text:
\begin{document}

\begin{abstract}
We obtain $C^2$ a priori estimates for solutions of the nonlinear second-order elliptic equation related to the geometric problem of finding a strictly locally convex hypersurface with prescribed curvature and boundary in a space form. Under the assumption of a strictly locally convex subsolution, we establish existence results by using degree theory arguments.
\end{abstract}

\maketitle

\section {\large Introduction}

In this paper, we stay in $(n+1)$ dimensional space form $N^{n+1}(K)$ ($n \geq 2$) with constant sectional curvature $K = 0$, $1$ or $- 1$, which can be modeled as follows. In Euclidean space $\mathbb{R}^{n + 1}$, fix the origin $0$ and let $\mathbb{S}^n$ denote the unit sphere centered at $0$. Choose the spherical coordinates $(z, \rho)$ in $\mathbb{R}^{n + 1}$ with $z \in \mathbb{S}^n$. Define the new metric on $\mathbb{R}^{n+1}$ by
\[ \bar{g} = d \rho^2 + \phi^2(\rho)\, \sigma \]
where $\sigma$ is the standard metric on $\mathbb{S}^n$ induced from $\mathbb{R}^{n + 1}$. Then $(\mathbb{R}^{n + 1}, \bar{g})$ is a model of $N^{n + 1}(K)$ for $K = 0$ if we choose $\phi(\rho) = \rho$ where $\rho \in [\, 0, \infty)$, for $K = 1$ if $\phi(\rho) = \sin(\rho)$ where $\rho \in [\, 0, \pi/2)$, and for $K = - 1$ if $\phi(\rho) = \sinh(\rho)$ where $\rho \in [\, 0, \infty)$, which correspond to the Euclidean space $\mathbb{R}^{n + 1}$, the upper hemisphere $\mathbb{S}_{+}^{n + 1}$ and the hyperbolic space $\mathbb{H}^{n + 1}$ respectively. Let $V = \phi(\rho) \,\frac{\partial}{\partial \rho}$ be the conformal Killing field in $ N^{n+1}(K)$. It is well known that $V$ is the position vector field in Euclidean space.

Given a disjoint collection $\Gamma = \{ \Gamma_1, \ldots, \Gamma_m \}$
of closed smooth embedded $(n - 1)$ dimensional submanifolds, a smooth symmetric function $f$ of $n$ variables and a smooth positive function $\psi$ defined on $N^{n+1}(K)$, it is a fundamental question in differential geometry to seek a strictly locally convex hypersurface $\Sigma$ with the prescribed curvature
\begin{equation}  \label{eq1-0}
f (\kappa [ \Sigma ]) = \psi( V )
\end{equation}
and boundary
\begin{equation} \label{eq1-2}
\partial \Sigma = \Gamma
\end{equation}
where $\kappa [\Sigma] = (\kappa_1, \ldots, \kappa_n)$ denotes the principal curvatures of $\Sigma$ at $V$ with respect to the outward unit normal $\nu
$. We call a hypersurface $\Sigma$ strictly locally convex if all its principal curvatures $\kappa_i > 0$ everywhere in $\Sigma$.

Equation \eqref{eq1-0} arises in various geometric problems. If we do not impose boundary condition \eqref{eq1-2} and consider closed hypersurfaces, there is a vast literature in this direction. When requiring the convexity of the hypersurfaces, the Gauss curvature case was studied by Oliker \cite{Oliker} while the most current breakthrough is due to Guan-Ren-Wang \cite{GRW15}, where the authors studied convex hypersurfaces with prescribed Weingarten curvature in $\mathbb{R}^{n + 1}$ for general $\psi$ depending on both $V$ and $\nu$. For starshaped compact hypersurfaces, we refer the readers to \cite{BLO} for the introductory material, and see Jin-Li \cite{JL05} for Weingarten curvature in hyperbolic space, \cite{BLO, LO02} for Weingarten curvature in elliptic space, Spruck-Xiao \cite{SX15} for scalar curvature in space forms for general $\psi$, Chen-Li-Wang \cite{CLW18} for Weingarten curvature in warped product spaces for general $\psi$.

For the Dirichlet problem, important examples include the classical Plateau problem concerning the mean curvature as well as the corresponding problem for Gauss curvature (see \cite{CNSI, GS93, Guan95, Guan98, GS02}). The Dirichlet problem in the general setting \eqref{eq1-0}--\eqref{eq1-2} was first studied by Caffarelli-Nirenberg-Spruck \cite{CNSV} for vertical graphs over strictly convex domains in $\mathbb{R}^n$ with constant boundary data. Since then, there have been significant progresses, among which, we mention Guan-Spruck \cite{GS04} and Trudinger-Wang \cite{TW02} for general locally convex hypersurfaces in $\mathbb{R}^{n+1}$ which may not be graphs, Su \cite{Su16} for strictly locally convex radial graphs in $\mathbb{R}^{n+1}$ and Cruz \cite{Cruz} for starshaped radial graphs with prescribed Weingarten curvature in $\mathbb{R}^{n+1}$.

As in \cite{GS04}, the curvature function $f$ is assumed to be defined on the open symmetric convex cone $\Gamma_n^+ \equiv \{ \lambda \in \mathbb{R}^n \vert \, \lambda_i > 0, i = 1, \ldots, n \}$ satisfying the fundamental structure conditions
\begin{equation} \label{eq1-5}
f_i (\lambda) \equiv \frac{\partial f(\lambda)}{\partial \lambda_i} > 0 \quad \mbox{in} \quad  \Gamma_n^+,\quad i = 1, \ldots, n
\end{equation}
\begin{equation} \label{eq1-6}
f \,\,\mbox{is} \,\,\mbox{concave}\,\, \mbox{in} \,\, \Gamma_n^+
\end{equation}
\begin{equation} \label{eq1-7}
f > 0 \quad \mbox{in} \quad \Gamma_n^+, \quad f = 0 \quad \mbox{on} \quad \partial\Gamma_n^+
\end{equation}
In addition, $f$ is assumed to satisfy the technical conditions
\begin{equation} \label{eq1-10}
\sum f_i (\lambda) \lambda_i \geq \sigma_0 \quad \mbox{on} \quad \{ \lambda \in \Gamma_n^+ \vert \, \psi_0 \leq f(\lambda) \leq \psi_1 \}
\end{equation}
for any $\psi_1 > \psi_0 > 0$, where $\sigma_0$ is a positive constant depending only on $\psi_0$ and $\psi_1$, and for any $C > 0$ and any compact set $E \subset \Gamma_n^+$ there exists $R = R( E, C ) > 0$ such that
\begin{equation} \label{eq1-11}
f(\lambda_1, \ldots, \lambda_{n - 1}, \lambda_n + R) \geq C \quad \forall \,\,\lambda \in E
\end{equation}
Examples satisfying \eqref{eq1-5}--\eqref{eq1-11} include a large family $f = \sum f_l$ where
\[ f_l = S_n^{\frac{1}{n N_l}}  \prod\limits_{i = 1}^{N_l - 1} \Big( c_i + \sum\limits_{k = 1}^{n - 1} c_{i, k} S_{n, k}^{\frac{1}{n - k}}\Big)^{\frac{1}{N_l}} \]
where $c_i$, $c_{i, k} \geq 0$ are constants, $c_i + \sum_k c_{i, k} > 0$ for each $i$, $S_k$ is the $k$th elementary symmetric function, $S_0 = 1$ and $S_{k, l} = S_k / S_l \,(0 \leq l < k \leq n)$. However, the pure curvature quotient $S_{n, k}^{1 / (n - k)}$ does not satisfy \eqref{eq1-11}.

In this paper, we are interested in strictly locally convex hypersurfaces embedded in $N^{n+1}(K)$ which can be represented as radial graphs over a domain in $\mathbb{S}^n$. Assuming $\Gamma$ to be the boundary of a smooth positive radial graph $\varphi$ in $N^{n + 1}(K)$ defined on a smooth domain $\Omega \subset \mathbb{S}^n$, we thus have $\Gamma = \{ (z, \varphi(z))\, | z \in \partial \Omega \}$ and look for a smooth strictly locally convex radial graph $\Sigma = \{ (z, \rho(z))\, | z \in \Omega \}$ satisfying the Dirichlet problem
\begin{equation} \label{eq1-1}
f(\kappa [ \rho ]) = \psi(z, \rho) \quad \mbox{in} \quad \Omega
\end{equation}
\begin{equation} \label{eq1-3}
\rho = \varphi \quad \mbox{on} \quad \partial \Omega
\end{equation}
where $\kappa[\rho]$ denotes the principal curvatures of the graph of $\rho$ and we use the same $\psi$ for the smooth positive function on the right hand side.
For $C^0$ estimates, we assume that
\begin{equation} \label{eq1-9}
\Omega \,\,\, \mbox{does}\,\,\mbox{not}\,\,\mbox{contain}\,\,\mbox{any}\,\,\mbox{hemisphere}.
\end{equation}

We obtain the following $C^2$ estimates:
\begin{thm} \label{Theorem1-1}
Under assumption \eqref{eq1-5}--\eqref{eq1-11} and \eqref{eq1-9}, suppose $\Gamma$  can span  a $C^2$ positive radial graph $\overline{\rho}$ in $N^{n + 1}(K)$ which is strictly locally convex in a neighborhood of $\Gamma$. Then for any $C^4$ strictly locally convex radial graph $\rho$ satisfying \eqref{eq1-1}-\eqref{eq1-3}  with $\rho \leq \overline{\rho}$ in $\Omega$, we have
\[ \Vert \rho \Vert_{C^2(\overline{\Omega})} \,\leq \, C \]
where $C$ depends only on $\Omega$, $\Vert\psi\Vert_{C^2}$, $\Vert \overline{\rho}\Vert_{C^1(\overline{\Omega})}$,  $\Vert \varphi \Vert_{C^4(\overline{\Omega})}$, $\inf \psi$, $\inf_{\partial\Omega}\overline{\rho}$ and the convexity of $\overline{\rho}$.
\end{thm}

We remark that for $C^2$ estimates, it is necessary in Theorem \ref{Theorem1-1} to assume $\overline{\rho}$ to be strictly locally convex near its boundary. To establish existence results, as in \cite{GS93, Guan95, Guan98, GS02, GS04, Su16}, we further require that $\overline{\rho}$ is a strictly locally convex subsolution. Since there are topological obstructions to the existence of strictly locally convex hypersurfaces spanning a given $\Gamma$ (see \cite{Ro93}), the existence of a subsolution allows the arbitrary geometry of $\Gamma$.
Using Theorem \ref{Theorem1-1}, we can prove the following existence results.

\begin{thm} \label{Theorem1-2}
Under assumption \eqref{eq1-5}--\eqref{eq1-11} and \eqref{eq1-9}, assume in addition that there exists a smooth strictly locally convex radial graph $\overline{\rho}$ satisfying
\begin{equation} \label{eq1-4}
\begin{aligned}
f (\kappa [ \overline{\rho} ])\, \geq & \,\, \psi(z, \overline{\rho}) \quad & \mbox{in} \quad &\Omega \\
\overline{\rho}\, = & \,\, \varphi \quad &\mbox{on} \quad &\partial \Omega
\end{aligned}
\end{equation}
Then there exists a smooth strictly locally convex radial graph $\Sigma = \{ (z, \rho(z))\,\vert\, z \in \Omega \}$ in space form $N^{n+1}(K)$  satisfying the Dirichlet problem \eqref{eq1-1}-\eqref{eq1-3} with $\rho \leq \overline{\rho}$ in $\overline{\Omega}$ and uniformly bounded principal curvatures
\[ 0 < K_0^{-1} \leq \kappa_i \leq K_0  \quad \mbox{on} \quad \Sigma. \]
where $K_0$ is a uniform positive constant depending only on $\Omega$, $\Vert\psi\Vert_{C^2}$, $\Vert \overline{\rho}\Vert_{C^1(\overline{\Omega})}$,  $\Vert \varphi \Vert_{C^4(\overline{\Omega})}$, $\inf \psi$, $\inf_{\partial\Omega}\overline{\rho}$ and the convexity of $\overline{\rho}$.
\end{thm}

In Euclidean space $\mathbb{R}^{n+1}$, Theorem \ref{Theorem1-2} was proved in \cite{GS93} for constant Gauss curvature assuming the existence of a strictly locally convex strict subsolution and was extended in \cite{Guan95} for general $\psi$ depending also on the gradient term.
These existence results are established via the theory of Monge-Amp\`ere type equations on $\mathbb{S}^n$. The linearized operators may have nontrivial kernels, which call for extra efforts for the proof of existence since one can not directly use continuity method. In \cite{GS93}, the authors established the existence results for equations with ${\partial\psi} / {\partial u} \leq 0$ by monotone iteration approach. In \cite{Guan95} the author rederived $C^2$ estimates for a wider class of equations which allows the application of degree theory to the proof of existence for general $\psi$ (the proof also need the existence result in \cite{GS93}). In \cite{Guan98}, Guan obtained the existence results for Monge-Amp\`ere equations with general $\psi$ over smooth bounded domains in $\mathbb{R}^{n}$ by assuming the existence of a subsolution (improving the results in \cite{CNSI} where the authors assumed the strict convexity of the domain) and stated that the strict subsolution assumption in \cite{GS93, Guan95} can be weakened to a subsolution. More recently, Su \cite{Su16} proved Theorem \ref{Theorem1-2} in $\mathbb{R}^{n + 1}$ assuming the existence of a strict subsolution, where the author reformulated \eqref{eq1-1} in a form with invertible linearized operator and thus continuity method and degree theory can be directly applied without extra $C^2$ estimates.

The novelty of this paper lies in: first, it provides a unified approach for $C^2$ estimates by transformation (see \eqref{eq3-14}). Second, for proving existence by degree theory, it generalizes Su's idea ( see \cite{Su16} ) to $\mathbb{H}^{n+1}$ and weaken the strict subsolution assumption. Besides, it creates a new continuity process starting from $\mathbb{R}^{n+1}$ to $\mathbb{S}^{n+1}_+$ and hence the existence in $\mathbb{S}^{n+1}_+$ is proved.

This paper is organized as follows: in section 2, we reformulate equation \eqref{eq1-1} in two different ways: one is used for deriving $C^2$ boundary estimates in section 3 and the other is for proving existence in $\mathbb{R}^{n+1}$ and $\mathbb{H}^{n+1}$ in section 5. Section 4 is devoted to global $C^2$ estimates. Section 6 is for existence in $\mathbb{S}^{n+1}_+$.

\vspace{5mm}

\section{Strictly locally convex radial graphs in space forms and reformulations of equation \eqref{eq1-1}}

\vspace{3mm}

Throughout this paper, we are interested in hypersurface $\Sigma \subset N^{n + 1} (K)$ that can be represented as a smooth radial graph over a smooth domain $\Omega \subset \mathbb{S}^n$, i.e.
\[ \Sigma  = \{ (z, \rho(z))\,| z \in \Omega \} \]
We note that the range for $\rho = \rho(z)$ is $ (0, \rho_U^K)$ where
\begin{equation} \label{eq3-15}
\rho_U^K = \left\{ \begin{aligned} & \infty,\quad\quad\mbox{if} \quad K = 0 \quad \mbox{or}\quad - 1 \\
& \frac{\pi}{2},\quad\quad\mbox{if} \quad K = 1
\end{aligned}
\right.
\end{equation}

Following the notations in \cite{SX15}, we introduce the following geometric quantities on $\Sigma$. Let $\nabla'$ denote the covariant derivatives with respect to some local orthonormal frame $e_1, \ldots, e_n$ on $\mathbb{S}^n$ (while $\nabla$ will be reserved for the covariant derivatives with respect to some local orthonormal frame $E_1, \ldots, E_n$ on $\Sigma$). The induced metric, its inverse, unit outer normal, and second fundamental form on $\Sigma$ are given respectively by
\begin{equation} \label{eq3-1}
 g_{ij} = \phi^2\, \delta_{ij} + \rho_i \rho_j
\end{equation}
\begin{equation} \label{eq3-2}
 g^{ij} = \frac{1}{\phi^2} \big( \delta_{ij} - \frac{\rho_i \rho_j}{\phi^2 + |\nabla'\rho|^2} \big)
\end{equation}
\begin{equation} \label{eq3-4}
 \nu = \frac{- \nabla' \rho + \phi^2 \,\frac{\partial}{\partial \rho}}{\sqrt{\phi^4 + \phi^2 |\nabla' \rho|^2}}
\end{equation}
\begin{equation} \label{eq3-5}
 h_{ij} = \frac{\phi}{\sqrt{\phi^2 + |\nabla' \rho|^2}} \,\big( - \nabla'_{ij} \rho + \frac{2 \phi'}{\phi}\,\rho_i \rho_j + \phi \phi' \delta_{ij} \big)
\end{equation}
where $\rho_i = \rho_{e_i} = \nabla'_{e_i} \rho = \nabla'_{i} \rho $, $\rho_{ij} = \nabla'_{e_j} \nabla'_{e_i} \rho =  \nabla'_{e_j e_i} \rho = \nabla'_{ji} \rho = \nabla'_{ij} \rho$, and higher order covariant derivatives are interpreted in this manner. We thus have $\nabla'\rho = \rho_k \, e_k$ (while in Section 4, $\rho_i$ may denote $\nabla_{E_i} \rho$, which is the covariant derivative with respect to $E_1, \ldots, E_n$).

The principal curvatures $\kappa_1, \ldots, \kappa_n$ of the radial graph $\rho$ are the eigenvalues of the real symmetric matrix $ \{ a_{ij} \}$:
\[ a_{ij} = \gamma^{i k} \,h_{k l}\, \gamma^{l j} \]
with $\{ \gamma^{ik} \}$ and its inverse $\{ \gamma_{ik} \}$ given respectively by
\begin{equation} \label{eq3-6}
 \gamma^{ik} = \frac{1}{\phi} ( \delta_{ik} - \frac{\rho_i \,\rho_k}{\sqrt{\phi^2 + |\nabla' \rho|^2} ( \phi + \sqrt{\phi^2 + |\nabla' \rho|^2} )} )
\end{equation}
\begin{equation} \label{eq3-7}
\gamma_{ik} = \phi \,\delta_{ik} + \frac{\rho_i \rho_k}{ \phi + \sqrt{\phi^2 + |\nabla' \rho|^2} }
\end{equation}
Note that $\{ \gamma_{ik} \}$ is the square root of the metric, i.e., $\gamma_{ik} \gamma_{kj} = g_{ij}$.

\begin{defn}
A hypersurface $\Sigma$ is strictly locally convex if its principal curvatures are all positive, i.e.
$\kappa_i > 0$ for $i = 1, \ldots, n$ everywhere on $\Sigma$;
or, equivalently, the symmetric matrix $\{ a_{ij }\}$ (or $\{ h_{ij} \}$) is positive definite everywhere in $\Omega$.

A $C^2$ function $\rho$ is strictly locally convex if the hypersurface $\Sigma$ represented by $\rho$ is strictly locally convex.
\end{defn}
For simplicity, throughout this paper  $a_{ij} > 0 $ (or $\geq 0$ ) means that the symmetric matrix $\{ a_{ij} \}$ is positive definite (or positive semi-definite); and $a_{ij} \geq b_{ij}$ means that the symmetric matrices $\{a_{ij}\}$ and $\{b_{ij}\}$ satisfy $ a_{ij} - b_{ij} \geq 0 $.
Now we will transform $\rho$ into other variables for deriving a priori estimates and proving the existence.

\vspace{5mm}

\subsection{Reformulation for deriving a priori estimates}~

\vspace{3mm}

We do the following transformation
\begin{equation} \label{eq3-14}
\rho = \zeta ( u ) \, = \,\left\{ \begin{aligned} & \frac{1}{u},\quad\quad & \mbox{if} \quad  K = 0  \\
& \mbox{arccot}\, u,\quad\quad & \mbox{if} \quad  K = 1  \\
& \frac{1}{2} \ln \big( \frac{ u + 1 }{u - 1} \big) ,\quad\quad & \mbox{if} \quad  K = - 1
\end{aligned}
\right.
\end{equation}
In view of \eqref{eq3-15}, the range for $u$ is  $(u_L^K, \infty)$ with
\begin{equation} \label{eq3-16}
u_L^K = \left\{ \begin{aligned} & 0,\quad\quad\mbox{if} \quad K = 0 \quad \mbox{or}\quad 1 \\
& 1,\quad\quad\mbox{if} \quad K = -1
\end{aligned}
\right.
\end{equation}
Then the formula \eqref{eq3-1}, \eqref{eq3-2}, \eqref{eq3-6}, \eqref{eq3-7} and \eqref{eq3-5} can be expressed as
\begin{equation} \label{eq3-8}
g_{ij} = \phi^2 \,\delta_{ij} + \zeta'^2(u)\, u_i u_j
\end{equation}

\begin{equation} \label{eq3-9}
g^{ij} = \frac{1}{\phi^2} \Big( \delta_{ij} - \frac{\zeta'^2(u) u_i u_j}{\phi^2 + \zeta'^2(u) |\nabla' u|^2} \Big)
\end{equation}

\begin{equation} \label{eq3-10}
\gamma^{ik} =  \,\frac{1}{\phi}\, \Big( \delta_{ik} - \frac{ \zeta'^2(u) u_i u_k }{\sqrt{\phi^2 + \zeta'^2(u) |\nabla' u|^2 } ( \phi + \sqrt{\phi^2 + \zeta'^2(u) |\nabla' u|^2 } )}\Big)
\end{equation}

\begin{equation} \label{eq3-11}
 \gamma_{ik} =  \phi \,\delta_{ik} + \frac{\zeta'^2(u) u_i u_k}{ \phi + \sqrt{\phi^2 + \zeta'^2(u) |\nabla' u|^2 } }
\end{equation}

\begin{equation} \label{eq3-13}
h_{ij} =  \frac{- \zeta'(u) \phi}{\sqrt{\phi^2 + \zeta'^2 |\nabla' u|^2}} ( \nabla'_{ij} u + u \,\delta_{ij} )
\end{equation}
Hence
\begin{equation} \label{eq3-18}
a_{ij} = \frac{- \zeta'(u) \phi}{\sqrt{\phi^2 + \zeta'^2 |\nabla' u|^2}} \gamma^{ik}\, ( \nabla'_{kl} u + u \,\delta_{kl} )\,\gamma^{lj}
\end{equation}
It is easy to see that  $\Sigma$ (or $u$) is strictly locally convex if and only if
\begin{equation} \label{eq3-17}
 \nabla'_{ij} u + \, u \,\delta_{ij}  > 0 \quad\mbox{in} \quad \Omega
\end{equation}

Under transformation \eqref{eq3-14},
the Dirichlet problem \eqref{eq1-1}-\eqref{eq1-3}
is equivalent to
\begin{equation} \label{eq2-11}
f(\kappa[ u ]) = \, \psi(z, u)  \quad  \mbox{in} \quad \Omega
\end{equation}
\begin{equation} \label{eq2-12}
\quad \,\, u =  \,\varphi  \quad \quad \mbox{on} \quad \partial \Omega
\end{equation}
Here we still use $\psi$ for the function on the right hand side, and $\varphi$ for the boundary value.
Denote $\kappa[u] = (\kappa_1, \ldots, \kappa_n) = \lambda( A [ u ] )$ where $\lambda(A)$ denotes the eigenvalues of $A$ and $A[u] = \{ a_{ij} \}$ with $a_{ij}$ given by \eqref{eq3-18}.
Define the function $F$ by $ F( A ) = f(\lambda( A ))$ and the function $G$ by
\[ G(r, p, u) \,= \, F ( A ( r, p, u ) )\]
where $A( r, p, u )$ is obtained from $A[u]$ with $(\nabla'^2 u, \nabla' u, u)$ replaced by $( r, p, u )$. Therefore
equation \eqref{eq2-11} can be rewritten in the following form
\begin{equation} \label{eq2-13}
G(\nabla'^2 u, \nabla' u, u) =  \,\psi(z, u)  \quad  \mbox{in} \quad \Omega
\end{equation}
Denote
\[ F^{ij} ( A ) = \frac{\partial F}{\partial a_{ij}} ( A ), \quad F^{ij, kl} (A) = \frac{\partial^2 F}{\partial a_{ij} \partial a_{kl}} (A)\]
\[ G^{ij}(r, p, u) = \frac{\partial G}{\partial r_{ij}}(r, p, u), \quad G^i(r, p, u) = \, \frac{\partial G}{\partial p_i}(r, p, u),  \quad G_u(r, p, u) = \, \frac{\partial G}{\partial u}(r, p, u)\]
\[ \psi_u(z, u) = \, \frac{\partial \psi}{\partial u}(z, u) \]
As mentioned in \cite{GS04}, the function $F$ possesses the following properties.
First, the matrix $\{ F^{ij} (A) \}$ is symmetric with eigenvalues $f_1, \ldots, f_n$.
By \eqref{eq1-5}, $F^{ij} (A) > 0$ whenever $\lambda(A) \in \Gamma_n^+$, and by \eqref{eq1-6} we know that $F$ is a concave function of $A$, i.e., the symmetric matrix $F^{ij, kl} (A) \leq 0 $ whenever $\lambda(A) \in \Gamma_n^+$.
The function $G$ satisfies similar structure conditions as $F$. In fact, from \eqref{eq3-18} we have
\begin{equation} \label{eq2-14}
 G^{ij} = \frac{\partial G}{\partial u_{ij}}  = \frac{\partial F}{\partial a_{kl}} \frac{\partial a_{kl}}{\partial u_{ij}} =  \frac{ - \phi \zeta'(u)}{\sqrt{\phi^2 + \zeta'^2(u) |\nabla' u|^2}} F^{kl} \gamma^{ik} \gamma^{jl}
\end{equation}
Thus the symmetric matrix $G^{ij} > 0$ if and only if $F^{ij} > 0$, which in particular implies that equation \eqref{eq2-13} is elliptic for strictly locally convex solutions.
Also by \eqref{eq3-18} we can calculate
\[ \frac{\partial^2 G}{\partial u_{ij} \partial u_{kl}} = \frac{\partial a_{pq}}{\partial u_{ij}} \,\frac{\partial^2 F}{\partial a_{pq} \partial a_{rs}}\,\frac{\partial a_{rs}}{\partial u_{kl}}\]
which implies that $G$ is concave with respect to $\{ u_{ij} \}$ for strictly locally convex $u$.

We next compute $G^s$ and $G_u$, which will be needed in section 3.
\begin{lemma} \label{Lemma2-1}
Denote $w = \sqrt{ \phi^2  + \zeta'^2(u) |\nabla' u|^2}$. Then
\begin{equation} \label{eq2-15}
 G^s \, = \, - \frac{2 \zeta'^2 ( w \,\gamma^{is}\, u_q + \phi \,\gamma^{q s} u_i )}{ w ( \phi + w )} F^{ij} a_{q j} - \frac{\zeta'^2\, u_s}{w^2} \, F^{ij} a_{ij}
 \end{equation}
\begin{equation} \label{eq2-31}
\begin{aligned}
G_u = -2 \Big( \phi \phi' \zeta' g^{iq} + \frac{\zeta' \zeta'' u_i u_q}{w^2}\Big) F^{ij} a_{q j} + \Big( \frac{\phi' \zeta'}{\phi} - \frac{\phi \phi' \zeta'}{w^2} + \frac{\phi^2 \zeta''}{\zeta'\, w^2} \Big) F^{ij} a_{ij} - \frac{\phi \zeta'}{w} F^{ij} g^{ij}
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
We first prove \eqref{eq2-15}. Note that
\begin{equation}  \label{eq2-16}
G^s \, = \frac{\partial F}{\partial a_{ij}} \frac{\partial a_{ij}}{\partial u_s} = F^{ij} \big( 2\, \frac{\partial \gamma^{ik}}{\partial u_s} \, h_{kl}\,\gamma^{lj} + \gamma^{ik}\,\frac{\partial h_{kl}}{\partial u_s}\,\gamma^{lj} \big)
\end{equation}
where
\begin{equation} \label{eq2-17}
\frac{ \partial \gamma^{ik}}{\partial u_s} = - \gamma^{ip}\, \frac{\partial \gamma_{pq}}{\partial u_s} \, \gamma^{qk}
\end{equation}
Direct calculations from \eqref{eq3-11} and \eqref{eq3-10} yield
\begin{equation} \label{eq2-18}
\frac{\partial \gamma_{pq}}{\partial u_s} \,=
   \frac{ \zeta'^2(u) ( \delta_{ps} u_q + \delta_{q s} u_p ) }{ \phi + w } - \frac{\zeta'^4( u )\, u_p u_q u_s}{( \phi + w )^2 w}
= \frac{\zeta'^2(u) ( \delta_{ps} u_q + \phi\, u_p \gamma^{q s}) }{ \phi + w  }
\end{equation}
and
\begin{equation} \label{eq2-19}
\gamma^{ip} \,u_p =  \frac{ u_i }{w}
\end{equation}
Besides, from \eqref{eq3-13} and \eqref{eq3-18} we have
\begin{equation} \label{eq2-20}
\gamma^{ik}\,\frac{\partial h_{kl}}{\partial u_s}\,\gamma^{lj} = - \frac{\zeta'^2(u) \,u_s}{ w^2 } \, a_{ij}
\end{equation}
Taking \eqref{eq2-17}--\eqref{eq2-20} into \eqref{eq2-16}, the formula \eqref{eq2-15} is proved.

The formula \eqref{eq2-31} can be proved similarly. In fact,
\begin{equation}  \label{eq2-36}
G_u \, = \frac{\partial F}{\partial a_{ij}} \frac{\partial a_{ij}}{\partial u} = F^{ij} \big( 2\, \frac{\partial \gamma^{ik}}{\partial u} \, h_{kl}\,\gamma^{lj} + \gamma^{ik}\,\frac{\partial h_{kl}}{\partial u}\,\gamma^{lj} \big)
\end{equation}
where
\begin{equation*}
\frac{ \partial \gamma^{ik}}{\partial u} = - \gamma^{ip}\, \frac{\partial \gamma_{pq}}{\partial u} \, \gamma^{qk}
\end{equation*}
From \eqref{eq3-11} we have
\begin{equation*}
\begin{aligned}
 \frac{\partial \gamma_{ik}}{\partial u} = & \, \phi' \zeta' \delta_{ik} + \frac{2 \zeta' \zeta'' u_i u_k}{\phi + w } - \frac{\zeta'^2(u) u_i u_k}{( \phi + w )^2} \big( \phi' \zeta'(u) + \frac{\phi \phi' \zeta' + \zeta'\zeta''|\nabla' u|^2}{w} \big)
 \\
= & \,\phi' \zeta' \delta_{ik} + \frac{ \zeta' u_i u_k}{\phi + w} \big( 2 \zeta'' - \frac{\zeta'}{\phi + w} \big( \phi' \zeta'  +  \frac{\phi \phi' \zeta' + \zeta'\zeta''|\nabla' u|^2}{w}   \big)\, \big) \\
= & \,\phi' \zeta' \delta_{ik} + \frac{ \zeta' u_i u_k}{\phi + w} \big( 2 \zeta'' - \frac{\zeta'^2 \zeta'' |\nabla' u|^2}{(\phi + w) w} - \frac{\phi' \zeta'^2}{w} \big) \\
= & \,\phi' \zeta' \delta_{ik} + \frac{ \zeta' u_i u_k}{\phi + w} \big(  \frac{w + \phi}{w} \zeta''  - \frac{\phi' \zeta'^2}{w} \big)
\end{aligned}
\end{equation*}
In view of \eqref{eq3-10}, the above formula becomes
\begin{equation} \label{eq2-37}
 \frac{\partial \gamma_{ik}}{\partial u} = \phi \phi' \zeta' \gamma^{ik} + \frac{\zeta' \zeta'' u_i u_k}{w}
\end{equation}
Direct calculation from \eqref{eq3-13} yields
\begin{equation} \label{eq2-38}
\frac{\partial h_{ij}}{\partial u} = ( - \frac{\phi' \zeta'^2}{w} + \frac{\phi^2 \phi' \zeta'^2}{w^3} - \frac{\phi^3 \zeta''}{w^3} ) (\nabla'_{ij} u + u \delta_{ij}) - \frac{\phi\, \zeta'}{w} \delta_{ij}
\end{equation}
Inserting \eqref{eq2-37} and \eqref{eq2-38} into \eqref{eq2-36} and in view of \eqref{eq3-18} and \eqref{eq2-19} we obtain
\eqref{eq2-31}.
\end{proof}
\begin{cor} \label{GsGu}
Suppose that we have the $C^1$ bounds for strictly locally convex solutions $u$ of \eqref{eq2-11}:
\begin{equation*}
 u_L^K < C_0^{-1} \leq u \leq C_0, \quad\quad |\nabla' u| \leq C_1 \quad\mbox{in} \quad \overline{\Omega}
\end{equation*}
Then
\[| G^s | \, \leq \, C \quad \mbox{and} \quad
| G_u | \,\leq \, C ( 1 + \sum G^{ii} )
\]
\end{cor}
\begin{proof}
Note that $\{ F^{ij} (A) \}$ and $A$ can be diagonalized simultaneously by an orthonormal transformation. Consequently, the eigenvalues of the matrix $\{ F^{ij}(A)\} A$, which is not necessarily symmetric, are given by
\[ \lambda( \{ F^{ij}(A)\} A ) = ( f_1 \kappa_1, \ldots, f_n \kappa_n ) \]
In particular we have
\[ F^{ij}\, a_{ij} = \sum f_i \kappa_i \]
In addition,
for a bounded matrix $B = \{b_{ij}\}$, i.e. $|b_{ij}| \leq C$ for all $1 \leq i, j \leq n$ we have
\[ \vert b_{i k} F^{i j } a_{k j} \vert \, \leq C \sum f_i \kappa_i \]
Thus from \eqref{eq2-15} and \eqref{eq2-31} we have
\[ \vert G^s \vert \,\leq \, C  \sum f_i \kappa_i\quad \mbox{and} \quad \vert G_u \vert \,\leq \,C ( \sum f_i \kappa_i + \sum f_i ) \]
Finally, by the concavity of $f$ and $f(0) = 0$ we can derive that $ \sum f_i \kappa_i \leq \psi \leq C$. Also, in view of \eqref{eq2-14} we have
$\sum f_i \leq C \sum G^{ii}$. Hence the corollary is proved.
\end{proof}

\vspace{5mm}

\subsection{ Reformulation for proving existence }~

\vspace{3mm}

For the proof of the existence in Section 5, we do the following transformation.
\begin{equation} \label{eq6-2}
u = \eta ( v ) \, = \,\left\{ \begin{aligned} & e^v,\quad\quad & \mbox{if} \quad  K = 0  \\
& \sinh v,\quad\quad & \mbox{if} \quad  K = 1  \\
& \cosh v,\quad\quad & \mbox{if} \quad  K = - 1
\end{aligned}
\right.
\end{equation}
In view of \eqref{eq3-16}, the range for $v$ is $(v_L^K, \infty)$ with
\begin{equation} \label{eq6-19}
v_L^K = \left\{ \begin{aligned} & - \infty,\quad\quad &\mbox{if} \quad K = 0  \\
& 0,\quad\quad &\mbox{if} \quad K = 1 \,\, \mbox{or}\,\, -1
\end{aligned}
\right.
\end{equation}
The formula \eqref{eq3-10} and \eqref{eq3-13} can consequently be transformed into
\begin{equation} \label{eq6-3}
\gamma^{ik} =  \,\eta'(v) \,\Big( \delta_{ik} - \frac{ v_i v_k }{\sqrt{1 +  |\nabla' v|^2 } ( 1 + \sqrt{ 1 +  |\nabla' v|^2 } )} \Big)
\end{equation}
\begin{equation} \label{eq6-4}
h_{ij} = \, \frac{1}{ \eta'^2(v) \sqrt{1 + |\nabla' v|^2}} \Big( \eta'(v) \nabla'_{ij} v + \eta(v) v_i v_j +  \eta(v) \,\delta_{ij} \Big)
\end{equation}
Denoting
\[ w = \sqrt{1 + |\nabla' v|^2}\quad \mbox{and} \quad \tilde{\gamma}^{ik} = \delta_{ik} - \frac{ v_i v_k }{w ( 1 + w )}\]
we therefore have
\begin{equation} \label{eq6-1}
\begin{aligned}
a_{ij} =  & \, \frac{1}{ w } \, \tilde{\gamma}^{ik} \Big( \eta'(v) \nabla'_{kl} v + \eta(v) v_k v_l +  \eta(v) \,\delta_{kl}\Big)\, \tilde{\gamma}^{lj} \\
= & \, \frac{1}{w} \Big(\, \eta(v)\, \delta_{ij} \,+ \,\eta'(v)\, \tilde{\gamma}^{ik}\, \nabla'_{kl} v  \,\tilde{\gamma}^{lj} \,\Big)
\end{aligned}
\end{equation}
It is easy to see that $\Sigma$ (or $v$) is strictly locally convex if and only if
\begin{equation} \label{eq6-6}
 \eta(v)\, \delta_{ij} \,+ \,\eta'(v)\, \tilde{\gamma}^{ik} \,\nabla'_{kl} v \, \tilde{\gamma}^{lj}  > 0 \quad \mbox{in} \quad \Omega
\end{equation}

Under transformation \eqref{eq6-2}, we now reformulate \eqref{eq2-11}--\eqref{eq2-12} in terms of $v$,
\begin{equation} \label{eq2-24}
f(\kappa[ v ]) = \, \psi(z, v)  \quad  \mbox{in} \quad \Omega
\end{equation}
\begin{equation} \label{eq2-25}
v =  \,\varphi  \quad \quad \mbox{on} \quad \partial \Omega
\end{equation}
Here we still use $\psi$ for the right function and $\varphi$ for the boundary value. At this time,
$\kappa[v] = (\kappa_1, \ldots, \kappa_n) = \lambda( A [ v ] )$ and $A[v] = \{ a_{ij} \}$ with $a_{ij}$ given by \eqref{eq6-1}. Replacing $(\nabla'^2 v, \nabla' v, v)$ in $A[v]$ by $( r, p, v )$ we can define $A( r, p, v )$. Then we can
define $\mathcal{G}$ by
$\mathcal{G}(r, p, v) \,= \, F ( A ( r, p, v ) )$.
Thus, equation \eqref{eq2-24} can be rewritten as
\begin{equation} \label{eq2-26}
\mathcal{G}(\nabla'^2 v, \nabla' v, v) =  \,\psi(z, v)  \quad  \mbox{in} \quad \Omega
\end{equation}
Denote
\[ \mathcal{G}^{ij}(r, p, v) = \frac{\partial \mathcal{G}}{\partial r_{ij}} (r, p, v), \quad \mathcal{G}^{i} (r, p, v) = \frac{\partial \mathcal{G}}{\partial p_{i}}(r, p, v), \quad \mathcal{G}_v (r, p, v) = \, \frac{\partial \mathcal{G}}{\partial v} (r, p, v) \]
By \eqref{eq6-1}, we notice that equation \eqref{eq2-26} is elliptic for strictly locally convex $v$, and $\mathcal{G}$ is concave with respect to $\nabla'^2 v $ for strictly locally convex $v$.

\vspace{5mm}

\section{ Second order boundary estimates}

\vspace{5mm}

In this section, we derive the $C^2$ a priori estimates for strictly locally convex solutions $u$ to the Dirichlet problem \eqref{eq2-13}-\eqref{eq2-12} with $u \geq \underline{u}$ in $\Omega$
\begin{equation} \label{eq4-2}
\Vert u \Vert_{C^{2}(\overline{\Omega})} \,\leq\, C
\end{equation}
which, together with $u \geq C_0^{-1} > u_L^K$, implies an upper bound for all the principal curvatures by \eqref{eq3-18}. By assumption \eqref{eq1-7}, the principal curvatures admit a uniform positive lower bound, i.e.,
\begin{equation} \label{eq4-3}
0 < K_0^{-1} \leq \kappa_i \leq K_0 \quad \mbox{in} \quad \Omega
\end{equation}
which implies the uniform ellipticity of the linearized operator. Then $C^{2, \alpha}$ estimates can be established by Evans-Krylov theory \cite{Evans,Krylov}
\begin{equation} \label{eq4-1}
\Vert u \Vert_{C^{2,\alpha}(\overline{\Omega})} \,\leq\, C
\end{equation}
and higher-order regularity follows from classical Schauder theory.

The following $C^1$ estimates have been established in \cite{GS93} which was originally stated for $\mathbb{R}^{n + 1}$, but it also works in space forms.
\begin{lemma} \label{Lemma5-1}
Under assumption \eqref{eq1-9}, for any strictly locally convex function $u$ with $ u \geq \underline{u}$ in $\Omega$ and $u = \underline{u}$ on $\partial\Omega$ we have
\begin{equation} \label{eq5-1}
 u_L^K < C_0^{-1} \leq u \leq C_0, \quad\quad |\nabla' u| \leq C_1 \quad\mbox{in} \quad \overline{\Omega}
\end{equation}
where $C_0$ depends only on $\Omega$, $\sup_{\partial\Omega}\underline{u}$ and $\inf_{\Omega}\underline{u}$; $C_1$ depends in addition on $\sup_{\partial\Omega}\vert\nabla'\underline{u}\vert$.
\end{lemma}

In next section, we will derive global curvature estimates, which is equivalent to the global estimates for $|\nabla'^2 u|$ on $\overline{\Omega}$ from its bound on $\partial\Omega$. Therefore in this section we focus on the boundary estimates:
\begin{equation} \label{eq3-32}
| \nabla'^2 u | \leq C \quad\mbox{on} \quad \partial\Omega
\end{equation}

Consider any fixed point $z_0 \in \partial \Omega$. Choose a local orthonormal frame field $e_1, \ldots, e_n$ around $z_0$ on $\Omega$, which is obtained by parallel translation of a local orthonormal frame field on $\partial\Omega$ and the interior, unit, normal vector field to $\partial \Omega$, along the geodesics perpendicular to $\partial \Omega$ on $\Omega$. Assume that $e_n$ is the parallel translation of the unit normal field on $\partial \Omega$.

Since $u = \varphi$ on $\partial \Omega$,
\[ \nabla'_{\alpha\beta} (u - \varphi) = - \nabla'_n ( u - \varphi )\,\Gamma_{\alpha\beta}^n, \quad \alpha, \beta < n \quad\mbox{on}\quad \partial\Omega \]
where $\Gamma_{ij}^k$ are the Christoffel symbols of $\nabla'$ with respect to the frame $e_1, \ldots, e_n$ on $\mathbb{S}^n$. We thus obtain
\begin{equation} \label{eq3-27}
| \nabla'_{\alpha \beta} u (z_0) | \leq C, \quad \alpha, \beta < n
\end{equation}

Let $\rho(z)$ and $d(z)$ denote the distances from $z \in \overline{\Omega}$ to $z_0$ and $\partial \Omega$ on $\mathbb{S}^n$, respectively.
Set
\[ \Omega_\delta = \{ z \in \Omega : \rho(z) < \delta \}  \]
Choose $\delta_0 > 0$ sufficiently small such that $\rho$ and $d$ are smooth in $\Omega_{\delta_0}$, on which, we have
\[ |\nabla' d| = 1, \quad\quad - C\, I \leq \nabla'^2\, d \leq C \,I, \quad
\quad |\nabla' \rho| = 1, \quad\quad I \,\leq \,\nabla'^2 \,\rho^2 \,\leq \, 3 I \]
where $C$ depends only on $\delta_0$ and the geometric quantities of $\partial \Omega$, and
\[ \nabla'^2 \underline{u} + \underline{u} \, I \geq 4 \, c_0 \, I  \]
for some constant $c_0 > 0$ in view of the strict local convexity  of $\underline{u}$ and \eqref{eq3-17}.

We will need the following barrier function
\[ \Psi = A v + B \rho^2 \]
with
\[ v = u - \underline{u} + \epsilon\, d - \frac{N}{2} \,d^2 \]
and the linearized operator associated with equation \eqref{eq2-13}
\begin{equation} \label{eq2-21}
 L  = G^{ij}\, \nabla'_{i j} + G^i \,\nabla'_i
\end{equation}
to estimate the mixed tangential normal and pure normal second derivatives at $z_0$.
By direct calculation and Corollary \ref{GsGu} we have
\begin{equation} \label{eq3-20}
\begin{aligned}
 L v = \,& ( G^{ij} \nabla'_{ij} + G^i \nabla'_i  ) ( u - \underline{u} + \epsilon \,d - \frac{N}{2} d^2) \\
     = \, & G^{ij} \nabla'_{ij} ( u - \underline{u} - \frac{N}{2} \,d^2 ) + \epsilon \,G^{ij} \nabla'_{ij}  d + G^i  \nabla'_i ( u - \underline{u} + \epsilon\, d - \frac{N}{2} \,d^2 ) \\
     \leq \, & G^{ij} \Big( \nabla'_{ij} u - \,\big( \nabla'_{ij}(\underline{u} + \frac{N}{2} \,d^2 ) - 2 c_0 \delta_{ij} \big) \Big) \\ & - 2 c_0 \sum G^{ii}
       + C \epsilon \sum G^{ii} + C  ( 1 +  \epsilon +  N \delta )
\end{aligned}
\end{equation}
Since $G(\nabla'^2 u, \nabla' u, u)$ is concave  with respect to $\nabla'^2 u$,
\begin{equation} \label{eq3-21}
\begin{aligned}
& G^{ij} \Big( \nabla'_{ij} u -\, \big( \nabla'_{ij} (\underline{u} + \frac{N}{2} \,d^2 ) - 2 c_0 \delta_{ij}\big) \Big) \\ \leq &\,\, G( \nabla'^2 u, \nabla' u, u ) - G\Big( \nabla'^2\big( \underline{u} + \frac{N}{2} \,d^2 \big) - 2 c_0 I, \nabla' u, u \Big)
\end{aligned}
\end{equation}
Note that
\begin{equation*}
\begin{aligned}
&  \nabla'^2\big( \underline{u} + \frac{N}{2} \,d^2 \big ) - 2 c_0 I + u I \\
= \quad &  \nabla'^2 \underline{u} + \underline{u}\, I +  N d \nabla'^2 d + N \nabla' d \otimes \nabla' d - 2 c_0 I + ( u - \underline{u} ) I \\
\geq \quad & 2 c_0 I - C N \delta I + N \nabla' d \otimes \nabla' d := \mathcal{H}
\end{aligned}
\end{equation*}
We thus have
\begin{equation} \label{eq3-22}
\begin{aligned}
& G\Big( \nabla'^2\big( \underline{u} + \frac{N}{2} \,d^2 \big) - 2 c_0 I, \nabla' u, u \Big)  \\
= \,\, & F \Big( \frac{- \phi \,\zeta'(u)}{\sqrt{\phi^2 + \zeta'^2 |\nabla' u|^2 }}\, g^{-1/2} \big( \nabla'^2( \underline{u} + \frac{N}{2} \,d^2 ) - 2 c_0 I + u I \big)\,\, g^{-1/2} \Big) \\
\geq  &\,\, F \Big( \frac{- \phi\, \zeta'(u)}{\sqrt{\phi^2 + \zeta'^2 |\nabla' u|^2 }} \,g^{-1/2} \mathcal{H}\, g^{-1/2} \Big)
\\ =  & \,\,F \Big( \frac{- \phi\, \zeta'(u)}{\sqrt{\phi^2 + \zeta'^2 |\nabla' u|^2 }} \,\mathcal{H}^{1/2} \,g^{-1} \,\mathcal{H}^{1/2} \Big) \\
\geq & \,\,F \Big( \frac{- \phi\, \zeta'(u)}{\sqrt{\phi^2 + \zeta'^2 |\nabla' u|^2 }}\,\mathcal{H}^{1/2} \, \frac{1}{ \phi^2 + \zeta'^2(u) |\nabla' u|^2} \,I \,\mathcal{H}^{1/2} \Big) \\
=  & \,\,F \Big( \frac{- \phi\, \zeta'(u)}{(\phi^2 + \zeta'^2 |\nabla' u|^2 )^{3/2}} \,\mathcal{H} \Big)\,
\geq \, \, F ( \tilde{c}\, \mathcal{H} )
\end{aligned}
\end{equation}
where $\tilde{c}$ is a positive constant depending only on $C_0$ and $C_1$.

Combining \eqref{eq3-20}--\eqref{eq3-22} we have
\begin{equation} \label{eq3-23}
 L v \leq   \, - F ( \tilde{c}\,\mathcal{H} )   + (   C \epsilon  - 2 c_0 ) \,\sum G^{ii} + C ( 1 + \epsilon + N \delta )
\end{equation}
where $\mathcal{H} = \mbox{diag} \Big( 2 c_0 - C N \delta,\,\, \ldots,\,\, 2 c_0 - C N \delta,\,\, 2 c_0 - C N \delta + N\Big)$.
By \eqref{eq1-11}, we can choose $N$ sufficiently large and $\epsilon$, $\delta$ sufficiently small with $\delta$ depending on $N$ such that
\[  C \epsilon  \leq c_0, \quad  C N \delta \leq c_0, \quad  -F ( \tilde{c}\,\mathcal{H} )   + C  + 2 c_0  \leq - 1, \]
Therefore, \eqref{eq3-23} becomes
\begin{equation} \label{eq3-25}
 L v \leq - c_0 \sum G^{ii} - 1
\end{equation}
We then choose $\delta \leq \frac{2 \epsilon}{N}$ such that
\[ v \,\geq \, 0 \quad \mbox{in} \quad \Omega_\delta \]
A direct consequence of \eqref{eq3-25} is
\begin{equation} \label{eq3-29}
L \Psi \, = \, A \,L v \,+\, B\,L( \rho^2 ) \,\leq \,A ( - c_0 \sum G^{ii} - 1 ) + B C ( 1 + \sum G^{ii} ) \quad\mbox{in}\,\,\Omega_{\delta}
\end{equation}
which will be used later.  Besides, we also need to estimate $L ( \nabla'_k u )$. For this, we first apply the formula
\begin{equation*}
\nabla'_{ij} ( \nabla'_k u ) = \,\nabla'_k \nabla'_{ij} u + \Gamma^{l}_{ik} \nabla'_{jl} u + \Gamma_{jk}^l \nabla'_{il} u + \nabla'_{k} \Gamma_{ij}^l \,u_l
\end{equation*}
to obtain
\begin{equation} \label{eq3-30}
\begin{aligned}
L ( \nabla'_k u ) =\,& G^{ij} \nabla'_{ij} ( \nabla'_k u ) + G^i \, \nabla'_i (\nabla'_k u) \\
=  \,& \big( G^{ij}\nabla'_k \nabla'_{ij} u +  G^i \, \nabla'_{k i} u \big) \, + G^{ij} \Gamma^{l}_{ik} \nabla'_{jl} u + G^{ij} \Gamma_{jk}^l \nabla'_{il} u \\ & + G^{ij} \nabla'_{k} \Gamma_{ij}^l \,u_l   +  G^i \, \Gamma_{ik}^l \,u_l
\end{aligned}
\end{equation}
By \eqref{eq2-14} and \eqref{eq3-18} we have
\begin{equation*}
G^{ij} \Gamma^{l}_{ik} ( \nabla'_{jl} u + \,u \,\delta_{jl}) =\, F^{st} \gamma^{is} \gamma^{jt} \Gamma^{l}_{ik}  \cdot  \gamma_{jp}\, a_{pq}\,\gamma_{ql}\\
=\,( \gamma^{is} \Gamma^{l}_{ik} \gamma_{q l } ) \,F^{st} \,a_{t q}
\end{equation*}
The term $G^{ij} \Gamma_{jk}^l \nabla'_{il} u$ can be computed similarly. Taking the covariant derivative of \eqref{eq2-13} and applying Corollary \ref{GsGu} we have
\[ | G^{ij}\nabla'_k \nabla'_{ij} u +  G^i \, \nabla'_{k i} u | \leq C + | (\psi_u - G_u) u_k | \leq C (1 + \sum G^{ii}) \]
From all these above, \eqref{eq3-30} can be estimated as
\begin{equation} \label{eq3-19}
\vert L ( \nabla'_k u ) \vert \leq  \, C ( 1 + \sum G^{ii} )
\end{equation}

For fixed $\alpha < n$, choose $B$ sufficiently large such that
\[ \Psi \pm \nabla'_\alpha (u - \varphi) \geq 0 \quad \mbox{on} \quad \partial\Omega_\delta   \]
From  \eqref{eq3-29} and \eqref{eq3-19}
\[ L ( \Psi \pm \nabla'_\alpha (u - \varphi) ) \leq A ( - c_0 \sum G^{ii} - 1 ) + B C ( 1 + \sum G^{ii} )  \]
Then choose $A$ sufficiently large such that
\[ L ( \Psi \pm \nabla'_\alpha (u - \varphi) ) \leq 0 \quad \mbox{in} \quad \Omega_\delta \]
Applying the maximum principle we have
\[ \Psi \pm \nabla'_\alpha (u - \varphi) \geq 0  \quad \mbox{in} \quad \Omega_\delta \]
which implies
\begin{equation} \label{eq3-28}
\vert \nabla'_{\alpha n} u (z_0) \vert \leq C
\end{equation}

It remains to estimate the double normal derivative $\nabla'_{n n} u $ on $\partial\Omega$. By the strict local convexity of $u$, we only need to give an upper bound
\[ \nabla'_{n n} u  \leq C \quad \mbox{on} \quad \partial\Omega \]
The following proof is inspired by an idea of Trudinger \cite{Tru}. First we prove that
\begin{equation} \label{eq13}
 M := \min\limits_{z \in \partial\Omega}\,\min\limits_{\xi \in T_z( \partial \Omega ), |\xi| = 1} ( \nabla'_{\xi\xi} u + u ) \geq c_1
\end{equation}
for some constant $c_1 > 0$. Assume that $M$ is achieved at $z_1 \in \partial \Omega$ in the direction of $\xi_1$.  Let $e_1, \ldots, e_n$ be the local orthonormal frame field around $z_1$ on $\Omega \subset \mathbb{S}^n$ such that $e_1 (z_1) = \xi_1$. Thus we have
\begin{equation*}
\begin{aligned}
M = & \nabla'_{\xi_1 \xi_1} u (z_1) + u(z_1) =  \nabla'_{11} u (z_1) + u(z_1) \\
= & (\nabla'_{11} \underline{u} (z_1) + \underline{u}(z_1)) - ( u - \underline{u} )_n (z_1) \,\Gamma_{11}^n (z_1)
\end{aligned}
\end{equation*}
Assume $(u - \underline{u})_n (z_1) \,\Gamma_{11}^n (z_1) > \frac{1}{2} ( \nabla'_{11} \underline{u} (z_1) + \underline{u} (z_1))$, for, otherwise we are done. Since the function $\Gamma_{11}^n$ is continuous and $0 < ( u - \underline{u} )_n (z_1) \leq C$,
\[ \Gamma_{11}^n (z) \geq \frac{1}{2} \,\Gamma_{11}^n (z_1) \geq c_2 > 0 \quad \quad\mbox{on} \quad
 \Omega_\delta = \{ z \in \Omega | \,\mbox{dist}_{\mathbb{S}^n}(z_1, z) < \delta  \}  \]
for some small $\delta > 0$.
Now consider
\[ \Phi = \frac{\nabla'_{11} \varphi + \varphi - M }{\Gamma_{11}^n} - (u - \varphi)_n \]
Since $\nabla'_{11}(u - \varphi) = - (u - \varphi)_n\, \Gamma_{11}^n$ on $\partial\Omega$,
\[ \nabla'_{11} \varphi + \varphi - (u - \varphi)_n\, \Gamma_{11}^n = \nabla'_{11} u + u \geq M \]
Thus, $\Phi \geq 0$ on $\partial\Omega \cap \Omega_\delta$.
Also, by \eqref{eq3-19}
\begin{equation}\label{eq3-31}
L(\Phi) = \, L \Big( \frac{\nabla'_{11} \varphi + \varphi - M }{\Gamma_{11}^n} + \varphi_n \Big) - L u_n
\leq  \, C ( 1 + \sum G^{ii} )
\end{equation}
Now choose $B$ sufficiently large such that $\Psi + \Phi \geq 0$ on $\partial \Omega_\delta$. By \eqref{eq3-29} and \eqref{eq3-31} we then choose $A$ sufficiently large such that
$L(\Psi + \Phi) \leq 0$ in $\Omega_{\delta}$. Since $(\Psi + \Phi)(z_1) = 0$, we have $(\Psi + \Phi)_n (z_1) \geq 0$ and consequently
\[ u_{nn} (z_1) \leq C. \]
Together with \eqref{eq3-27} and \eqref{eq3-28}, a bound $|\nabla'^2 u (z_1)| \leq C$ can be obtained and hence a bound for all principle curvatures at $z_1$ by \eqref{eq3-18}. Therefore, the principle curvatures at $z_1$ admit a uniform positive lower bound by \eqref{eq1-7}, which in turn yields a positive lower bound for the eigenvalues of $\nabla'^2 u (z_1) + u(z_1) I $.  Hence \eqref{eq13} is proved.

By \eqref{eq13} and Lemma 1.2 in \cite{CNSIII} there exists a constant $R > 0$ depending on the estimates \eqref{eq3-27} and \eqref{eq3-28} such that if $u_{nn}(z_0) \geq R$ and $z_0 \in \partial \Omega$, then the eigenvalues $(\lambda_1, \ldots, \lambda_n)$ of $\nabla'^2 u (z_0) + u(z_0) I $ satisfy
\[ \frac{c_1}{2} \leq \lambda_\alpha \leq C, \quad \alpha = 1, \ldots, n - 1, \quad \lambda_n \geq \frac{R}{2} \]
Consequently
\[ \nabla'^2 u (z_0) + u(z_0) I \geq X^{-1} \Lambda X  \]
where $X$ is an orthogonal matrix and  $\Lambda = \mbox{diag}\Big( \frac{c_1}{2},\, \ldots,\, \frac{c_1}{2},\, \frac{R}{2} \Big)$.
Hence at $z_0$,
\begin{equation*}
\begin{aligned}
& G( \nabla'^2 u, \nabla' u, u ) (z_0) \\
= \,\, &F \Big(\frac{- \phi \zeta'(u)}{\sqrt{\phi^2 + \zeta'^2 |\nabla' u|^2 }} \, g^{-1/2} \big( \nabla'^2 u (z_0) + u(z_0) I \big)\, g^{-1/2} \Big) \\
\geq \,\, &F \Big(\frac{- \phi \zeta'(u)}{\sqrt{\phi^2 + \zeta'^2 |\nabla' u|^2 }} \, g^{-1/2} X^{-1} \Lambda X\, g^{-1/2} \Big) \\
=  \,\,& F \Big(\frac{- \phi \zeta'(u)}{\sqrt{\phi^2 + \zeta'^2 |\nabla' u|^2 }} \, \Lambda^{1/2} \,X \, g^{-1} \,X^{-1}\,\Lambda^{1/2} \Big) \\
\geq \, \,& F \Big( \frac{- \phi \zeta'(u)}{\sqrt{\phi^2 + \zeta'^2 |\nabla' u|^2 }} \, \Lambda^{1/2} \,X  \, \frac{1}{ \phi^2 + \zeta'^2(u) |\nabla' u|^2} \,I \,X^{-1}\,\Lambda^{1/2} \Big) \\
=   \,\,& F \Big( \frac{- \phi \zeta'(u)}{(\phi^2 + \zeta'^2 |\nabla' u|^2 )^{3/2}} \,\Lambda \Big) \geq \,F ( \tilde{c} \,\Lambda )
\end{aligned}
\end{equation*}
By \eqref{eq1-11} we can choose $R$ sufficiently large such that
$F ( \tilde{c} \,\Lambda ) > \sup_{\bar{\Omega} \times [C_0^{-1}, C_0]} \psi$. It follows that
\[ G( \nabla'^2 u, \nabla' u, u ) (z_0) > \psi(z_0, u(z_0))\]
which is a contradiction to equation \eqref{eq2-13}. Hence $\nabla'_{nn} u \leq R$ on $\partial\Omega$ and \eqref{eq3-32} is proved.

\vspace{5mm}

\section{Global curvature estimates}

\vspace{3mm}

The ideas for deriving global $C^2$ a priori estimates for starshaped compact or convex hypersurfaces can be found in \cite{JL05, SX15, GRW15} (see also \cite{CNSV} for vertical graphs). For strictly locally convex hypersurfaces, we synthesize the ideas in \cite{JL05, SX15} to estimate from above for the largest principal curvature $\kappa_{\max} = \max_{1 \leq i \leq n} \kappa_i$ of $\Sigma$, which, together with \eqref{eq5-1}, \eqref{eq3-32} and \eqref{eq3-5} implies an estimate for $\Vert \rho \Vert_{C^2(\overline{\Omega})}$.

\begin{thm} \label{Theorem 1}
Assume \eqref{eq1-5}, \eqref{eq1-6} and \eqref{eq1-10}. Let $ \Sigma = \{ ( z, \rho (z) ) |\, z \in \Omega \subset \mathbb{S}^n \} \subset N^{n + 1} (K)$ be a strictly locally convex $C^4$ hypersurface satisfying equation
\eqref{eq1-0} for some positive $C^2$ function $\psi$ defined on $N^{n + 1} (K)$. Suppose in addition that
\begin{equation*}
 0 < C_0^{-1} \leq \rho(z) \leq C_0 < \rho_U^K \quad \mbox{and}  \quad |\nabla' \rho| \leq C_1 \quad \mbox{on} \quad \overline{\Omega}
\end{equation*}
where $C_0$ and $C_1$ are uniform positive constants. Then there exists a constant $C$ depending only on $C_0$, $C_1$, $\Vert\psi\Vert_{C^2}$ and $\inf\psi$ such that
\[ \max\limits_{\substack{ z \in \Omega \\  i = 1, \ldots, n}}  \kappa_i (z)  \leq C \,( 1 +  \max\limits_{\substack{z \in \partial \Omega\\ i = 1, \ldots, n}}  \kappa_i (z) )  \]
\end{thm}

\begin{proof}
Since $\kappa_i > 0$ for all $i$ on $\Sigma$, it suffices to estimate from above for the largest principal curvature $\kappa_{\max}$ of $\Sigma$. To construct a test function, we will make use of the following ingredients:
\[  \Phi(\rho) = \int_0^{\rho} \phi(r) \,d r \]
and the support function
\[ \tau = \bar{g}( V,\, \nu ) = \langle V,\, \nu \rangle  \]
Note that $\tau$ has a positive lower bound
\begin{equation*}
\tau =  \,\Big\langle \phi(\rho) \frac{\partial}{\partial \rho}, \frac{- \nabla' \rho + \phi^2 \,\frac{\partial}{\partial \rho}}{\sqrt{\phi^4 + \phi^2 |\nabla' \rho|^2}} \Big\rangle  \geq \, 2 a > 0
\end{equation*}
Now define the test function
\begin{equation*}
\Theta = \ln \kappa_{\max} - \,\ln ( \tau - a ) + \beta\, \Phi
\end{equation*}
Assume $\Theta$ achieves its maximum value at ${\bf x}_0 = (z_0, \rho(z_0))\in \Sigma$.
Choose a local orthonormal frame $E_1, \ldots, E_n$ around ${\bf x}_0$ on $\Sigma$ such that $h_{ij}({\bf x}_0) = \kappa_i \,\delta_{ij}$, where $\kappa_1, \ldots, \kappa_n$ are the principal curvatures of $\Sigma$ at ${\bf x}_0$.  We may assume $\kappa_1 = \kappa_{\max} ({\bf x}_0) \geq 1$. Then, at ${\bf x}_0$,
\begin{equation} \label{eq2-3}
\frac{h_{11i}}{h_{11}}  -  \frac{{\tau}_i}{\tau - a} + \beta \,\Phi_i = 0
\end{equation}

\begin{equation} \label{eq2-4}
\frac{h_{11ii}}{h_{11}} - \frac{h_{11i}^2}{h_{11}^2} \,- \frac{{\tau}_{ii}}{\tau - a} + \Big(\frac{{\tau}_i}{\tau - a}\Big)^2 + \, \beta \,\Phi_{ii}\, \leq 0
\end{equation}
By Codazzi equation and Gauss equation we have
\begin{equation} \label{eq2-9}
\nabla_l h_{ij} = \nabla_j h_{il}
\end{equation}
and
\begin{equation} \label{eq2-5}
 h_{iill} = h_{llii} + \kappa_l \kappa_i^2 - \kappa_l^2 \kappa_i + K (\kappa_i - \kappa_l)
\end{equation}

In the rest of this section all computations are evaluated at ${\bf x}_0$.
Under the local orthonormal frame $E_1, \ldots, E_n$, equation \eqref{eq1-0} appears as
\begin{equation} \label{eq2-32}
 F(h) = f(\lambda (h)) = \psi(V) \quad\quad\mbox{where}\quad h = \{ h_{ij} \}
\end{equation}
Covariantly differentiate \eqref{eq2-32} twice we have
\begin{equation} \label{eq2-6}
 F^{ii} h_{iil} = \phi'\, d_V\psi (E_l)
\end{equation}
and
\begin{equation} \label{eq2-7}
F^{ii} h_{ii11} + F^{ij,\, kl} h_{ij1} h_{kl1} \geq - C \kappa_1
\end{equation}
Here we have used the property of the conformal Killing field $V$,
\[ \nabla_{E_l} V = \phi' \,E_l  \]
Combining \eqref{eq2-4}, \eqref{eq2-5} and \eqref{eq2-7} we have
\begin{equation} \label{eq2-8}
\begin{aligned}
& - \frac{1}{\kappa_1} F^{ij, kl} h_{ij1} h_{kl1} - \,\frac{1}{\kappa_1^2} \sum f_i  h_{11i}^2 - \sum f_i \kappa_i^2 \, + ( \kappa_1 - \frac{K}{\kappa_1} ) \sum f_i \kappa_i \\
& \,+ K \, \sum f_{i} - C \,- \frac{1}{\tau - a} \sum F^{ii} {\tau}_{ii} + \,\frac{1}{(\tau - a)^2} \sum f_i {\tau}_i^2 \,+ \,\beta
  \sum F^{ii} \Phi_{ii}  \leq 0
\end{aligned}
\end{equation}
Now we partition $\{1, \ldots, n\}$ into two parts,
\[ I = \{ j: \,f_j \leq 2 f_1 \}, \quad \quad  J = \{ j: \,f_j >  2 f_1 \}\]
By \eqref{eq2-3}, for any $\epsilon > 0$,
\begin{equation} \label{eq2-33}
 \frac{1}{\kappa_1^2} \sum\limits_{i\in I} f_i  h_{11i}^2 \,\leq  \, C (1 + \epsilon^{-1}) \beta^2 f_1 \sum \Phi_i^2  + \frac{(1 + \epsilon)}{(\tau - a)^2} \sum f_i {\tau}_i^2
\end{equation}
Taking \eqref{eq2-33}, \eqref{eq2-6} and the following equations (see Lemma 2.2 and Lemma 2.6 in \cite{GL13} for the proof)
\[ \Phi_i = \phi(\rho) \,\rho_i, \quad\quad \Phi_{ii} = \phi' - \tau \,\kappa_i\]
\[ {\tau}_i = \phi(\rho)\, \rho_i\, \kappa_i \]
\[ {\tau}_{ii} = \phi(\rho)\,\sum\limits_m \rho_m\, h_{iim} + \phi'(\rho)\,\kappa_i - \tau \,\kappa_i^2 \]
into \eqref{eq2-8} yields
\begin{equation} \label{eq2-10}
\begin{aligned}
&   - \frac{1}{\kappa_1} F^{ij, kl} h_{ij1} h_{kl1} - \,\frac{1}{\kappa_1^2} \sum\limits_{i \in J} f_i \, h_{11i}^2 \\  & + \,\big( \frac{a}{\tau - a} - \frac{C \epsilon}{( \tau - a )^2} \big) \sum f_{i} \kappa_i^2 \,- C \beta^2 (1 + \epsilon^{-1})\, f_1 \\
& + ( \kappa_1 - \frac{K}{\kappa_1} - \tau \beta - \frac{\phi'}{\tau - a} ) \sum f_i \kappa_i \,+ ( \beta \phi' + K ) \sum f_{i} \,- C - \frac{C}{a}  \leq 0
\end{aligned}
\end{equation}
Using an inequality due to Andrews \cite{And} and Gerhardt \cite{Ger} and applying \eqref{eq2-9}
\begin{equation*}
- \frac{1}{\kappa_1} F^{ij, kl} h_{ij1} h_{kl1} \geq  \frac{1}{\kappa_1} \sum\limits_{i \neq j} \frac{f_i - f_j}{\kappa_j - \kappa_i} h_{ij1}^2
\geq \frac{2}{\kappa_1} \sum\limits_{i \geq 2} \frac{f_i - f_1}{\kappa_1 - \kappa_i} h_{11i}^2 \geq \frac{1}{ \kappa_1^2} \sum\limits_{i \in J} f_i \, h_{11i}^2
\end{equation*}
where the fractions are interpreted as limits whenever the denominators are zero. Inserting it into \eqref{eq2-10}, applying assumption \eqref{eq1-10}, choosing $\epsilon = {a^2}/{(2 C)}$ and $\beta = u_L^K$ we obtain
\begin{equation*}
\frac{a^2}{2 (\tau - a)^2}  \sum f_{i} \kappa_i^2 \, - C \beta^2 ( 1 + \frac{1}{a^2} ) \, f_1 \,
+ ( \kappa_1 - 1 - \tau \beta - \frac{\phi'}{\tau - a} ) \,\sigma_0 \,- C - \frac{C}{a} \leq 0
\end{equation*}
Since $\sum f_{i} \kappa_i^2  \geq f_1 \kappa_1^2 $, a uniform upper bound for $\kappa_1$ follows easily from the above inequality. Consequently, we obtain a uniform upper bound for $\kappa_{\max}$ on $\Sigma$.
\end{proof}

\vspace{5mm}

\section{Existence in $\mathbb{R}^{n+1}$ and $\mathbb{H}^{n+1}$}

\vspace{3mm}

In this section, we will use classical continuity method and degree theory developed by Y. Y. Li \cite{Li89} to prove the existence of solution to the Dirichlet problem \eqref{eq2-26}--\eqref{eq2-25}. Under the transformation $\overline{\rho} = \zeta ( \underline{u} )$ and $\underline{u} = \eta ( \underline{v} )$, the subsolution condition \eqref{eq1-4} can be expressed as
\begin{equation} \label{eq2-29}
\left\{ \begin{aligned}
\mathcal{G}(\nabla'^2 \underline{v}, \nabla' \underline{v}, \underline{v}) \geq & \,\psi(z, \underline{v})  \quad  \mbox{in} \quad \Omega\\
\underline{v} = & \,\varphi  \quad \quad \mbox{on} \quad \partial \Omega \end{aligned} \right.
\end{equation}
Assume that $\underline{v}$ is not a solution of \eqref{eq2-26}, for otherwise we are done.
We consider the following two auxiliary equations.

\begin{equation} \label{eq6-8}
\left\{ \begin{aligned} \mathcal{G} (\nabla'^2 v, \nabla' v, v) \, =  & \, \big( t \epsilon + ( 1 - t ) \frac{\underline{\psi} (z)}{ \xi (\underline{v}) } \big)  \,\xi (v) \quad\quad & \mbox{in} \quad \Omega \\ v \,  = & \, \underline{v} \quad \quad & \mbox{on} \quad \partial\Omega \end{aligned} \right.
\end{equation}
and
\begin{equation} \label{eq6-9}
\left\{ \begin{aligned} \mathcal{G} (\nabla'^2 v, \nabla' v, v) \, =  & \,  t \, \psi(z, v) + ( 1 - t ) \,\epsilon \,\xi (v) \quad\quad & \mbox{in} \quad \Omega \\ v \,  = & \, \underline{v} \quad \quad & \mbox{on} \quad \partial\Omega \end{aligned} \right.
\end{equation}
where $t \in [0, 1]$, $\underline{\psi} (z) = \mathcal{G}( \nabla'^2 \underline{v}, \nabla' \underline{v}, \underline{v} ) (z)$, $\epsilon$ is a small positive constant such that
\begin{equation} \label{eq6-20}
\underline{\psi} (z) \,>  \epsilon \,\xi(\underline{v}) \quad\mbox{in}\quad \Omega
\end{equation}
and $\xi(v) = e^{2v}$ if $K = 0$ while $\xi(v) = \sinh v$ if $K = - 1$. The existence results in $\mathbb{R}^{n+1}$ was given in \cite{Su16} where the author assumed the existence of a strict subsolution. In this section, we will consider the case when $K = 0$ or $K = -1$ with the subsolution assumption.

\begin{lemma} \label{Lemma6-1}
Let $\psi(z)$ be a positive function defined on $\Omega$. For $z \in \Omega$ and a strictly locally convex function $v$ near $z$, if
\[\mathcal{G}(\nabla'^2 v, \nabla' v, v) (z) =  F(a_{ij}[v])(z) = f (\kappa [v])(z) = \psi(z) \,\xi(v)(z) \]
then
\[ \mathcal{G}_v (\nabla'^2 v, \nabla' v, v) (z) - \,\psi(z) \,\xi'(v)(z)  < 0\]
\end{lemma}
\begin{proof}
The proof can be found in \cite{Sui1}.
\end{proof}

\begin{lemma}  \label{Lemma6-2}
For any fixed $t \in [0, 1]$, if $\underline{V}$ and $v$ are strictly locally convex subsolution and solution to \eqref{eq6-8}, then $v \geq \underline{V}$.  Thus the Dirichlet problem \eqref{eq6-8} has at most one strictly locally convex solution.
\end{lemma}
\begin{proof}
The proof is similar to Lemma 3.40 in \cite{Sui2}.
\end{proof}

\begin{lemma} \label{Lemma6-3}
Let $v$ be a strictly locally convex solution of \eqref{eq6-9}. If $v \geq \underline{v}$ in $\Omega$, then
$v > \underline{v}$ in $\Omega$ and ${\bf n}(v - \underline{v}) > 0$ on $\partial\Omega$, where ${\bf n}$ is the interior unit normal to $\partial\Omega$.
\end{lemma}

\begin{proof}
By \eqref{eq2-29} and \eqref{eq6-20} we know that $\underline{v}$ is a strict subsolution of \eqref{eq6-9} when $t \in [0, 1)$, while it is a subsolution but not a solution of \eqref{eq6-9} when $t = 1$. It is relatively easy to prove the conclusion when $t \in [0, 1)$, following the ideas in \cite{Su16}. For the case $t = 1$:
\begin{equation*}
\left\{ \begin{aligned} \mathcal{G} (\nabla'^2 v, \nabla' v, v) \, =  & \,   \psi(z, v) \quad & \mbox{in} \quad \Omega \\ v \,  = & \, \underline{v} \quad \quad & \mbox{on} \quad \partial\Omega \end{aligned} \right.
\end{equation*}
we will make use of the maximum principle which was originally discovered in \cite{Serrin}, while more precisely stated for our purposes in \cite{GNN} (see section 1.3, p. 212). Because the maximum principle and Hopf lemma there are designed for domains in Euclidean spaces, we need to rewrite the above equation in a local coordinate system of $\mathbb{S}^n$. For convenience, we first transform the above equation back under the transformation \eqref{eq6-2} into a form as \eqref{eq2-13}:
\begin{equation} \label{eq5-7}
\left\{ \begin{aligned}
G(\nabla'^2 u, \nabla' u, u) = & \,\psi(z, u)  \quad \quad  &\mbox{in} \quad & \Omega \\
u = & \,\underline{u}  \quad \quad & \mbox{on} \quad & \partial\Omega \end{aligned} \right.
\end{equation}
Recall that $G(\nabla'^2 u, \nabla' u, u) = F(A[u])$ where $A[u] = \{ \gamma^{ik} h_{kl} \gamma^{lj} \}$. Since at this time we do not use local orthonormal frame on $\mathbb{S}^n$, but rather a local coordinate system of $\mathbb{S}^n$, $\gamma^{ik}$ and $h_{kl}$ will appear differently (comparing with \eqref{eq3-10} and \eqref{eq3-13}). Also,
condition \eqref{eq2-29} (i.e. \eqref{eq1-4}) can be rewritten as
\begin{equation*}
\left\{ \begin{aligned}
G(\nabla'^2 \underline{u}, \nabla' \underline{u}, \,\underline{u}) \geq & \,\,\psi(z, \underline{u})  \quad  &\mbox{in} \quad \Omega\\
\underline{u} = & \,\,\varphi  \quad \quad &\mbox{on} \quad \partial \Omega \end{aligned} \right.
\end{equation*}
Note that $\underline{u}$ is not a solution of \eqref{eq5-7}.

(i) We first show that if a strictly locally convex solution $u$ of \eqref{eq5-7} satisfies $u \geq \underline{u}$ in $\Omega$, then $u > \underline{u}$ in $\Omega$. Let $N \notin \Omega$ be the north pole of $\mathbb{S}^n$. Take the radial projection of $\mathbb{S}^n \setminus \{N\}$ onto $\mathbb{R}^n \times \{ -1 \} \subset \mathbb{R}^{n+1}$ and let $\tilde{\Omega}$ be the image of $\Omega$. We thus have a coordinate system $(x_1, \ldots, x_n)$ on $\mathbb{R}^n \times \{ -1 \} \cong \mathbb{R}^n$.  The metric on $\mathbb{S}^n$, its inverse, and the Christoffel symbols are given respectively by
\[ \sigma_{ij} = \frac{16}{\mu^2} \,\delta_{ij}, \quad \mu = 4 + \sum x_i^2, \quad \sigma^{ij} = \frac{\mu^2}{16}\, \delta_{ij} \]
\[ {\Gamma}_{ij}^k = - \frac{2}{\mu} \,(\delta_{ik} x_j + \delta_{jk} x_i - \delta_{ij} x_k) \]
Consequently, the metric on $\Sigma$, its inverse and the second fundamental form on $\Sigma$ are given respectively by (c.f. \cite{SX15})
\begin{equation*}
g_{ij} = \phi^2 \,\sigma_{ij} + \zeta'^2(u)\, u_i u_j
\end{equation*}
\begin{equation*}
g^{ij} = \frac{1}{\phi^2} \Big( \sigma^{ij} - \frac{\zeta'^2(u) u^i u^j}{\phi^2 + \zeta'^2(u) |\nabla' u|^2} \Big), \quad u^i = \sigma^{ik} u_k
\end{equation*}
\begin{equation*}
h_{ij} =  \frac{- \zeta'(u) \phi}{\sqrt{\phi^2 + \zeta'^2(u) |\nabla' u|^2}} ( \nabla'_{ij} u + u \,\sigma_{ij} )
\end{equation*}
The entries of the symmetric matrices $\{\gamma_{ik}\}$ and $\{\gamma^{ik}\}$ depend only on $x_1, \ldots, x_n$, $u$ and the first derivatives of $u$.

Now, setting $\tilde{u} = \mu u$ and by straightforward calculation we have
\begin{equation} \label{eq5-10}
\nabla'_{ij} u + u \,\sigma_{ij} = \frac{1}{\mu} \tilde{u}_{ij} + \frac{2 \delta_{ij}}{\mu^2} ( \tilde{u} - \sum\limits_k x_k \tilde{u}_k )
\end{equation}
and \eqref{eq5-7} can be transformed into the following form:
\begin{equation*}
\left\{ \begin{aligned}
\tilde{G}(D^2\tilde{u}, D \tilde{u}, \tilde{u}, x_1, \ldots, x_n)  = F\Big( A\Big[ \frac{\tilde{u}}{\mu} \Big]\Big) = & \,\tilde{\psi}(x_1, \ldots, x_n, \tilde{u})  \quad \quad  &\mbox{in} \quad & \tilde{\Omega} \\
\tilde{u} = & \,\mu \,\underline{u}  \quad \quad & \mbox{on} \quad & \partial\tilde{\Omega} \end{aligned} \right.
\end{equation*}
where $\tilde{u}_i = \frac{\partial\tilde{u}}{\partial x_i}$, $D \tilde{u} = ( \tilde{u}_1, \ldots, \tilde{u}_n )$, $\tilde{u}_{ij} = \frac{\partial^2 \tilde{u}}{\partial x_i \partial x_j}$ and $D^2 \tilde{u} = \{ \tilde{u}_{ij} \}$.

In view of \eqref{eq5-10} and \eqref{eq2-14} we know that
\[ \frac{\partial \tilde{G}}{\partial \tilde{u}_{ij}} = \frac{\partial F}{\partial a_{kl}} \,\frac{\partial a_{kl}}{\partial \tilde{u}_{ij}} = \frac{1}{\mu} \, \frac{\partial G}{\partial u_{ij}}  \]
Also, the function $\tilde{\underline{u}} = \mu \underline{u}$ satisfies
\begin{equation*}
\left\{ \begin{aligned}
\tilde{G}(D^2 \tilde{\underline{u}}, D \tilde{\underline{u}}, \tilde{\underline{u}}, x_1, \ldots, x_n) \geq & \,\tilde{\psi}(x_1, \ldots, x_n, \tilde{\underline{u}})  \quad  \mbox{in} \quad \tilde{\Omega}\\
\tilde{\underline{u}} = & \,\mu\underline{u}  \quad \quad \mbox{on} \quad \partial \tilde{\Omega} \end{aligned} \right.
\end{equation*}
Hence we can apply the Maximum Principle (see p. 212 of \cite{GNN}) to conclude that $\tilde{u} > \tilde{\underline{u}}$ in $\tilde{\Omega}$, which immediately yields $u > \underline{u}$ in $\Omega$.

(ii) To prove ${\bf n}(u - \underline{u}) > 0$ on $\partial\Omega$, we pick an arbitrary point $z_0 \in \partial\Omega$ and assume $z_0$ to be the north pole of $\mathbb{S}^n \subset \mathbb{R}^{n + 1}$. We introduce a local coordinate system about $z_0$ by taking the radial projection of the upper hemisphere onto the tangent hyperplane of $\mathbb{S}^n$ at $z_0$ and identifying this hyperplane to $\mathbb{R}^n$. Denote the coordinates by $(y_1, \ldots, y_n)$  and assume that the positive $y_n$-axis is the interior normal direction to $\partial\Omega \subset \mathbb{S}^n$ at $z_0$. In this coordinate system, the metric on $\mathbb{S}^n$, its inverse, and the Christoffel symbols are given respectively by (see \cite{Oliker, GS93})
\[ \sigma_{ij} = \frac{1}{\mu^2} \Big( \delta_{ij} - \frac{y_i y_j}{\mu^2} \Big), \quad \mu = \sqrt{1 + \sum y_i^2}  \]
\[ \sigma^{ij} = \mu^2 ( \delta_{ij} + y_i y_j ) \]
\[ {\Gamma}_{ij}^k = - \frac{\delta_{ik} y_j + \delta_{jk} y_i}{\mu^2} \]
The metric $g_{ij}$, its inverse $g^{ij}$ and the second fundamental form $h_{ij}$ on $\Sigma$ have the form as above.
The entries of the symmetric matrices $\{\gamma_{ik}\}$ and $\{\gamma^{ik}\}$  depend only on $y_1, \ldots, y_n$, $u$ and the first derivatives of $u$.

Now set $\tilde{u} = \mu u$. By straightforward calculation we have
\begin{equation} \label{eq5-8}
\nabla'_{ij} u + u \,\sigma_{ij} = \mu^{- 1} \tilde{u}_{ij}
\end{equation}
and equation \eqref{eq5-7} can be transformed into an equation defined in an open neighborhood of $0$ on $\mathbb{R}^n$, which is the radial projection of a neighborhood of $z_0$ on $\mathbb{S}^n$:
\begin{equation*}
\tilde{G}(D^2\tilde{u}, D \tilde{u}, \tilde{u}, y_1, \ldots, y_n)  = F\Big( A\Big[ \frac{\tilde{u}}{\mu} \Big]\Big) =  \,\tilde{\psi}(y_1, \ldots, y_n, \tilde{u})
\end{equation*}
where $\tilde{u}_i = \frac{\partial\tilde{u}}{\partial y_i}$, $D \tilde{u} = ( \tilde{u}_1, \ldots, \tilde{u}_n )$, $\tilde{u}_{ij} = \frac{\partial^2 \tilde{u}}{\partial y_i \partial y_j}$ and $D^2 \tilde{u} = \{ \tilde{u}_{ij} \}$.
In view of \eqref{eq5-8} and \eqref{eq2-14} we know that
\[ \frac{\partial \tilde{G}}{\partial \tilde{u}_{ij}} = \frac{\partial F}{\partial a_{kl}} \,\frac{\partial a_{kl}}{\partial \tilde{u}_{ij}} = \frac{1}{\mu} \, \frac{\partial G}{\partial u_{ij}}  \]
Applying Lemma H (see p. 212 of \cite{GNN}) we find that $(\tilde{u} - \tilde{\underline{u}})_n (0) > 0$ and equivalently ${\bf n}(u - \underline{u}) (z_0) > 0$.
\end{proof}

\begin{thm} \label{Theorem6-1}
For any $t \in [0, 1]$, the Dirichlet problem \eqref{eq6-8} has a unique strictly locally convex solution.
\end{thm}
\begin{proof}
The proof is the same as in \cite{Sui1}.
\end{proof}

\begin{thm} \label{Theorem6-2}
For any $t \in [0, 1]$, the Dirichlet problem \eqref{eq6-9} has a strictly locally convex solution. In particular, \eqref{eq2-26}--\eqref{eq2-25} has a strictly locally convex solution.
\end{thm}

\begin{proof}
The proof is the same with \cite{Sui1} except slight modifications.
\end{proof}

\vspace{5mm}

\section{Existence in \, $\mathbb{S}^{n + 1}_{+}$}

\vspace{5mm}

For any $\epsilon > 0$,  we want to prove the existence of a strictly locally convex solution to the Dirichlet problem when $K = 1$,
\begin{equation} \label{eq7-2}
\left\{
\begin{aligned}
G[u] := G( \nabla'^2 u, \nabla' u, u ) \,\,= & \,\, \psi(z, u) - \epsilon  \quad\quad &\mbox{in} \quad \Omega \\
u \,\, = & \,\, \varphi   &\mbox{on} \quad \partial\Omega
\end{aligned} \right.
\end{equation}
Then a strictly locally convex solution to  \eqref{eq2-13}--\eqref{eq2-12} follows from the uniform ($\epsilon$-independent) $C^2$ estimates (established in Section 3 and 4) and approximation.

As we have seen from last section, there does not exist an auxiliary equation in $\mathbb{S}^{n+1}_+$ with an invertible linearized operator. Hence we want to build a continuity process starting from an auxiliary equation in $\mathbb{R}^{n+1}$.

For this, we first consider a continuous version of \eqref{eq3-18}. For $t \in [0, 1]$, denote
\[a^t_{ij} = \frac{- (\zeta^t)' \phi^t}{\sqrt{(\phi^t)^2 + (\zeta^t)'^2 |\nabla' u|^2}} (\gamma^t)^{ik}\, ( \nabla'_{kl} u + u \,\delta_{kl} )\,(\gamma^t)^{lj}\]
where
\[(\gamma^t)^{ik} =   \,\frac{1}{\phi^t}\, \Big( \delta_{ik} - \frac{ (\zeta^t)'^2(u) u_i u_k }{\sqrt{(\phi^t)^2 + (\zeta^t)'^2(u) |\nabla' u|^2 } ( \phi^t + \sqrt{(\phi^t)^2 + (\zeta^t)'^2(u) |\nabla' u|^2 } )}\Big)\]
and
\[ \phi^t(\rho) = \frac{\sin (t \rho)}{t}, \quad \quad  \zeta^t(u) = \frac{1}{t} \, \mbox{arccot} \frac{u}{t} \]
Note that these geometric quantities on $\Sigma$ correspond to the background metric
\[ \bar{g}^t = d \rho^2 + (\phi^t)^2(\rho)\, \sigma \]
which provides a deformation process from $\mathbb{R}^{n+1}$ ($t = 0$) to $\mathbb{S}^{n+1}_+$ ($t = 1$).
Define
\[ G^t [ u ] \, = \, G^t( \nabla'^2 u, \nabla' u, u ) \, = \,F( a^t_{ij} ) \]
Hence $G^1 = G$.
The following property is true by direct calculation.
\begin{prop}
$G^t [u]$ is increasing with respect to $t$.
\end{prop}
\begin{proof}
\[ a_{ij}^t = \,\Big(1 + \frac{|\nabla' u|^2}{u^2 + t^2} \Big)^{- \frac{1}{2}}\, \,\tilde{\gamma}^{ik}\,\big( \nabla'_{kl} u \,+\, u \,\delta_{kl} \big)\,\tilde{\gamma}^{lj} \]
where
\[ \tilde{\gamma}^{ik} \,=\, \delta_{ik} - \frac{u_i \,u_k}{\sqrt{u^2 + t^2 + |\nabla' u|^2} \,\big( \sqrt{u^2 + t^2} + \sqrt{u^2 + t^2 + |\nabla' u|^2} \big)} \]

\[ \begin{aligned}
\,\frac{\partial}{\partial t} \,G^t[ u ]  = \,& \Big( 1 + \frac{|\nabla' u|^2}{u^2 + t^2} \Big)^{- 3/2}F^{ij}  \cdot \\ &  \Big(  \frac{t |\nabla' u|^2}{(u^2 + t^2)^2}  \tilde{\gamma}^{ik} +  2 \Big( 1 + \frac{|\nabla' u|^2}{u^2 + t^2} \Big) \frac{\partial \tilde{\gamma}^{ik}}{\partial t} \Big) \big( \nabla'_{kl} u + u \delta_{kl} \big) \tilde{\gamma}^{lj}  \end{aligned} \]
The inverse $(\tilde{\gamma}_{ik})$ of $(\tilde{\gamma}^{ik})$ is given by
\[ \tilde{\gamma}_{ik} \,=\, \delta_{ik} + \frac{u_i \,u_k}{ u^2 + t^2 + \sqrt{ (u^2 + t^2)^2 + (u^2 + t^2) |\nabla' u|^2} }  \]
Since
\[  \frac{\partial \tilde{\gamma}^{ik}}{\partial t} \,=\, -  \tilde{\gamma}^{ip}\,\frac{\partial \tilde{\gamma}_{pq}}{\partial t} \, \tilde{\gamma}^{qk} \]
\[ \frac{\partial \tilde{\gamma}_{pq}}{\partial t} = \, - \frac{u_p\, u_q \,t}{(u^2 + t^2)^{3/2} \sqrt{u^2 + t^2 + |\nabla' u|^2}}\]
and
\[ \tilde{\gamma}^{ik} \,u_k \,=\, \frac{\sqrt{u^2 + t^2}}{\sqrt{u^2 + t^2 + |\nabla' u|^2}} \, u_i \]
therefore,
\[ \begin{aligned} \,\frac{\partial}{\partial t}\, G^t[ u ] = & \frac{t}{\sqrt{u^2 + t^2} \big( u^2 + t^2 + |\nabla' u|^2 \big)^{3/2} }    F^{ij} \cdot \\ & \Big( |\nabla' u|^2 \delta_{iq} + 2 u_i u_q \Big) \tilde{\gamma}^{qk}\, \big( \nabla'_{kl} u + u \delta_{kl} \big)\, \tilde{\gamma}^{lj} \geq 0
\end{aligned} \]
\end{proof}

Recall that we have assumed a strictly locally convex subsolution.
\[\left\{
\begin{aligned}
G[\underline{u}] \,\,\geq & \,\,\psi(z, \underline{u})  \quad\quad &\mbox{in} \quad \Omega \\
\underline{u} \,\, = & \,\, \varphi   &\mbox{on} \quad \partial\Omega
\end{aligned} \right.
\]
Choose $\epsilon$ small such that
\[ \epsilon < \min\{   \min\limits_{\overline{\Omega}} G^0[\underline{u}], \,\, \min\limits_{\overline{\Omega}}\psi(z, \underline{u}) \} \]
By continuity, for $t \in [1 - \delta_1, \,1]$ where $\delta_1$ is a sufficiently small positive constant depending on $\epsilon$, we have
\begin{equation} \label{eq7-3}
\left\{
\begin{aligned}
G^t[\underline{u}] \,\,> & \,\, \psi(z, \underline{u}) - \,\frac{\epsilon}{2}  \quad\quad &\mbox{in} \quad \Omega \\
\underline{u} \,\, = & \,\, \varphi   &\mbox{on} \quad \partial\Omega
\end{aligned} \right.
\end{equation}

Denote $\mathcal{G}^t [\, v\, ] \,:= \mathcal{G}^t( \nabla'^2 v, \nabla' v, v )\, =: \, G^t [\, e^v \,]$. Consider the continuity process,
\begin{equation} \label{eq7-1}
\left\{ \begin{aligned}
\mathcal{G}^t [\, v\, ] \,  =  & \, \, \big( 1 - T(t) \big) \,\delta_2 \, e^{2 v} \, + \, T(t) \,\big( \psi(z, e^v) - \epsilon \big) \quad\quad & \mbox{in} \quad \Omega \\ v \,  = & \, \,\ln \varphi \quad \quad & \mbox{on} \quad \partial\Omega \end{aligned} \right.
\end{equation}
where $\delta_2$ is a small positive constant such that
\[ \delta_2  \,\max\limits_{\overline{\Omega}} \underline{u}^2 \,<\, \frac{\epsilon}{2} \]
and $T(t)$ is a smooth strictly increasing function with $T(0) = 0$,  $T(1) = 1$ satisfying
\[  \min\limits_{\overline{\Omega}} \,G^0 [\underline{u}] \, > \,2 \,\,T(1 - \delta_1)\, \,\max\limits_{ \overline{\Omega}} \psi(z, \underline{u})  \]

\begin{prop}
$\underline{v} = \ln \underline{u}$ is a strict subsolution of \eqref{eq7-1} for any $t\in [0, 1]$.
\end{prop}
\begin{proof}
For $t \in [1 - \delta_1,  1]$,
\[ \begin{aligned}
\mathcal{G}^t [ \underline{v} ]  \,= &\, G^t [\underline{u}]  \, >   \, \psi(z, \underline{u}) - \frac{\epsilon}{2} \,> \, \delta_2  \, \underline{u}^2 +   \, \big( \psi(z, \underline{u}) - \epsilon \big) \\
\,\geq  & \, \big( 1 - T(t) \big) \, \delta_2  \, e^{2 \underline{v}} \, + \,  T(t)\,\, \big( \psi(z, e^{\underline{v}}) - \epsilon \big)
\end{aligned} \]
For $t \in [0, 1 - \delta_1]$,
\[ \begin{aligned}
\mathcal{G}^t [ \underline{v} ]  \,= & \, G^t [ \underline{u} ] \, \geq  \, G^0 [\underline{u}] \, > \, \frac{\epsilon}{2} \, +  \,T(1 - \delta_1) \, \psi(z, \underline{u}) \\ \geq &
 \,  \big( 1 - T(t) \big) \, \delta_2  \, \underline{u}^2 \, + \,  T(t)\,\, \big( \psi(z, \underline{u}) - \epsilon \big) \\ = &\, \big( 1 - T(t) \big) \, \delta_2  \, e^{2 \underline{v}} \, + \,  T(t)\,\, \big( \psi(z, e^{\underline{v}}) - \epsilon \big)
\end{aligned}
\]
\end{proof}

Now we can obtain the existence results in $\mathbb{S}^{n+1}_+$.

\begin{thm} \label{Theorem7-1}
For any $t \in [0, 1]$, the Dirichlet problem \eqref{eq7-1} has a strictly locally convex solution. In particular, \eqref{eq7-2} has a strictly locally convex solution when $K = 1$.
\end{thm}

\begin{proof}
The $C^{2, \alpha}$ estimates for strictly locally convex solutions $v$ of \eqref{eq7-1} with $v \geq \underline{v}$ is equivalent to the $C^{2, \alpha}$ estimates for strictly locally convex solutions to the Dirichlet problem
\[
\left\{ \begin{aligned}
G^t [\, u\, ] \,  =  & \, \, \big( 1 - T(t) \big) \,\delta_2 \, u^2 \, + \, T(t) \,\big( \psi(z, u) - \epsilon \big) \quad\quad & \mbox{in} \quad \Omega \\ u \,  = & \, \,\varphi \quad \quad & \mbox{on} \quad \partial\Omega \end{aligned} \right.
\]
This can be established by changing $\phi$ and $\zeta$ into $\phi^t$ and $\zeta^t$ in the previous proof.  Then $C^{4, \alpha}$ estimates follows by classical Schauder theory. Thus we have the $t$-independent uniform estimates,
\begin{equation} \label{eq7-4}
\Vert v \Vert_{C^{4,\alpha}(\overline{\Omega})} < C_4 \quad \quad \mbox{and} \quad \quad
C_2^{-1}\, I <   \,\{ v_{ij} \,+\,v_i \, v_j\, + \,\delta_{ij} \} \,  < C_2\, I    \quad \mbox{in} \quad \overline{\Omega}
\end{equation}
Consider the subspace of $C^{4,\alpha}( \overline{\Omega} )$ given by
\[ C_0^{ 4, \alpha} (\overline{\Omega}) := \{ w \in C^{ 4, \alpha}( \overline{\Omega} ) \,| \,w = 0 \,\, \mbox{on} \,\, \partial\Omega \} \]
and the bounded open subset
\[ \mathcal{O} := \left\{ w \in C_0^{4, \alpha} (\overline{\Omega}) \,\left\vert\,\begin{footnotesize}\begin{aligned} & w > 0 \,\,\mbox{in}\,\,\Omega, \quad\quad \nabla'_{\bf n}\, w > 0 \,\,\mbox{on}\,\, \partial\Omega,\\ & C_2^{-1} \,I <  \, \{ (\underline{v} + w)_{ij} \,+\,(\underline{v} + w)_i \, (\underline{v} + w)_j\, + \,\delta_{ij} \} \,  < C_2 \, I   \,\, \mbox{in} \,\, \overline{\Omega} \\ & \Vert w {\Vert}_{C^{4,\alpha}(\overline{\Omega})} < C_4 + \Vert\underline{v}\Vert_{C^{4,\alpha}(\overline{\Omega})} \end{aligned}\end{footnotesize} \right.\right\} \]
Construct a map
$\mathcal{M}_t (w):  \,\mathcal{O} \times [ 0, 1 ] \rightarrow C^{2,\alpha}(\overline{\Omega})$,
\[ \mathcal{M}_t ( w ) = \mathcal{G}^t [ \underline{v} + w ] \, - \big( 1 - T(t) \big) \,\delta_2 \, e^{2 (\underline{v} + w )} - \, T(t) \, \big( \psi(z, \,e^{\underline{v} + w}) - \epsilon \big)\]
At $t = 0$, by Theorem \ref{Theorem6-1} for the case $K = 0$, there is a unique solution $v^0$ to
\eqref{eq7-1}.
By Lemma \ref{Lemma6-2} and Lemma \ref{Lemma6-3} we have $w^0 := v^0 - \underline{v}> 0$ in $\Omega$ and $\nabla'_{\bf n}\, w^0 > 0$ on $\partial\Omega$.  Moreover, $w^0$ satisfies \eqref{eq7-4} and thus $w^0 \in \mathcal{O}$.
Also, Lemma \ref{Lemma6-3} and  \eqref{eq7-4} implies that $\mathcal{M}_t( w ) = 0$ has no solution on $\partial\mathcal{O}$ for any $t \in [0, 1]$.
Besides, $\mathcal{M}_t$ is uniformly elliptic on $\mathcal{O}$ independent of $t$. Therefore,
$\deg (\mathcal{M}_t, \mathcal{O}, 0)$, the degree of $\mathcal{M}_t$ on $\mathcal{O}$ at $0$,
is well defined and independent of $t$.
Hence it suffices to compute $\deg (\mathcal{M}_0, \mathcal{O}, 0)$.

Note that $\mathcal{M}_0 ( w ) = 0$ has a unique solution $w^0 \in \mathcal{O}$. The Fr\'echet derivative of $\mathcal{M}_0$ with respect to $w$ at $w^0$ is a linear elliptic operator from $C^{4, \alpha}_0 (\overline{\Omega})$ to $C^{2, \alpha}(\overline{\Omega})$,
\begin{equation}
\mathcal{M}_{0, w} |_{w^0} ( h )  =  \,  (\mathcal{G}^0)^{ij}[ v^0 ] \nabla'_{ij} h  + (\mathcal{G}^0)^i [ v^0 ]  \nabla'_i h   + \big( (\mathcal{G}^0)_v [v^0]  - \,2 \,\delta_2 \, e^{2 v^0} \big) h
\end{equation}
By Lemma \ref{Lemma6-1}
\[ (\mathcal{G}^0)_v [v^0]  - \,2 \,\delta_2 \, e^{2 v^0}\, < 0 \quad\mbox{in} \quad \Omega \]
Thus $\mathcal{M}_{0,w} |_{w^0}$ is invertible. Applying the degree theory in \cite{Li89},
\[ \deg (\mathcal{M}_0, \mathcal{O}, 0) = \deg( \mathcal{M}_{0, w} |_{w^0}, B_1, 0) = \pm 1 \neq 0 \]
where $B_1$ is the unit ball in $C_0^{4,\alpha}(\overline{\Omega})$. Thus
\[ \deg(\mathcal{M}_t, \mathcal{O}, 0) \neq 0 \quad\mbox{for}\,\,\mbox{all}\,\,t \in [0, 1]\]
and this theorem is proved.
\end{proof}

\vspace{3mm}

config: arxiv


TITLE: Sigma algebra generated by sets in the range of a function
QUESTION [1 upvotes]: This is Bartle's The Elements of Integration, Exercise 2.O.
Let $(X,\bf{X})$ be a measurable space and $f$ defined on $X$ to $Y$.Let $\bf A$ be a collection of subsets of $Y$ such that $\forall E\in \bf{A}$$: f^{-1}(E)\in \bf X$. Show that $\forall F\in \sigma(\bf{A})$$: f^{-1}(F)\in \bf X$, where $\sigma(\bf{A})$ is the sigma algebra generated by $\bf A$.
I thought of this solution:
Let $\bf Y$ be a collection of subsets of $Y$ defined as:
$\bf Y$$=\left\{E\subset Y: f^{-1}(E)\in\bf{X}\right\}$$\implies \bf Y$ is a sigma algebra.
By definition we know $\bf A$ $\subset \bf Y$. Since $\sigma(\bf{A})$ is the smallest sigma algebra that contains the sets in $\bf A$, then it must be the case that $\sigma(\bf A)$ $\subset \bf Y$, which implies that $\forall F\in \sigma(\bf{A})$$: f^{-1}(F)\in \bf X$, since if $F\in \sigma(\bf{A})$$\implies F\in \bf Y$.
My question is if it is true that $\sigma(\bf A)$ $\subset \bf Y$ by definition of these two sigma algebras.

REPLY [1 votes]: For each $E$ in $ \bf A$, we have $f^{-1} (E) \in \bf X$, so $\bf A \subset Y$.
This implies that $\sigma(\bf A) \subset \sigma(\bf Y) = \bf Y$.


TITLE: Has Goedel's Second Incompleteness Theorem been proven using Lawvere's Fixed Point Theorem?
QUESTION [7 upvotes]: This question is a request for assistance in surveying the existing literature on applications of Lawvere's Fixed Point Theorem (LFPT).
Yanofsky [0] has demonstrated several applications of LFPT to prove various limitative results, in particular Goedel's First Incompleteness Theorem, and alludes to its applicability for proofs of the Second Incompleteness Theorem:

Goedel’s second incompleteness theorem about the unprovability within
arithmetic of the consistency of arithmetic. This theorem is a simple
consequence of the first incompleteness theorem. However Kreisal has
a direct model theoretic proofs that uses a diagonal method (see, e.g.,
page 860 of Smorynski’s article in [1].) This proof seems amenable to our
scheme.

Has there been any work conducted to further this vein of inquiry? Can the LFPT be used to prove Goedel's Second Incompleteness Theorem?
[0] https://arxiv.org/abs/math/0305282
[1] Handbook of mathematical logic. North-Holland Publishing Co., Amsterdam, 1977. Edited by Jon Barwise, With the cooperation of H. J. Keisler,
K. Kunen, Y. N. Moschovakis and A. S. Troelstra, Studies in Logic and the
Foundations of Mathematics, Vol. 90.

REPLY [11 votes]: As suggested by the OP, I'm turning my comment into an affirmative answer. WARNING: The presentation of Joyal's proof in the paper I cited contains an incorrect conclusion about Joyal's sentence (that it is undecidable from mere consistency). I have modified my answer accordingly.
Both Gödel's first and second incompleteness theorems have been proven by André Joyal mimicking the arithmetization of metamathematics in Gödel's original proof with internal reasoning inside the initial arithmetic universe. This latter is none other than (the pretopos completion of) the syntactic category, in coherent logic, of the sequents expressing the axioms of primitive recursive arithmetic ($PRA$); in particular the type theoretic treatment of arithmetic universes is not really needed in the proof. Joyal gives an alternative construction of the initial arithmetic universe by defining the category of primitive recursive predicates, which corresponds to taking a different site of definition of the classifying topos (Coste's construction).
The proof has been very recently made available on arxiv in the paper of van Dijk/Oldenziel, although it's been part of categorical folklore since the seventies. Once an internal initial arithmetic universe has been shown to exist inside the initial arithmetic universe, its externalization provides the Gödel numbering of formulas, and one can define functorially the provability predicate $Prov(x)$. Then the proof of the incompleteness theorems proceeds by building a self-referential sentence similar to Gödel's sentence "I am not provable", exactly as Gödel did, using that this sentence is the fixed point of $\neg Prov(x)$, and an argument essentially equivalent to applying Lawvere's fixed point theorem provides its construction. The sentence is however slightly changed to "I am provably false" (which I call Joyal's sentence, the fixed point of $Prov(\neg x)$). It is well known that a fixed point for $\neg Prov(x)$ is equivalent to $\neg Prov(\bot)$, i.e., to $Con(PRA)$, and similarly Joyal also proves categorically that Joyal's sentence is equivalent to $Incon(PRA)$.
The second incompleteness then follows from the observation that Joyal's sentence cannot be provably false (though for certain consistent but $\omega$-inconsistent recursive extensions of PRA it can very well be provably true; here's what goes wrong with the conclusion of the cited paper).
While Lawvere's fixed point theorem is quite trivial, Gödel's incompleteness theorems are much deeper, more complex and subtle than just the fixed point lemma, since they contain the arithmetization of syntax, which Joyal has shown to correspond precisely to internal reasoning inside the arithmetic universe (it is certainly not trivial at all to internally construct an arithmetic universe inside an arithmetic universe). His proof thus deserves to be much widely known than it is.


TITLE: Process for checking reducibility of polynomials of degree 2 or lower in $\mathbb{Z}_2$
QUESTION [0 upvotes]: How might I verify that polynomials of degree 2 or lower and with coefficients from $\mathbb{Z}_2$ are reducible or not?
I figure I can write out all possible polynomials because there are only 2 possible coefficients and 3 possible exponents, but I want to know and understand a procedure to figuring out if it is reducible.
For example, can I just check for roots in $\mathbb{Z}_2$? E.g., $f(x) = x^2 + x + 1$, then try $f(0)$ and $f(1)$ to see if either equals 0? It seems like having roots would guarantee it is reducible because if the degree is 2 or less, any factor must be linear, which is equivalent to a root.

REPLY [1 votes]: A polynomial $f\in K[X]$ of degree $2$ and $3$ over a field $K$ is reducible if and only if has a root in $K$. This answers your question for $K=\Bbb F_2$.
Reference: Irreducible polynomial means no roots?


TITLE: Lebesgue Integral and Gradient
QUESTION [0 upvotes]: I am currently studying on Mixed Method to solve Stokes Equation. I encounter this problem when it says $p:\Omega \to \mathbb{R}$ has a zero-mean over a domain $\Omega \subset \mathbb{R}^n$, then $\int_{\Omega}p dx = 0$. May I know how should I understand this statement?
Also, given $\int_{\Omega}p dx = 0$ and $\nabla p = 0$ over $\Omega$, why does this imply $p=0$? I know these may be basic knowledge on Lebesgue Integral, but I hope someone can explain it clearly. Thanks.

REPLY [0 votes]: For Lebesgue measures, nonempty open sets (as a domain $\Omega$ must be) have positive measure. The mean of a function $f$ on $\Omega$ is defined to be $\frac{1}{\mu(\Omega)}\int_{\Omega} f\ dx$, so if this equals zero, the desired integral must of course equal zero.
To show the second statement, $\nabla f = 0$ on $\Omega$ implies that $f$ is constant on $\Omega$. If you want to justify that, suppose $f(x)\ne f(y)$. Then we can find a path $\gamma$ in $\mathbb{R}^n$ connecting $x$ to $y$ such that the path is piecewise composed of segments in which only one coordinate changes. Then, apply the mean value theorem to each of these segments, as you would in single variable calculus. Having justified this, if $f = C$ on $\Omega$, then $0 = \int_{\Omega} f\ dx = \mu(\Omega)C$ implies that $C = 0$ since $\mu(\Omega) \ne 0$.


TITLE: A pair of mutually "bounding" sequences.
QUESTION [3 upvotes]: Given $({a_n})_{n=1}^{\infty}$, $({b_n})_{n=1}^{\infty}$ convergent sequences
and where
$$\{n\in\mathbb{N}\mid a_n\le b_n\}\quad\text{and}\quad\{n\in\mathbb{N}\mid b_n\le a_n\}$$
are both unbounded, prove that
$$\lim \limits_{n\to \infty}a_n=\lim \limits_{n\to \infty}b_n$$
I would like to know how I can prove it using simple calculus theorem(I only know the definition of limit, arithmetics of limits and the Squeeze Theorem).
Thank you very much for your time and help.

REPLY [1 votes]: Here's a somewhat different take that uses the arithmetic of sequences in a central way.
Since $\lim_{n\to\infty} a_n =a$ and  $\lim_{n\to\infty} b_n = b$, we have by arithmetic of sequences that $\lim_{n\to\infty} \{a_n - b_n\} = a-b$. By hypothesis, $\{a_n-b_n\}$ has infinitely many positive terms and infinitely many negative terms. If $a-b$ is positive, then the negative terms can't get arbitrarily close to $a-b$ as they would have to do.
Formally, for every $N$ there is some $n\ge N$ for which $a_n-b_n<0$ and thus $|(a-b) - (a_n-b_n)| \ge a-b$. Thus $ \{a_n-b_n\}\not\to a-b$, contradiction.
On the other hand, if $a-b$ is negative, then the positive terms can't get arbitrarily close to $a-b$, again a contradiction. (The formal statement is almost exactly like the one above.) Therefore $a-b=0$.


TITLE: Is the sequences$\{S_n\}$ convergent?
QUESTION [4 upvotes]: Let $$S_n=e^{-n}\sum_{k=0}^n\frac{n^k}{k!}$$
Is the sequences$\{S_n\}$ convergent?
The following is my answer,but this is not correct. please give some hints.
For all $x\in\mathbb{R}$, $$\lim_{n\rightarrow\infty}\sum_{k=0}^n\frac{x^k}{k!}=e^x.$$
then
$$\lim_{n\rightarrow\infty}e^{-n}\sum_{k=0}^n\frac{n^k}{k!}=1.$$

REPLY [0 votes]: Ramanujan showed this limit to be $\frac12$.
I gave a reference to this result
in a previous answer of mine,
but I can't find it right now.
Added later:
Here is a reference:
http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=B04924E79B2361751E7AE86C5AF43688?doi=10.1.1.217.7589&rep=rep1&type=pdf
This is called Ramanujan's Q-function.


TITLE: What is the lowest energy gamma ray?
QUESTION [0 upvotes]: If we use the "origin-based" definition of gamma rays (i.e. photons emitted from nuclear decay, regardless of wavelength), which decay produces the lowest-energy, longest-wavelength emission? This comment Are the Properties of the EM Spectrum Fluid? alludes to low-energy gammas in what would usually be considered the ultraviolet band; are there or could there be any in the visible?

REPLY [4 votes]: In addition to Ben Crowell's answer:
Thorium 229 has a nuclear isomer Thorium 229m that has an energy 8.28±0.17 eV above the ground state. When it decays it does so via an UV wavelength photon (which is a honorary gamma by its origin). As far as I know this is the lowest energy emission known to regularly occur.


TITLE: A few problems on series
QUESTION [0 upvotes]: The following are two series problems which I couldn't do any help would be appreciated
1) Find a series $ \sum_{n=0}^\infty a_n $
 of positive terms such that  $ \sum_{n=0}^\infty a_n $ diverges and $ \sum_{n=0}^\infty a_n^2 $= $10^{-133}$
2) Find a series $ \sum_{n=0}^\infty a_n $
 of positive terms such that  $ \sum_{n=0}^\infty a_n $ = $ \sum_{n=0}^\infty a_n^2 $= 7

REPLY [2 votes]: Fot 2) take $a_n=\lambda\,r^n$ with $\lambda>0$ and $0<r<1$. Then, using the formula for the sum of a geometric series,
$$
\sum_{n=0}^\infty a_n=\frac{\lambda}{1-r}=7,\quad \sum_{n=0}^\infty a_n^2=\frac{\lambda^2}{1-r^2}=t.
$$
Solve for $\lambda$ and $r$.


TITLE: Calculating mean and standard deviation of very large sample sizes
QUESTION [4 upvotes]: I have about 3.5GB worth of data that I need to get statistics from, all split up into about 71MB files, with an (approximately) equal number of samples.  From this data, I would like to gather mean and standard deviation.  Parsing the entirety of the standard deviation is probably a bad idea, since it's 3.5GB.
However, I know that with mean, I can at least (with some accuracy, since they're approximately the same size), take the average for each file, and then take the average of each of those sub-averages.
With standard deviation, it's a little more tricky, though.  I've taken some time to run tests and found out that the standard deviation of a large sample size seems to be approximately similar to the average of the standard deviations of equivalently sized smaller chunks of samples.  Does this actually hold true, or was that just a coincidence within the few tests that I've run?  If it does hold true, then can I calculate what my percent error is probably going to be?  Finally, is there a more accurate way to test for standard deviation that doesn't require me mulling over 3.5GB of data at a time?

REPLY [1 votes]: You seem to have about $f=50$ files.  If you know the mean $\mu_i$ and variance $\sigma_i^2$ (square of the standard deviation) and number of elements $n_i$ of each file, then your overall mean should be
$$\mu = \frac{\sum_{i=1}^{f} n_i \mu_i }{\sum_{i=1}^{f} n_i}$$
and your overall variance
$$\sigma^2 = \frac{\sum_{i=1}^{f} n_i \left(\sigma_i^2 +(\mu_i-\mu)^2\right) }{\sum_{i=1}^{f} n_i}.$$
If you have forgotten to collect the number of elements of each file but are confident they are each the same then you can use
$$\mu = \frac{1}{f}\sum_{i=1}^{f} \mu_i $$
which is the mean of the means, and
$$\sigma^2 = \frac{1}{f}\sum_{i=1}^{f} \left(\sigma_i^2 +(\mu_i-\mu)^2\right) .$$
The wrong thing to do would be taking the average of the standard deviations (or even the average of the variances), since this would ignore the effect of the differences in the means and so produce a result which was too small.


TITLE: Elliptic equation on riemannian manifolds
QUESTION [5 upvotes]: Let $ M $ be a compact Riemannian manifold with or without boundary) and let $ \Delta $ be the metric laplacian. I want to study the differential operator $ -\Delta +q $ where $ q $ is a smooth function on $ M $. Now the first thing to do is a development of the theory of weak solutions for this differential operator. I think that all existence results in Chapter 6 , paragraph 2 of Evans' book 'Partial differentail equation' can be reformulate without changement for our case if we are able to prove the Poincarè inequality in the setting of a compact Riemannian manifold. Therefore the first question is
is the poincare inequality true for a compact riannian manifold with boundary?
(i think so)
Now the second problem is the proof of the well- known regularity theorem in our setting (chapter 6.3 of Evans book). Can anyone suggest me a good reference for it? I know the Aubin book but it seems not so clear at this point.\
Thanks

REPLY [2 votes]: If you are willing to assume that $q \geq 0$, then this problem has been extensively studied by Mitrea and Taylor (cf. also the references therein for their other papers on this topic that go back to 1999). If you want $q$ to be an arbitrary smooth function, then I doubt that there is any comprehensive study to date.


TITLE: inductive limit
QUESTION [0 upvotes]: Consider spaces
$$E_n=\{x=\{x_k\}_{k\in\mathbb{N}}\mid x_j=0,\quad j>n\},\quad x_k\in\mathbb{R}$$ endowed with $\|\cdot\|_\infty$ norms.
Let $E$ be an inductive limit of these spaces. This set consists of finite sequences.
Is it true that for a basis of neighbourhoods in $E$ one can take sets
$$U_f=\{x=\{x_k\}_{k\in\mathbb{N}}\mid |x_k|\le f_k\}$$ here $f=\{f_k\}_{k\in\mathbb{N}}$ is a sequence of positive numbers.

REPLY [0 votes]: You can verify that the universal property of inductive limits holds. Also, I believe it is better to consider $U_f=\left\{x=(x_k)_k:|x_k|< f_k\right\}$, so that each $U_f$ will be open (this won't hold if we use $\leq$).
First of all, let's verify that the class $\left\{U_f:f=(f_k)_k\subseteq(0,\infty)\right\}$ is a basis for some l.c. (locally convex) topology on $E$: Given $f=(f_k)_k$, $f_k>0$, let $p_f:E\to\mathbb{R}$ be given by $p_f(x_k)=\sup_{k\in\mathbb{N}}|x_k/f_k|$. Each $p_f$ is a seminorm on $E$ and $U_f=p_f^{-1}(-\infty,1)$. Also, $p_f^{-1}(-\infty,\varepsilon)=U_{\varepsilon f}$, so $\left\{U_f\right\}$ is a basis for the locally convex topology induced by $\left\{p_f\right\}$.
Hence, we consider $E$ with the topology generated by the sets $U_f$. It remains only to show that the universal property holds:

If $W$ is a l.c. TVS and $T:E\to W$ is linear and each restriction $T|_{E_n}:E_n\to W$ is continuous, then $T$ is also continuous.

Let $q$ be a continuous seminorm on $W$ and $\varepsilon>0$. Given $k\in\mathbb{N}$, we can find $\delta_k>0$ such that if $x\in E_k$ and $\Vert x\Vert_{\infty}<\delta_k$, then $q(Tx)<2^{-k}\varepsilon$.
Consider the sequence $\delta=(\delta_k)_k$. Let $x=(x_k)\in U_\delta$, and choose $n$ such that $x\in E_n$ (since $E=\bigcup_n E_n$). Then we can write $x=(x_1,x_2,\ldots,x_n,0,0,\ldots)=\sum_{i=1}^ny_i$, where $y_i=(0,\ldots,x_i,0,\ldots)$. Notice that $y_i\in E_i$ and $\Vert y_i\Vert_\infty<\delta_i$. Then $q(Tx)\leq \sum_{i=1}^n q(Ty_i)<\sum_{i=1}^n 2^{-i}\varepsilon<\varepsilon$. This shows that $T:E\to W$ is continuous, and thus $E$, with the topology generated by the sets $U_f$, is the inductive limit of the $E_k$'s.


TITLE: Help with arrow notation
QUESTION [1 upvotes]: My apologies if this question is too dumb but I am not too familiar with the $\leadsto$ notation. I came across with this remark:
Can someone please tell me the meaning of $\leadsto$ in this case???
Thanks in advance
Phil

REPLY [2 votes]: It is defined for you in the final equation.
It means the matrix where $0$'s are appended at the bottom.
This is not a standard notation and it is not clear if we always append a single row of $0$'s or could it be multiple rows of zeros.