n\overline{j}}\partial u_{i\overline{l}}}$$ and $$D_{in}= \sum_{l=1}^n \sum_{j=1}^n (\partial _{x_j}(f)+i\partial _{x_{j+n}}(f) )\frac{\partial ^2F}{\partial u_{n\overline{j}}\partial u_{i \overline{l}}}[\partial _{x_{l}}+i\partial _{x_{l}+n}] .$$ Since$F$is the determinant function, then,$\frac{\partial F}{\partial u_{i\overline{j}}}$is independent of$u_{i \overline{l}}$and$u_{l\overline{j}}$for$l=1,\dots ,n$. Therefore$\frac{ \partial ^2F}{\partial u_{i\overline{j}}\partial u_{p\overline{q}}}$vanishes unless$i\neq p$,$j\neq q$. So, $$D_{in}=\sum_{(l,j)\neq (i,n),\, l,j\leq n}(\partial _{x_j}(f)+i\partial _{x_{j+n}}(f))\frac{\partial ^2F}{\partial u_{n \overline{j}}\partial u_{i\overline{l}}}[\partial _{x_{l}}+i\partial _{x_{l}+n}] .$$ We have$\varphi _{i\overline{j}}(0)=(1-\delta _i^{n})\sigma _i\delta _i^{j}$; Therefore, if$n\geq 3$and$(l,s)\neq (i,n)$, $$\frac{\partial ^2F}{\partial u_{n\overline{s}}\partial u_{i\overline{l}}} (\varphi _{i\overline{j}})(0)=0.$$ If$n=2$and$l=1$, then$s=1$and we also have $$\frac{\partial ^2F}{\partial u_{2\overline{1}}\partial u_{1\overline{1}}} (\varphi _{i\overline{j}})(0)=0.$$ So, \eqref{V.3} is proved for$m=1$. By a recursion on$m$, we deduce this lemma. \end{proof} On the other hand, we have by \eqref{III.5} $$\begin{gathered} \Phi ^{ij}(\varphi _{i\overline{j}})(0)=0, \quad\mbox{for }(i,j)\neq (n,n), \\ A_n(\varphi _{i\overline{j}})(0)=\prod^{n-1}_{i=1} \sigma _i>0, \\ A_i(\varphi _{i\overline{j}})(0)= \prod^{n-1}_{j\neq i,\, i=1} \sigma _i>0. \end{gathered} \label{V.4}$$ Or by the hypothesis,$\partial _x^{\beta }g(0)=0$for all$|\beta |\sigma -\tau $, \eqref{II.6} \eqref{IV.15}$_n)$and$\beta \geq 1$give $$\| w_n-\widetilde{w}_n\|_s\leq \beta \| w_n\|_s\leq \beta \sqrt{\widetilde{\varepsilon }}\mu _n^{s-\sigma }.$$ (e) \eqref{IV.18}$_n$: By \eqref{IV.12}, we have $$r_n=\underset{(1)}{\underbrace{[L_{G}( w_{n-1})-L_{G}(\widetilde{w}_{n-1})] u_n}}- \underset{(2)}{\underbrace{\theta _{n-1}\triangle u_n}}+\underset{( 3)}{\underbrace{Q_n}}$$ When$n=1$,$(1)=0$. In the case$n\geq 2$, since $$(1)= \int_0^{1}\frac{d}{dt}[L_{G}( \widetilde{w}_{n-1}+t(w_{n-1}-\widetilde{w}_{n-1}))u_n] dt,$$ by \eqref{II.1} and \eqref{IV.17}$_{n-1}$, we get $$|w_{n-1}-\widetilde{w}_{n-1}|_{2}\leq \beta \| w_{n-1}- \widetilde{w}_{n-1}\|_{2+n_{\ast }}\leq 2\beta ^2\sqrt{\widetilde{ \varepsilon }}\mu _{n-1}^{3+n_{\ast }-\sigma }.$$ But$2\beta ^2\sqrt{\widetilde{\varepsilon }}\leq 1$and$3+n_{\ast }\leq 4+2n_{\ast }\leq \sigma $, so,$|w_{n-1}-\widetilde{w}_{n-1}| _{2}\leq 1$. In the same way, \eqref{II.1}, \eqref{IV.5} and \eqref{IV.14}$_n$give $$|u_n|_{2}\leq \beta \| u_n\|_{2+n_{\ast }}\leq \beta \sqrt{\widetilde{\varepsilon }}\mu _{n-1}^{3+n_{\ast }-\sigma }\leq 1.$$ By \eqref{IV.16}$_{n-1}$, we also have$|\widetilde{w}_{n-1}|_{2}\leq 1. Hence, we can apply Proposition \ref{VI.1} to get \begin{align*} \| (1)\|_s\leq &\widetilde{\varepsilon } K_0\{[\| \varphi \|_{s+2}+\| \widetilde{w} _{n-1}\|_{s+2}+\| w_{n-1}\|_{s+2}+1] |w_{n-1}- \widetilde{w}_{n-1}|_{2}|u_n|_{2} \\ &\quad+(\| \varphi \|_{2+n_{\ast }}+\| \widetilde{w} _{n-1}\|_{2+n_{\ast }}+\| w_{n-1}\|_{2+n_{\ast}}+1)\\ &\quad \times (|w_{n-1}-\widetilde{w}_{n-1}|_{2}\| u_n\|_{s+2}+\| w_{n-1}-\widetilde{w}_{n-1}\| _{s+2}|u_n|_{2})\}. \end{align*} Using \eqref{II.3} and \eqref{IV.3}, we get for0\leq s\leq s_{\ast }$, $$\| \varphi \|_{s+2}\leq \beta |\varphi |_{s_{\ast}+2}\leq \beta \mu \leq \mu ^2.$$ By \eqref{II.2}, it suffices to prove \eqref{IV.18}$_n$for$s=0$and$s=s_{\ast }-2$. \noindent Case$s=0$: combining \eqref{II.1}, \eqref{IV.14}$_n$, \eqref{IV.15}$_{n-1}$and \eqref{IV.17}$_{n-1}, we have \begin{align*} \| (1)\|_0&\leq \widetilde{\varepsilon } K_0\{(\mu ^2+2\beta \sqrt{\widetilde{\varepsilon }}+2\sqrt{ \widetilde{\varepsilon }}+1)2\beta ^{3}\widetilde{\varepsilon }\mu _{n-1}^{4+2n_{\ast }-2\sigma } \\ & \quad +(\mu ^2+2\beta \sqrt{\widetilde{\varepsilon }}+2\sqrt{ \widetilde{\varepsilon }}+1)4\beta ^{3}\widetilde{\varepsilon }\mu _{n-1}^{4+n_{\ast }-2\sigma }\}, \end{align*} which using \eqref{IV.5} and\sigma \geq 4+2n_{\ast}\geq 4+n_{\ast }$gives$\| (1)\|_0\leq \widetilde{\varepsilon }K_0\mu_{n-1}^{-\sigma }$. \noindent Case$s=s_{\ast }-2$: \eqref{IV.5} and$s_{\ast }\geq \sigma +\tau $, as in the previous case, imply $$\| (1)\|_{s_{\ast }-2}\leq \widetilde{\varepsilon } K_0\mu _{n-1}^{s_{\ast }-2-\sigma }.$$ By \eqref{II.2}, we obtain for$0\leq s\leq s_{\ast }-2$, $$\| (1)\|_s\leq \beta \widetilde{\varepsilon }K_0\mu _{n-1}^{s-\sigma }.$$ Next, $$\| (2)\|_s\leq \theta _{n-1}\|u_n\|_{s+2}.$$ If$n=1$combining \eqref{IV.5}, \eqref{IV.9} and \eqref{IV.14}$_n$, we obtain $$\| (2)\|_s\leq |G(0)|_0\| u_1\|_{s+2}\leq \widetilde{\varepsilon }\sqrt{ \widetilde{\varepsilon }}\mu ^{s+2-\sigma }\leq \widetilde{\varepsilon }\mu ^{s-\sigma }.$$ In the case$n\geq 2$: \eqref{IV.14}$_n$and \eqref{IV.20}$_{n-1}imply $$\| (2)\|_s\leq a_{3}\widetilde{\varepsilon }\mu _{n-1}^{-2}\mu _{n-1}^{s+2-\sigma }=a_{3}\widetilde{\varepsilon }\mu _{n-1}^{s-\sigma }.$$ Finally, since by \eqref{IV.13}, \begin{align*} (3)&=Q_n=G(w_{n-1}+u_n)-G(w_{n-1})-L_{G}(w_{n-1})u_n \\ & =\int_0^{1}(\int_0^{t}\frac{d}{dh}[L_{G}(w_{n-1}+hu_n)u_n] dh)dt\,. \end{align*} Then, using \eqref{II.1}, \eqref{IV.5} and \eqref{IV.15}_{n-1}$, we obtain $$|w_{n-1}|_{2}\leq \beta \| w_{n-1}\|_{2+n_{\ast }}\leq 2\beta \sqrt{\widetilde{\varepsilon }}\leq 1.$$ Since we proved that$|u_n|_{2}\leq 1, we can apply proposition \ref{VI.1} to have \begin{align*} \| (3)\|_s&\leq \widetilde{\varepsilon } K_0[(\| \varphi \|_{s+2}+\| u_n\| _{s+2}+\| w_{n-1}\|_{s+2}+1)|u_n|_{2}^2\\ &\quad +2|u_n|_{2}\| u_n\|_{s+2}( \| \varphi \|_{2+n_{\ast }}+\| u_n\|_{2+n_{\ast}} +\| w_{n-1}\|_{2+n_{\ast }}+1)]. \end{align*} Combining \eqref{II.1}, \eqref{IV.14}_n$and \eqref{IV.15}$_{n-1}$, we get \noindent For$s=0: \begin{align*} &\| (3)\|_0\\ &\leq \widetilde{\varepsilon }K_0\{(\mu ^2 +\sqrt{\widetilde{\varepsilon}}[\max (\mu ,\mu _{n-1})] ^{2-\sigma } +2\sqrt{\widetilde{\varepsilon }}+1)\beta ^2\widetilde{\varepsilon }[ \max (\mu ,\mu _{n-1})] ^{4+2n_{\ast }-2\sigma } \\ &\quad +8(\mu ^2+\sqrt{\widetilde{\varepsilon }}\beta [\max ( \mu ,\mu _{n-1})] ^{2+n_{\ast }-\sigma }+2\sqrt{\widetilde{ \varepsilon }}+1)\widetilde{\varepsilon }\beta [\max (\mu ,\mu _{n-1})] ^{4+n_{\ast }-2\sigma }\}, \end{align*} which combined with \eqref{IV.5} and\sigma \geq 4+2n_{\ast }$gives $$\| (3)\|_0\leq \widetilde{\varepsilon }K_0 [\max (\mu ,\mu _{n-1})] ^{-\sigma }.$$ For$s=s_{\ast }-2$; since$\sigma \geq 4+2n_{\ast }$, we also get $$\| (3)\|_{s_{\ast }-2}\leq \widetilde{\varepsilon } K_0[\max (\mu ,\mu _{n-1})] ^{s_{\ast }-2-\sigma}.$$ Then \eqref{II.2} shows that, for$0\leq s\leq s_{\ast }-2, $$\| (3)\|_s\leq \beta \widetilde{\varepsilon } K_0[\max (\mu ,\mu _{n-1})] ^{s-\sigma },$$ and we conclude that \begin{align*} \| r_n\|_s & \leq (2\beta K_0+a_{3}) \widetilde{\varepsilon }[\max (\mu ,\mu _{n-1})]^{s-\sigma } \\ & \leq 9K_0\mu ^{5}\widetilde{\varepsilon }[\max (\mu ,\mu _{n-1})] ^{s-\sigma } \\ & =a_1\widetilde{\varepsilon }[\max (\mu ,\mu _{n-1})] ^{s-\sigma }. \end{align*} (f) \eqref{IV.19}_n): By \eqref{IV.10} and \eqref{IV.11}, \begin{align*} g_n& =S_{n-1}R_{n-1}-S_nR_n+(S_{n-1}-S_n)G(0) \\ & =\underset{(4)}{\underbrace{( S_{n-1}R_{n-1}-S_nR_{n-1})}}-\underset{(5)}{ \underbrace{S_nr_n}}+\underset{(6)}{\underbrace{( S_{n-1}-S_n)G(0)}}. \end{align*} Cases=0$: \eqref{II.6}, \eqref{IV.11} and \eqref{IV.18}$_j$,$j\leq n-1, imply \begin{align*} \| (4)\|_0&\leq \| (I-S_{n-1})R_{n-1}\|_0+\| (I-S_n) R_{n-1}\|_0 \\ & \leq \beta \| R_{n-1}\|_{s_{\ast }-2}\mu _{n-1}^{2-s_{\ast }}+\beta \mu _n^{2-s_{\ast }}\| R_{n-1}\|_{s_{\ast}-2} \\ & \leq (\beta a_1\widetilde{\varepsilon }\mu _{n-1}^{2-s_{\ast }}+\beta a_1\widetilde{\varepsilon }\mu _n^{2-s_{\ast }})( \mu ^{s_{\ast }-2-\sigma }+\overset{n-1}{\underset{j=2}{\sum }}\mu _{j-1}^{s_{\ast }-2-\sigma }). \end{align*} Sinces_{\ast }-2>\sigma $and$\beta \leq \mu $, then $$\| (4)\|_0 \leq \beta a_1\widetilde{ \varepsilon }(\mu _{n-1}^{2-s_{\ast }}+\mu _n^{2-s_{\ast }}) \mu _{n-1}^{s_{\ast }-2-\sigma } \leq 2a_1\mu ^2\widetilde{\varepsilon }\mu _n^{-\sigma }.$$ On the other hand, combining \eqref{II.4}, \eqref{IV.18}$_n$,$\sigma 0$, we define the set$S_{\delta }$by $$S_{\delta }=\{x\in \overline{\Omega },d(x,S)<\delta\}.$$ \begin{lemma} \label{lmVII.3} Assume that$S$is a compact$C^{\infty }$submanifold of$\Omega $\ and$ \Omega \backslash S$is connected. Then there exists a function$\mu \in L^{\infty }(\Omega )$and a constant$C>0$such that$\mu =0$on$S$,$m_{\delta }=\inf_{\overline{\Omega }\backslash S_{\delta }}\mu >0$for any sufficiently small$\delta $and $$\int_{\Omega }\mu u^2dx\leq C\big\{\| Qu\|_0\| u\|_0+\frac{1}{2}\sup [b_{ij}^{ij}-2b] \|u\|_0^2\big\}, \label{VII.3}$$ for$u\in C^{s_{\ast },\tau }(\Omega )\cap H_0^{1}(\Omega )$. \end{lemma} \begin{proof} Standard techniques of elliptic operators give $$\int \lambda |Du|^2dx\leq C\big\{\| Qu\|_0\| u\|_0 +\frac{1}{2}\sup [b_{ij}^{ij}-2b]\| u\|_0^2\big\}.$$ Hence, it suffices to show that$\int \mu u^2dx\leq \int \lambda |Du|^2dx$. First, let us fix a point$p\in \overline{\Omega \backslash S}$arbitrarily. By virtue of the fundamental theorem of ordinary differential equations, we can construct a family of curves$c(t,x)\in C^{\infty }( [0,T_p] \times U_p)$such that$c(0,x)=x$,$c(t,x)\notin S$for$00$such that for all$u\in C^{s_{\ast },\tau}(\Omega )$, $$\| [\chi ,Q] u\|_s^2\leq C_s\big(A( 2)\| Qu\|_s\| u\|_s+\sum_{(i,j)\in \Lambda _s} A(i+2)^2\| u\|_j^2\big). \label{VII.6}$$ \end{lemma} \begin{proof} Let us consider a cut-off function$\widetilde{\chi }\in C_0^{\infty}(\Omega )$satisfying$0\leq \widetilde{\chi }\leq 1$and$\widetilde{\chi }=1$on$\cup_i\mathop{\rm supp}\partial _i\chi$, and define an operator$\widetilde{Q}=\widetilde{b}{}^{ij}D_{x_i}D_{x_j}+ \widetilde{b}$by$\widetilde{Q}=\widetilde{\chi }Q$. Since$[\chi ,\widetilde{Q}] u=[\chi ,Q] u$and$\| \widetilde{Q}u\|_s\leq C\| Qu\|_s$, it will suffice to prove \eqref{VII.6} for$\widetilde{Q}$. For$s=0$: The corollary to Lemma 1.7.1 in \cite{o1} shows that $$\big(\sum_{i,j} \widetilde{b}{}^{ij}u_j\big)^2\leq 2A(0)\widetilde{b}{}^{ij}u_iu_j.$$ which gives $$\| [\chi ,\widetilde{Q}] u\|_0^2\leq CA(0)\int \widetilde{b}{}^{ij}u_iu_j +CA(0)^2\|u\|_0^2,$$ Integrating by parts we have $$\int \widetilde{b}{}^{ij}u_iu_j =-\langle \widetilde{Q}u,u\rangle +\frac{1}{2}\langle (\widetilde{b}_{ij}^{ij}-2 \widetilde{b})u,u\rangle \leq \| \widetilde{Q}u\|_0\| u\|_0+CA(2)\| u\|_0^2,$$ which implies \eqref{VII.6}$_0$. Note that \eqref{VII.6}$_{s\geq 1}$follows from \eqref{VII.6}$_0$by induction with respect to$s$\end{proof} \subsection{Estimates near the degenerate points of$L$} For$t\geq t_0\geq 1$, we define $$V_{t}(0)=\{x\in \Omega ,|x_n|<\frac{1}{t} \}\cap B(0,\delta _1).$$ \begin{proposition} \label{propVII.6} For any integer$0\leq s\leq s_{\ast }$and any function$u\in C_0^{s_{\ast },\tau }(V_{t}(0))$, there exists a constant$C_s''=C_s''(n,\Omega,\varphi ,\delta _1)>0$such that \begin{gather} \| u\|_0\leq C_0^{''}t^{-1}\| L_{\nu}u\|_0, \label{VII.7} \\ \| u\|_s\leq C_s''t^{-1}(\| L_{\nu }u\|_s+\underset{(i,j)\in \Lambda _s}{\sum }A( i+2)\| u\|_j),\quad s\geq 1, \label{VII.8} \end{gather} where$\delta _1$is as in Lemma \ref{lmIII.1}. \end{proposition} \begin{proof} Let$v=(T-e^{tx_n})^{-1}u$, and$T>5e$a constant. A direct computation gives \begin{gather*} Qu=(T-e^{tx_n})Qv-te^{tx_n}\{2b^{nj}v_j+tb^{nn}v\}, \\ \int (T-e^{tx_n})^{-1}Qu.v=-I+II-III-IV, \end{gather*} where \begin{gather*} I= \int b^{ij}v_iv_j, \quad II=\frac{1}{2} \int \{b_{ij}^{ij}-2b\}v^2, \\ III=t^2 \int e^{tx_n}b^{nn}(T-e^{tx_n})^{-1}v^2, \quad IV=2t \int e^{tx_n}(T-e^{tx_n})^{-1}vb^{nj}v_j. \end{gather*} Using the Cauchy-Schwartz inequality, we get $$|IV|\leq \int b^{ij}v_iv_j+4t^2 \int e^{2tx_n}(T-e^{tx_n})^{-2}b^{nn}v^2.$$ Since $$e^{tx_n}(T-e^{tx_n})^{-1}-4e^{2tx_n}( T-e^{tx_n})^{-2}=e^{tx_n}(T-e^{tx_n})^{-2}(T-5e^{tx_n}),$$ it follows that $$t^2 \int e^{2tx_n}(T-e^{tx_n})^{-4}( T-5e^{tx_n})b^{nn}u^2\leq - \int (T-e^{tx_n})^{-2}Qu.u-II.$$ Also $$e^{-1}\leq e^{tx_n}\leq e, \quad (T-e^{-1})^{-1}\leq (T-e^{tx_n})^{-1}\leq (T-e)^{-1};$$ therefore, $$C_0t^2\inf_{V_{t}(0)} (b^{nn})\| u\|_0^2 \leq C\big\{\| Qu\|_0\| u\|_0+\frac{1}{2}\underset{V_{t}(0)}{\sup }[ b_{ij}^{ij}-2b] \| u\|_0^2\big\}. \label{VII.9}$$ To prove \eqref{VII.7}, we apply \eqref{VII.8}. So, for$u\in C_0^{s_{\ast },\tau }(V_{t}(0))$, we can write $$t\big\{tC_0\underset{V_{t}(0)}{\inf }(b^{nn})- \frac{C}{2}\sup_{V_{t}(0)}|b_{ij}^{ij}-2b|\big\}\| u\|_0^2\leq C\| Qu\|_0\| u\|_0,$$ with$Q=L_{\nu }$and$b^{nn}=(\Phi ^{nn}+4(\theta +\nu ))$. If$|w|_{3,\tau }\leq 1$,$|x|\leq \delta _0$and$\varepsilon \leq \varepsilon _1$, we have $$\Phi ^{nn}\geq \overset{n-1}{\underset{i=1}{\prod }}\sigma _i-M\delta _1-M\varepsilon _1=\alpha >0.$$ Taking$t\geq t_0=\max (\frac{4(C+1)A(2)}{\alpha C_0},1)$, \eqref{VII.7} is proved. To prove \eqref{VII.8}, we use \eqref{VII.7} and recursion on$s$. We now estimate$\| \chi u\|_s$. \end{proof} \begin{proposition} \label{propVII.7} For any cut-off function$\chi \in C_0^{\infty }(V_{t}( 0))$,$u\in C^{s_{\ast},\tau }(\Omega )\cap H_0^{1}(\Omega )$and$1\leq s\leq s_{\ast }$, \label{VII.10} \| \chi u\|_s\leq 2C_s''(\| L_{\nu}u\|_s+\| [\chi ,L_{\nu }] u\|_s +\sum_{j0$ such that $\mathop{\rm Supp}(1-\chi )\subset \overline{\Omega } \backslash B(0,\delta )$. Let us consider the cut-off functions: $\widetilde{\chi }$, $\widetilde{\widetilde{\chi }}\in C_0^{\infty }(\overline{\Omega } \backslash S)$, $0\leq \widetilde{\chi },\widetilde{\widetilde{\chi}}\leq 1$ and such that $\widetilde{\chi }=1$ on $\mathop{\rm supp}\partial_i\chi$ and $\widetilde{\widetilde{\chi }}=1$ on $\mathop{\rm supp}\widetilde{\chi }$. Let $\mu$ be the function given by Lemma \ref{lmVII.3} ($m_{\delta }$ depends only on $\varphi ,\Omega ,n$). By \eqref{VII.3}, there exists $C_0=C_0(\varphi,\Omega ,n)>0$ such that $$\| (1-\chi )u\|_0^2= \int_{\overline{\Omega } \backslash B(0,\delta )}u^2dx \leq \frac{1}{m_{\delta }} \int \mu u^2dx \leq C_0(\| u\|_0\| L_{\nu }u\|_0+B\| u\|_0^2),$$ where $B=\frac{1}{2}\sup [b_{ij}^{ij}-2b]$. By proposition \ref{VII.1}, $\sum_{ij}b_{ij}^{ij}=0$, and the hypothesis (A2) imply that $-2b\leq \varrho$. So, $B\leq \varrho$ and we have $$\| (1-\chi )u\|_0^2\leq C_1(\|u\|_0\| L_{\nu }u\|_0 +\varrho \| u\|_0^2).$$ Since $\mathop{\rm Supp}\widetilde{\widetilde{\chi }}\subset \overline{\Omega }\backslash\{0\}$, we also have by the same way, $$\| \widetilde{\widetilde{\chi }}u\|_0^2\leq C_1( \| u\|_0\| L_{\nu }u\|_0+\varrho \|u\|_0^2).$$ On the other hand, by \eqref{VII.8}, $$\| \chi u\|_0^2\leq C_{2}\| L_{\nu }\chi u\|_0^2 \leq C_{2}(\| L_{\nu }u\|_0^2+\| [\chi ,L_{\nu }] u\|_0^2),$$ but $\widetilde{\chi }L_{\nu }\widetilde{\widetilde{\chi }}u =\widetilde{\chi}L_{\nu }u$ and $[\chi ,L_{\nu }] u=[\chi ,\widetilde{\chi }L_{\nu }] \widetilde{\widetilde{\chi }}u$. Since $A(2)\leq M_0$ and $\nu \leq 1$, using Lemma \ref{lmVII.5}, we get \begin{align*} \| [\chi ,L_{\nu }] u\|_0^2=\| [\chi ,\widetilde{\chi }L_{\nu }] \widetilde{\widetilde{\chi }}u\|_0^2 &\leq C\big[\| \widetilde{\chi }L_{\nu }\widetilde{ \widetilde{\chi }}u\|_0\| \widetilde{\widetilde{\chi }} u\|_0+(M_0+1)^2\| \widetilde{\widetilde{\chi }}u\|_0^2\big] \\ & \leq C'\big(\| L_{\nu }u\|_0\| \widetilde{\widetilde{\chi }}u\|_0 +\| \widetilde{\widetilde{\chi }}u\|_0^2\big). \end{align*} Combining these inequalities with the fact that $\varrho <<1$, and using the inequality $\alpha \beta \leq \varepsilon \alpha ^2+\frac{1}{\varepsilon }\beta ^2$, we get \eqref{III.13} \end{proof} \begin{proof}[Proof of \eqref{III.14}] We have $\mathop{\rm supp}(1-\chi)\subset \overline{\Omega }\backslash B(0,\delta )$. Or $\varphi$ is strictly plurisubharmonic on $E=\mathop{\rm supp}(1-\chi )$, then for $\varepsilon \leq \varepsilon _{4}$ small enough, $L$ is uniformly elliptic on $E$. Using \eqref{VII.1} and the estimation $A(2)\leq M_0$, we have $$\| (1-\chi )u\|_1\leq C_1'(\| L_{\nu }u\|_0+(M_0+1)\| u\|_0 +\| [\chi ,L_{\nu }] u\|_0).$$ Applying Lemma \ref{lmVII.5}, we get $$\| [\chi ,L_{\nu }] u\|_0\leq C_0(\|L_{\nu }u\|_0+(M_0+1)\| u\|_0),$$ therefore, $$\| (1-\chi )u\|_1\leq C_1(M_0)(\| L_{\nu }u\|_0+\| u\|_0).$$ On the other hand, since $A(2)\leq M_0$, we get using \eqref{VII.10}, $$\| \chi u\|_1\leq C_1(M_0)(\| L_{\nu }u\|_1+\| [\chi ,L_{\nu }] u\|_1+\| u\|_0)\,.$$ But $\widetilde{\chi }L_{\nu }\widetilde{\widetilde{\chi }}u=\widetilde{\chi }L_{\nu }u$ and $[\chi ,L_{\nu }] u=[\chi ,\widetilde{\chi }L_{\nu }] \widetilde{\widetilde{\chi }}u$, so, since $A(2)\leq M_0$, Lemma \ref{lmVII.5} gives \begin{align*} \| [\chi ,L_{\nu }] u\|_1& \leq C_1( \| \widetilde{\chi }L_{\nu }\widetilde{\widetilde{\chi }}u\|_1 +(1+M_0)\| \widetilde{\widetilde{\chi }}u\|_1) \\ & \leq C_1(\| L_{\nu }u\|_1+(1+M_0) \| \widetilde{\widetilde{\chi }}u\|_1). \end{align*} Since $L_{\nu }$ is uniformly elliptic on $\mathop{\rm supp}\widetilde{\widetilde{\chi }}$ and $A(2)\leq M_0$, then we have by \eqref{VII.1}, $$\| \widetilde{\widetilde{\chi }}u\|_1 \leq C_1'(\| L_{\nu }u\|_0+(M_0+1)\|u\|_0 +\| [\widetilde{\widetilde{\chi }},L_{\nu }]u\|_0),$$ which using \eqref{VII.6} gives $$\| \widetilde{\widetilde{\chi }}u\|_1\leq C_1( M_0)(\| L_{\nu }u\|_1+\| u\|_0).$$ Combining these inequalities, we get \eqref{III.14}. \end{proof} The proof of \eqref{III.15} is identical to that of \eqref{III.14} using the inequalities \eqref{VII.1}, \eqref{VII.2}, \eqref{VII.6}, and \eqref{VII.10}. \subsection*{Acknowledgments} The author wishes to thank the anonymous referee for his or her helpful comments. \begin{thebibliography}{99} \bibitem{a1} S. 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