\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 81, pp. 1--17.\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 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/81\hfil Schouten equations with boundary] {Schouten tensor equations in conformal geometry with prescribed boundary metric} \author[O. C. Schn\"urer\hfil EJDE-2005/81\hfilneg] {Oliver C. Schn\"urer} \address{Oliver C. Schn\"urer \hfill\break Freie Universit\"at Berlin, Arnimallee 2-6, 14195 Berlin, Germany} \email{Oliver.Schnuerer@math.fu-berlin.de} \date{} \thanks{Submitted March 15, 2004. Published July 15, 2005.} \subjclass[2000]{53A30; 35J25; 58J32} \keywords{Schouten tensor; fully nonlinear equation; conformal geometry; \hfill\break\indent Dirichlet boundary value problem} \begin{abstract} We deform the metric conformally on a manifold with boundary. This induces a deformation of the Schouten tensor. We fix the metric at the boundary and realize a prescribed value for the product of the eigenvalues of the Schouten tensor in the interior, provided that there exists a subsolution. This problem reduces to a Monge-Amp\ere equation with gradient terms. The main issue is to obtain a priori estimates for the second derivatives near the boundary. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{notation}[theorem]{Notation} \newcommand{\abs}[1]{|#1|} \allowdisplaybreaks \section{Introduction} Let $(M^n,g_{ij})$ be an $n$-dimensional Riemannian manifold, $n\ge3$. The Schouten tensor $(S_{ij})$ of $(M^n,g_{ij})$ is defined as $$S_{ij}=\tfrac1{n-2}\big(R_{ij}-\tfrac1{2(n-1)}Rg_{ij}\big),$$ where $(R_{ij})$ and $R$ denote the Ricci and scalar curvature of $(M^n,g_{ij})$, respectively. Consider the manifold $(\tilde M^n,\,\tilde g_{ij}) =(M^n,\,e^{-2u}g_{ij})$, where we have used $u\in C^2(M^n)$ to deform the metric conformally. The Schouten tensors $S_{ij}$ of $g_{ij}$ and $\tilde S_{ij}$ of $\tilde g_{ij}$ are related by $$\tilde S_{ij}=u_{ij}+u_iu_j-\tfrac12\vert\nabla u\vert^2 g_{ij}+S_{ij},$$ where indices of $u$ denote covariant derivatives with respect to the background metric $g_{ij}$, moreover $\vert\nabla u\vert^2=g^{ij}u_iu_j$ and $(g^{ij}) =(g_{ij})^{-1}$. Eigenvalues of the Schouten tensor are computed with respect to the background metric $g_{ij}$, so the product of the eigenvalues of the Schouten tensor $(\tilde S_{ij})$ equals a given function $s:M^n\to\mathbb{R}$, if $$\label{s eqn} \frac{\displaystyle\det(u_{ij}+u_iu_j-\tfrac12\vert\nabla u\vert^2 g_{ij}+S_{ij})}{\displaystyle e^{-2nu}\det(g_{ij})}=s(x).$$ We say that $u$ is an admissible solution for \eqref{s eqn}, if the tensor in the determinant in the numerator is positive definite. At admissible solutions, \eqref{s eqn} becomes an elliptic equation. As we are only interested in admissible solutions, we will always assume that $s$ is positive. Let now $M^n$ be compact with boundary and $\underline{u}:M^n\to\mathbb{R}$ be a smooth (up to the boundary) admissible subsolution to \eqref{s eqn} $$\label{sub sol} \frac{\displaystyle\det(\underline{u}_{ij}+\underline{u}_i\underline{u}_j -\tfrac12\vert\nabla \underline{u}\vert^2 g_{ij}+S_{ij})} {\displaystyle e^{-2n\underline{u}}\det(g_{ij})}\ge s(x).$$ Assume that there exists a supersolution $\overline{u}$ to \eqref{s eqn} fulfilling some technical conditions specified in Definition \ref{sup def}. Assume furthermore that $M^n$ admits a strictly convex function $\chi$. Without loss of generality, we have $\chi_{ij}\ge g_{ij}$ for the second covariant derivatives of $\chi$ in the matrix sense. The conditions of the preceding paragraph are automatically fulfilled if $M^n$ is a compact subset of flat $\mathbb{R}^n$ and $\underline{u}$ fulfills \eqref{sub sol} and in addition $\det(\underline{u}_{ij})\ge s(x)e^{-2n\underline{u}}\det(g_{ij})$ with $\underline{u}_{ij}>0$ in the matrix sense. Then Lemma \ref{sup sol exi lem} implies the existence of a supersolution and we may take $\chi=\vert x\vert^2$. We impose the boundary condition that the metric $\tilde g_{ij}$ at the boundary is prescribed, $$\tilde g_{ij}=e^{-2\underline{u}}g_{ij}\quad\text{on } \partial M^n.$$ Assume that all data are smooth up to the boundary. We prove the following \begin{theorem} Let $M^n$, $g_{ij}$, $\underline{u}$, $\overline{u}$, $\chi$, and $s$ be as above. Then there exists a metric $\tilde g_{ij}$, conformally equivalent to $g_{ij}$, with $\tilde g_{ij} =e^{-2\underline{u}}g_{ij}$ on $\partial M^n$ such that the product of the eigenvalues of the Schouten tensor induced by $\tilde g_{ij}$ equals $s$. \end{theorem} This follows readily from the next statement. \begin{theorem}\label{main thm} Under the assumptions stated above, there exists an admissible function $u\in C^0(M^n)\cap C^\infty(M^n\setminus\partial M^n)$ solving \eqref{s eqn} such that $u=\underline{u}$ on $\partial M^n$. \end{theorem} Recently, in a series of papers, Jeff Viaclovsky studied conformal deformations of metrics on closed manifolds and elementary symmetric functions $S_k$, $1\le k\le n$, of the eigenvalues of the associated Schouten tensor, see e.\,g.\ \cite{ViaclovskyCAG} for existence results. Pengfei Guan, Jeff Viaclovsky, and Guofang Wang provide an estimate that can be used to show compactness of manifolds with lower bounds on elementary symmetric functions of the eigenvalues of the Schouten tensor \cite{GVWSchoutenCompact}. An equation similar to the Schouten tensor equation arises in geometric optics \cite{GuanWang,WangInverse}. Xu-Jia Wang proved the existence of solutions to Dirichlet boundary value problems for such an equation, similar to \eqref{s eqn}, provided that the domains are small. In \cite{OSRefl} we provide a transformation that shows the similarity between reflector and Schouten tensor equations. For Schouten tensor equations, Dirichlet and Neumann boundary conditions seem to be geometrically meaningful. For reflector problems, solutions fulfilling a so-called second boundary value condition describe the illumination of domains. Pengfei Guan and Xu-Jia Wang obtained local second derivative estimates \cite{GuanWang}. This was extended by Pengfei Guan and Guofang Wang to local first and second derivative estimates in the case of elementary symmetric functions $S_k$ of the Schouten tensor of a conformally deformed metric \cite{GuanLocalSchouten}. We will use the following special case of it \begin{theorem}[Pengfei Guan and Xu-Jia Wang/Pengfei Guan and Guofang Wang] \label{pfg} Suppose $f$ is a smooth function on $M^n\times\mathbb{R}$. Let $u\in C^4$ be an admissible solution of $$\log\det(u_{ij}+u_iu_j -\tfrac12\lvert\nabla u\rvert^2g_{ij}+S_{ij})=f(x,\,u)$$ in $B_r$, the geodesic ball of radius $r$ in a Riemannian manifold $(M^n,\,g_{ij})$. Then, there exists a constant $c=c(\Vert u\Vert_{C^0},\, f,\,S_{ij},\,r,\,M^n)$, such that $$\Vert u\Vert_{C^2(B_{r/2})}\le c.$$ \end{theorem} Boundary-value problems for Monge-Amp\ere equations have been studied by Luis Caffarelli, Louis Nirenberg, and Joel Spruck in \cite{CNS1} an many other people later on. For us, those articles using subsolutions as used by Bo Guan and Joel Spruck will be especially useful \cite{BGuanTrans,GuanSpruck,NehringCrelle,OSMathZ}. There are many papers addressing Schouten tensor equations on compact manifolds, see e.\,g.\ \cite{BrendleViaclovsky, ChangRad, ChangAnn, GVWSchoutenCompact, GuanWangSchoutenFlow, GuanLocalSchouten, GuofangPengfeiInequ, GurskyViaclovsky, GurskyViaclovskyFour, GurskyViaclovskyNegative, GurskyViaclovskyInvariant, HanLocal, LiCR, LiLiouvilleI, LiLiouvilleII, ALiYYLiSchouten, LiHarnack, LiCPAM,LiLiouville, LiMovingSperes, YanYanLiICM2002, MazzeoPacardSchouten, ViaclovskyCAG}. There, the authors consider topological and geometrical obstructions to solutions, the space of solutions, Liouville properties, Harnack inequalities, Moser-{}Tru\-din\-ger inequalities, existence questions, local estimates, local behavior, blow-up of solutions, and parabolic and variational approaches. If we consider the sum of the eigenvalues of the Schouten tensor, we get the Yamabe equation. The Yamabe problem has been studied on manifolds with boundary, see e.\,g.\ \cite{AmbrosettiYamabe, BrendleAsian, EscobarYamabe, LiYamabeBoundary, MaYamabeDirichlet}, and in many more papers on closed manifolds. The Yamabe problem gives rise to a quasilinear equation. For a fully nonlinear equation, we have to apply different methods. The present paper addresses analytic aspects that arise in the proof of a priori estimates for an existence theorem. This combines methods for Schouten tensor equations, e.\,g.\ \cite{GuanLocalSchouten,ViaclovskyCAG}, with methods for curvature equations with Dirichlet boundary conditions, e.\,g.\ \cite{CNS1,BGuanTrans}. We can also solve Equation \eqref{s eqn} on a non-compact manifold $(M^n,\,g_{ij})$. \begin{corollary} \label{coro1.4} Assume that there are a sequence of smooth bounded domains $\Omega_k$, $k\in\mathbb{N}$, exhausting a non-compact manifold $M^n$, and functions $\underline{u}$, $\overline{u}$, $s$, and $\chi$, that fulfill the conditions of Theorem \ref{main thm} on each $\Omega_k$ instead of $M^n$. Then there exists an admissible function $u\in C^\infty (M^n)$ solving \eqref{s eqn}. \end{corollary} \begin{proof} Theorem \ref{main thm} implies that equation \eqref{s eqn} has a solution $u_k$ on every $\Omega_k$ fulfilling the boundary condition $u=\underline{u}$ on $\partial\Omega_k$. In $\Omega_k$, we have $\underline{u}\le u_k\le\overline{u}$, so Theorem \ref{pfg} implies locally uniform $C^2$-estimates on $u_k$ on any domain $\Omega\subset M^n$ for $k>k_0$, if $\Omega\Subset\Omega_{k_0}$. The estimates of Krylov, Safonov, Evans, and Schauder imply higher order estimates on compact subsets of $M^n$. Arzel\`a-Ascoli yields a subsequence that converges to a solution. \end{proof} Note that either $s(x)$ is not bounded below by a positive constant or the manifold with metric $e^{-2u}g_{ij}$ is non-complete. Otherwise, \cite{GVWSchoutenCompact} implies a positive lower bound on the Ricci tensor, i.\,e.\ $\tilde R_{ij}\ge\frac1c\tilde g_{ij}$ for some positive constant $c$. This yields compactness of the manifold \cite{GroKliMey}. It is a further issue to solve similar problems for other elementary symmetric functions of the Schouten tensor. As the induced mean curvature of $\partial M^n$ is related to the Neumann boundary condition, this is another natural boundary condition. To show existence for a boundary value problem for fully nonlinear equations like Equation \eqref{s eqn}, one usually proves $C^2$-estimates up to the boundary. Then standard results imply $C^k$-bounds for $k\in\mathbb{N}$ and existence results. In our situation, however, we don't expect that $C^2$-estimates up to the boundary can be proved. This is due to the gradient terms appearing in the determinant in \eqref{s eqn}. It is possible to overcome these difficulties by considering only small domains \cite{WangInverse}. Our method is different. We regularize the equation and prove full regularity up to the boundary for the regularized equation. Then we use the fact, that local interior $C^k$-estimates (Theorem \ref{pfg}) can be obtained independently of the regularization. Moreover, we can prove uniform $C^1$-estimates. Thus we can pass to a limit and get a solution in $C^0(M^n)\cap C^\infty(M^n\setminus\partial M^n)$. To be more precise, we rewrite \eqref{s eqn} in the form $$\label{f eqn} \log\det(u_{ij}+u_iu_j-\tfrac12\vert\nabla u \vert^2g_{ij}+S_{ij})=f(x,u),$$ where $f\in C^\infty(M^n\times\mathbb{R})$. Our method can actually be applied to any equation of that form provided that we have sub- and supersolutions. Thus we consider in the following equations of the form \eqref{f eqn}. Equation \eqref{f eqn} makes sense in any dimension provided that we replace $S_{ij}$ by a smooth tensor. In this case Theorem \ref{main thm} is valid in any dimension. Note that even without the factor $\frac1{n-2}$ in the definition of the Schouten tensor, our equation is not elliptic for $n=2$ for any function $u$ as the trace $g^{ij}(R_{ij}-\frac12Rg_{ij})$ equals zero, so there has to be a non-positive eigenvalue of that tensor. Let $\psi:M^n\to[0,1]$ be smooth, $\psi=0$ in a neighborhood of the boundary. Then our strategy is as follows. We consider a sequence $\psi_k$ of those functions that fulfill $\psi_k(x)=1$ for $\mathop{\rm dist}(x,\partial M^n)>\tfrac2k$, $k\in\mathbb{N}$, and boundary value problems $$\label{psi eqn} \begin{gathered} \log\det(u_{ij}+\psi u_iu_j-\tfrac12\psi \vert\nabla u\vert^2g_{ij}+T_{ij}) =f(x,u)\quad \mbox{in}M^n,\\ u=\underline{u} \quad \mbox{on }\partial M^n. \end{gathered}$$ We dropped the index $k$ to keep the notation simple. The tensor $T_{ij}$ coincides with $S_{ij}$ on $\left\{x\in M^n:\mathop{\rm dist}(x,\partial M^n)>\tfrac2k\right\}$ and interpolates smoothly to $S_{ij}$ plus a sufficiently large constant multiple of the background metric $g_{ij}$ near the boundary. For the precise definitions, we refer to Section \ref{upp barr}. Our sub- and supersolutions act as barriers and imply uniform $C^0$-estimates. We prove uniform $C^1$-estimates based on the admissibility of solutions. Admissibility means here that $u_{ij}+\psi u_iu_j-\tfrac12\psi \vert\nabla u\vert^2+T_{ij}$ is positive definite for those solutions. As mentioned above, we can't prove uniform $C^2$-estimates for $u$, but we get $C^2$-estimates that depend on $\psi$. These estimates guarantee, that we can apply standard methods (Evans-Krylov-Safonov theory, Schauder estimates for higher derivatives, and mapping degree theory for existence, see e.\,g.\ \cite{GT,BGuanTrans,LiExist,Taylor3}) to prove existence of a smooth admissible solution to \eqref{psi eqn}. Then we use Theorem \ref{pfg} to get uniform interior a priori estimates on compact subdomains of $M^n$ as $\psi=1$ in a neighborhood of these subdomains for all but a finite number of regularizations. These a priori estimates suffice to pass to a subsequence and to obtain an admissible solution to \eqref{f eqn} in $M^n\setminus\partial M^n$. As $u^k=u=\underline{u}$ for all solutions $u^k$ of the regularized equation and those solutions have uniformly bounded gradients, the boundary condition is preserved when we pass to the limit and we obtain Theorem \ref{main thm} provided that we can prove $\| u^k\|_{C^1(M^n)}\le c$ uniformly and $\| u^k\|_{C^2(M^n)} \le c(\psi)$. These estimates are proved in Lemmata \ref{unif C1} and \ref{int C2}, the crux of this paper. \begin{proof}[Proof of Theorem \ref{main thm}] For admissible smooth solutions to \eqref{psi eqn}, the results of Section \ref{C0 sec} imply uniform $C^0$-estimates and Section \ref{C1 sec} gives uniform $C^1$-estimates. The $C^2$-estimates proved in Section \ref{C2 sec} depend on the regularization. The logarithm of the determinant is a strictly concave function on positive definite matrices, so the results of Krylov, Safonov, Evans, \cite[14.13/14]{Taylor3}, and Schauder estimates yield $C^l$-estimates on $M^n$, $l\in\mathbb{N}$, depending on the regularization. Once these a priori estimates are established, existence of a solution $u^k$ for the regularized problem \eqref{psi eqn} follows as in \cite[Section 2.2]{BGuanTrans}. On a fixed bounded subdomain $\Omega_\varepsilon:= \{x:\mathop{\rm dist}(x,\, \partial M^n)\ge\varepsilon\}$, $\varepsilon>0$, however, Theorem \ref{pfg} implies uniform $C^2$-estimates for all $k\ge k_0=k_0(\varepsilon)$. The estimates of Krylov, Safonov, Evans, and Schauder yield uniform $C^l$-estimates on $\Omega_{2\varepsilon}$, $l\in\mathbb{N}$. Recall that we have uniform Lipschitz estimates. So we find a convergent sequence of solutions to our approximating problems. The limit $u$ is in $C^{0,\,1}(M^n)\cap C^\infty(M^n\setminus\partial M^n)$. \end{proof} The rest of the article is organized as follows. We introduce supersolutions and some notation in Section \ref{upp barr}. We mention $C^0$-estimates in Section \ref{C0 sec}. In Section \ref{C1 sec}, we prove uniform $C^1$-estimates. Then the $C^2$-estimates proved in Section \ref{C2 sec} complete the a priori estimates and the proof of Theorem \ref{main thm}. The author wants to thank J\"urgen Jost and the Max Planck Institute for Mathematics in the Sciences for support and Guofang Wang for interesting discussions about the Schouten tensor. \section{Supersolutions and Notation} \label{upp barr} Before we define a supersolution, we explain more explicitly, how we regularize the equation. For fixed $k\in\mathbb{N}$ we take $\psi_k$ such that $$\psi_k(x)=\begin{cases} 0 & \mathop{\rm dist}(x,\partial M^n)<\tfrac 1k,\\ 1 & \mathop{\rm dist}(x,\partial M^n)>\tfrac 2k \end{cases}$$ and $\psi_k$ is smooth with values in $[0,1]$. Again, we drop the index $k$ to keep the notation simple. We fix $\lambda\ge0$ sufficiently large so that $$\label{mod sub sol} \log\det(\underline{u}_{ij}+\psi\underline{u}_i\underline{u}_j-\tfrac12 \psi\vert\nabla\underline{u}\vert^2g_{ij}+S_{ij}+\lambda(1-\psi) g_{ij})\ge f(x,\underline{u})$$ for any $\psi=\psi_k$, independent of $k$. As $\log\det(\cdot)$ is a concave function on positive definite matrices, \eqref{mod sub sol} follows for $k$ sufficiently large, if $$\log\det(\underline{u}_{ij}+\underline{u}_i\underline{u}_j-\tfrac12 \vert\nabla\underline{u}\vert^2g_{ij}+S_{ij})\ge f(x,\underline{u}) \quad\text{on }M^n$$ and $$\log\det(\underline{u}_{ij}+S_{ij}+\lambda g_{ij}) \ge f(x,\underline{u})\quad\text{near~}\partial M^n,$$ provided that the arguments of the determinants are positive definite. We define \begin{definition}[supersolution]\label{sup def} \rm A smooth function $\overline{u}:M^n\to\mathbb{R}$ is called a supersolution, if $\overline{u}\ge\underline{u}$ and for any $\psi$ as considered above, $$\log\det(\overline{u}_{ij}+\psi\overline{u}_i\overline{u}_j-\tfrac12 \psi\vert\nabla\overline{u}\vert^2g_{ij}+S_{ij}+\lambda(1-\psi) g_{ij})\le f(x,\underline{u})$$ holds for those points in $M^n$ for which the tensor in the determinant is positive definite. \end{definition} \begin{lemma}\label{sup sol exi lem} If $M^n$ is a compact subdomain of flat $\mathbb{R}^n$, the subsolution $\underline{u}$ fulfills \eqref{sub sol} and in addition $$\det(\underline{u}_{ij})\ge s(x)e^{-2n\underline{u}} \det(g_{ij})$$ holds, where $\underline{u}_{ij}>0$ in the matrix sense, then there exists a supersolution. \end{lemma} \begin{proof} In flat $\mathbb{R}^n$, we have $S_{ij}=0$. The inequality $$\label{middle} \frac{\displaystyle\det(\underline{u}_{ij}+\psi\underline{u}_i\underline{u}_j -\tfrac12\psi\vert\nabla \underline{u}\vert^2 g_{ij})} {\displaystyle e^{-2n\underline{u}}\det(g_{ij})}\ge s(x)$$ is fulfilled if $\psi$ equals $0$ or $1$ by assumption. As above, \eqref{middle} follows for any $\psi\in[0,1]$. Thus \eqref{mod sub sol} is fulfilled for $\lambda=0$. Let $\overline{u}=\sup\limits_{M^n}\underline{u}+1+\varepsilon\vert x\vert^2$ for $\varepsilon>0$. It can be verified directly that $\overline{u}$ is a supersolution for $\varepsilon>0$ fixed sufficiently small. \end{proof} Our results can be extended to topologically more interesting manifolds, that may not allow for a globally defined convex function. \begin{remark} \rm Assume that all assumptions of Theorem \ref{main thm} are fulfilled, but the convex function $\chi$ is defined only in a neighborhood of the boundary. Then the conclusion of Theorem \ref{main thm} remains true. \end{remark} \begin{proof} We have employed the globally defined convex function $\chi$ only to prove interior $C^2$-estimates for the regularized problems. On the set $$\left\{x:\mathop{\rm dist}(x,\,\partial M^n)\ge\varepsilon\right\},\quad \varepsilon>0,$$ Theorem \ref{pfg} implies $C^2$-estimates. In a neighborhood $$U=\left\{x:\mathop{\rm dist}(x,\,\partial M^n)\le2\varepsilon\right\}$$ of the boundary, we can proceed as in the proof of Lemma \ref{int C2}. If the function $W$ defined there attains its maximum over $U$ at a point $x$ in $\partial U\cap M^n$, i.\,e. $\mathop{\rm dist}(x,\,\partial M^n)=2\varepsilon$, $W$ is bounded and $C^2$-estimates follow, otherwise, we may proceed as in Lemma \ref{int C2}. \end{proof} \subsection*{Notation} We set \begin{align*} w_{ij}=&u_{ij}+\psi u_iu_j-\tfrac12\psi\vert\nabla u\vert^2 g_{ij}+S_{ij}+\lambda(1-\psi) g_{ij}\\ =&u_{ij}+\psi u_iu_j-\tfrac12\psi\vert\nabla u\vert^2 g_{ij}+T_{ij} \end{align*} and use $(w^{ij})$ to denote the inverse of $(w_{ij})$. The Einstein summation convention is used. We lift and lower indices using the background metric. Vectors of length one are called directions. Indices, sometimes preceded by a semi-colon, denote covariant derivatives. We use indices preceded by a comma for partial derivatives. Christoffel symbols of the background metric are denoted by $\Gamma^k_{ij}$, so $u_{ij}=u_{;ij}=u_{,ij}-\Gamma^k_{ij}u_k$. Using the Riemannian curvature tensor $(R_{ijkl})$, we can interchange covariant differentiation $$\label{interchange} \begin{split} u_{ijk}=&u_{kij}+u_a g^{ab}R_{bijk},\\ u_{iklj}=&u_{ikjl}+u_{ka}g^{ab}R_{bilj} +u_{ia}g^{ab}R_{bklj}. \end{split}$$ We write $f_z=\frac{\partial f}{\partial u}$ and $\mathop{\rm tr}w=w^{ij}g_{ij}$. The letter $c$ denotes estimated positive constants and may change its value from line to line. It is used so that increasing $c$ keeps the estimates valid. We use $(c_j)$, $(c^k)$, \ldots{} to denote estimated tensors. \section{Uniform $C^0$-Estimates} \label{C0 sec} The techniques of this section are quite standard, but they simplify the $C^0$-estimates used before for Schouten tensor equations, see \cite[Proposition 3]{ViaclovskyCAG}. Here, we interpolate between the expressions for the Schouten tensors rather than between the functions inducing the conformal deformations. We wish to show that we can apply the maximum principle or the Hopf boundary point lemma at a point, where a solution $u$ touches the subsolution from above or the supersolution from below. Note that $u$ can touch $\overline{u}$ from below only in those points, where $\overline{u}$ is admissible. We did not assume that the upper barrier is admissible everywhere. But at those points, where it is not admissible, $u$ cannot touch $\overline{u}$ from below. More precisely, at such a point, we have $\nabla u=\nabla\overline{u}$ and $D^2 u\le D^2\overline{u}$. If $\overline{u}$ is not admissible there, we find $\xi\in\mathbb{R}^n$ such that $0\ge(\overline{u}_{ij}+\psi\overline{u}_i\overline{u}_j-\frac12\psi\abs{\nabla\overline{u}}g_{ij} +T_{ij})\xi^i\xi^j$. This implies that $0\ge(u_{ij}+\psi u_iu_j-\frac12\psi\abs{\nabla u}g_{ij}+T_{ij}) \xi^i\xi^j$, so $u$ is not admissible there, a contradiction. The idea, that the supersolution does not have to be admissible, appears already in \cite{CGScalar}. Without loss of generality, we may assume that $u$ touches $\underline{u}$ from above. Here, touching means $u=\underline{u}$ and $\nabla u=\nabla\underline{u}$ at a point, so our considerations include the case of touching at the boundary. It suffices to prove an inequality of the form $$\label{ell inequ} 0\le a^{ij}(\underline{u}-u)_{ij}+b^i(u-\underline{u})_i+d(\underline{u}-u)$$ with positive definite $a^{ij}$. The sign of $d$ does not matter as we apply the maximum principle only at points, where $u$ and $\underline{u}$ coincide. Define $$S_{ij}^\psi[v]=v_{ij}+\psi v_iv_j-\tfrac12\psi\vert\nabla v\vert^2g_{ij}+T_{ij}.$$ We apply the mean value theorem and get for a symmetric positive definite tensor $a^{ij}$ and a function $d$ \begin{align*} 0\le&\log\det S^\psi_{ij}[\underline{u}]-\log\det S^\psi_{ij}[u] -f(x,\underline{u})+f(x,u)\\ =&\int\limits_0^1\frac{d}{dt}\log\det\left\{ tS^\psi_{ij}[\underline{u}]+(1-t)S^\psi_{ij}[u]\right\}dt -\int\limits_0^1\frac d{dt}f(x,t\underline{u}+(1-t)u)dt\\ =&a^{ij}((\underline{u}_{ij}+\psi\underline{u}_i\underline{u}_j -\tfrac12\psi\vert\nabla\underline{u}\vert^2g_{ij}) -(u_{ij}+\psi u_iu_j-\tfrac12\psi \vert\nabla u\vert^2g_{ij} ))\\ &+d\cdot(\underline{u}-u). \end{align*} The first integral is well-defined as the set of positive definite tensors is convex. We have $\vert\nabla\underline{u}\vert^2 -\vert\nabla u\vert^2=\langle\nabla(\underline{u}-u),\nabla(\underline{u}+u)\rangle$ and \begin{align*} a^{ij}(\underline{u}_i\underline{u}_j-u_iu_j)=&a^{ij}\int\limits_0^1\frac d{dt} ((t\underline{u}_i+(1-t)u_i)(t\underline{u}_j+(1-t)u_j))dt\\ =&2a^{ij}\int\limits_0^1(t\underline{u}_j+(1-t)u_j)dt \cdot(\underline{u}-u)_i, \end{align*} so we obtain an inequality of the form \eqref{ell inequ}. Thus, we may assume in the following that we have $\underline{u}\le u\le\overline{u}$. %%%%%%%%%%%%%%%%%%%%%%%%% \section{Uniform $C^1$-Estimates} \label{C1 sec} \begin{lemma}\label{unif C1} An admissible solution of \eqref{psi eqn} has uniformly bounded gradient. \end{lemma} \begin{proof} We apply a method similar to \cite[Lemma 4.2]{OSMathZ}. Let $$W=\tfrac12\log\vert\nabla u\vert^2+\mu u$$ for $\mu\gg1$ to be fixed. Assume that W attains its maximum over $M^n$ at an interior point $x_0$. This implies at $x_0$ $$0=W_i=\frac{u^ju_{ji}}{\vert\nabla u\vert^2}+\mu u_i$$ for all $i$. Multiplying with $u^i$ and using admissibility gives \begin{align*} 0=&u^iu^ju_{ij}+\mu\vert\nabla u\vert^4\\ \ge&-\psi\vert\nabla u\vert^4+\tfrac12\psi\vert\nabla u\vert^4 -c\vert\nabla u\vert^2-\lambda\vert\nabla u\vert^2 +\mu\vert\nabla u\vert^4. \end{align*} The estimate follows for sufficiently large $\mu$ as $\lambda$, see \eqref{mod sub sol}, does not depend on $\psi$. If $W$ attains its maximum at a boundary point $x_0$, we introduce normal coordinates such that $W_n$ corresponds to a derivative in the direction of the inner unit normal. We obtain in this case $W_i=0$ for $i0$ to be fixed sufficiently small, where $\psi=0$. Thus the equation takes the form $$\label{bdry eqn} \log\det(u_{ij}+T_{ij})=\log\det(u_{,ij} -\Gamma^k_{ij}u_k+T_{ij})=f(x,u).$$ Assume furthermore that $\delta>0$ is chosen so small that the distance function to $\partial M^n$ is smooth in $\Omega_\delta$. The constant $\delta$, introduced here, depends on $\psi$ and tends to zero as the support of $\psi$ tends to $\partial M^n$. We differentiate the boundary condition tangentially \label{bdry d1} 0=(u-\underline{u})_{,t}(\hat x, \omega(\hat x)) +(u-\underline{u})_{,n}(\hat x, \omega(\hat x)) \omega_{,t}(\hat x),\quad t0$and$\mu\gg1$to be chosen $$\vartheta:=(u-\underline{u})+\alpha d-\mu d^2.$$ The function$\vartheta$will be the main part of our barrier. As$\underline{u}$is admissible, there exists$\varepsilon>0$such that $$\underline{u}_{,ij}-\Gamma^l_{ij}\underline{u}_l+T_{ij}\ge 3\varepsilon g_{ij}.$$ We apply the definition of$L$$$\label{L theta est} \begin{split} L\vartheta=&w^{ij}(u_{,ij}-\Gamma^l_{ij}u_l+T_{ij}) -w^{ij}(\underline{u}_{,ij}-\Gamma^l_{ij}\underline{u}_l+T_{ij})\\ &+\alpha w^{ij}d_{,ij}-\alpha w^{ij}\Gamma^l_{ij}d_l\\ &-2\mu d w^{ij}d_{,ij}-2\mu w^{ij}d_id_j +2\mu d w^{ij}\Gamma^l_{ij}d_l \end{split}$$ We have$w^{ij}(u_{,ij}-\Gamma^l_{ij}u_l+T_{ij}) =w^{ij}w_{ij}=n$. Due to the admissibility of$\underline{u}$, we get$-w^{ij}(\underline{u}_{,ij}-\Gamma^l_{ij} \underline{u}_l+T_{ij})\le-3\varepsilon\mathop{\rm tr} w^{ij}$. We fix$\alpha>0$sufficiently small and obtain $$\alpha w^{ij}d_{,ij}-\alpha w^{ij}\Gamma^l_{ij}d_l \le\varepsilon\mathop{\rm tr} w^{ij}.$$ Obviously, we have $$-2\mu dw^{ij}d_{,ij}+2\mu dw^{ij}\Gamma^l_{ij}d_l \le c\mu\delta\mathop{\rm tr} w^{ij}.$$ To exploit the term$-2\mu w^{ij}d_id_j$, we use that$\vert d_i-\delta^n_i\vert\le c\cdot\vert x-x_0\vert\le c\cdot\delta$, so $$-2\mu w^{ij}d_id_j\le-\mu w^{nn}+c\mu\delta \max\limits_{k,\,l}\left\lvert w^{kl}\right\rvert.$$ As$w^{ij}$is positive definite, we obtain by testing$\begin{pmatrix}w^{kk} & w^{kl}\\ w^{kl} & w_{ll} \end{pmatrix}$with the vectors$(1,1)$and$(1,-1)$that$| w^{kl}|\le\mathop{\rm tr} w^{ij}$. Thus \eqref{L theta est} implies $$\label{L theta est1} L\vartheta\le-2\varepsilon\mathop{\rm tr} w^{ij}-\mu w^{nn}+c+c\mu\delta\mathop{\rm tr} w^{ij}$$ We may assume that$(w^{ij})_{i,\,j