text:
\section{Introduction}
Suppose $u$ is some function defined in a bounded domain $\Omega\subset \mathbb{R}^n$ and $\varphi$ is some given function defined on the boundary $\partial\Omega$. In this paper, we concern the  Dirichlet problem
of degenerate $k$-Hessian equations with non-homogenous boundary functions ($k \geq 2$)
\begin{equation}
\label{1-1}
\left\{ \begin{aligned}
   \sigma_k \big(\lambda(D^{2} u)\big) & = f  \;\;\mbox{ in }~ \Omega, \\
                 u &= \varphi  \;\;\mbox{ on }~ \partial \Omega,
\end{aligned} \right.
\end{equation}
where $f \geq 0$ in $\Omega$ and
$\sigma_{k}$ are the elementary symmetric functions
\[
\sigma_{k} (\lambda) = \sum_ {i_{1} < \ldots < i_{k}}
\lambda_{i_{1}} \ldots \lambda_{i_{k}},\ \ k = 1, \ldots, n,
\]
and $\lambda(D^2u)$ is the eigenvalue vector of the hessian $D^2u$.
The Poisson equation
and Monge-Amp\`{e}re equation
fall into the form of \eqref{1-1} as $k = 1$ and $k = n$ respectively. We call a function
$u \in C^2 (\Omega)$ is $k$-convex if $\lambda (D^2 u) \in \overline \Gamma_k$ in $\Omega$, where $\Gamma_k$ is
the G{\aa}rding's cone
\[
\Gamma_{k} = \{\lambda \in \mathbb{R}^{n}: \sigma_{j} (\lambda) > 0, j = 1, \ldots, k\}.
\]
A bounded domain $\Omega$ in $\mathbb{R}^n$ is called uniformly $(k-1)$-convex, if there is a positive constant
$K$ such that for each $x \in \partial \Omega$,
\[
(\kappa_1 (x), \ldots, \kappa_{n - 1} (x), K) \in \Gamma_k,
\]
where $\kappa_1 (x), \ldots, \kappa_{n - 1} (x)$ are the principal curvatures of $\partial \Omega$ at
$x$.

The central issue of the degenerate equations is the existence of $C^{1,1}$ solution, which major needs  the a prior $C^{1,1}$ estimates of the solutions. Unlike the non degenerate equations, the establishment of $C^{1,1}$ estimate always requires some regularity of $f$ near the set of $f=0$. In \cite{GL1} and \cite{GL2}, Guan-Li have studied the degenerate Weyl problem and degenerate Gauss curvature measure problem, which both are degenerate Monge-Amp\`ere type equations. They found that the conditions
\begin{eqnarray}\label{cond1}
\Delta (f^{1/(n-1)})\geq -A \text{ and } |D (f^{1/(n-1)})|\leq A
\end{eqnarray}
for some constant $A$, are sufficient to get the global $C^{1,1}$ bound. Soon after, for the Dirichlet problem of the degenerate Monge-Amp\`ere equations, Guan \cite{Guan97} found that the above conditions also are sufficient to get the $C^{1,1}$ boundary estimates, if we only consider the homogenous boundary problem. For the non homogenous boundary problem,  Guan-Trudinger-Wang \cite{GTW99}  established the $C^{1,1}$ boundary estimate, if one require
\begin{eqnarray}\label{cond2}
f^{1/(n-1)}\in C^{1,1}(\overline{\Omega}).
\end{eqnarray}
It is not difficult to see, \eqref{cond2} implies \eqref{cond1}.

In view of the above results, a natural and interesting question is that can we establish the $C^{1,1}$ estimates for $k$-convex solutions of \eqref{1-1}, including boundary estimates  and global estimates, by using the condition
\begin{eqnarray}\label{condition}
f^{1/(k-1)}\in C^{1,1}(\overline{\Omega}).
\end{eqnarray}
Note that if $k=n$,  \eqref{condition} is \eqref{cond2}.  This question is first proposed by Ivochkina-Trudinger-Wang \cite{ITW2004},
which is not solved until now.

Let's review some related research of the above question.
Write $\widetilde{f} := f^{1/(k-1)}$. Dong \cite{Dong06} has considered the homogenous Dirichlet problem \eqref{1-1} with homogenous boundary condition $\varphi \equiv 0$. He established
the $C^2$ estimates by \eqref{condition} and
\begin{equation}
\label{strong}
|D \widetilde{f}| \leq C \widetilde{f}^{1/2} \mbox{ on } \overline \Omega
\end{equation}
for some positive constant $C$. In \cite{JW}, by using \eqref{condition}, the authors obtain the  $C^{1,1}$ estimate  for  the Dirichlet problem of degenerate $k$-curvature equations with homogenous boundary condition, which can be view as a generalization of \cite{Dong06}.
If we require $f^{1/k} \in C^{1,1} (\overline \Omega)$, which is a little stronger than \eqref{condition}, for non homogenous boundary problem,
the $C^{1, 1}$ regularity has been established by Krylov \cite{K94a, K94b, K95a, K95b}, seeing an alternative proof
by Ivochkina-Trudinger-Wang \cite{ITW2004}.
In \cite{Wang95},  Wang gave an example, which shows that the condition \eqref{condition} is optimal for the solutions of $k$-Hessian equations being in $C^{1,1}(\overline{\Omega})$. Therefore, condition \eqref{condition} should be sharp.
For more reference, the reader may see \cite{CNS86, Guan98, WX14}  and the reference therein.

The main results of this paper is to give the boundary estimate for convex solutions of \eqref{1-1}  with non homogenous boundary functions, only using condition \eqref{condition}. Note that, for $k$-Hessian equations, the assumption of convex  solutions has been used in \cite{GRW,GQ, MSY}.  Our result partially answers Ivochkina-Trudinger-Wang's question.

\begin{theorem}
\label{conj2}
Suppose $\Omega$ is uniformly convex with $\partial \Omega \in C^{3, 1}$, $\varphi \in C^{3,1} (\partial \Omega)$,
$f > 0$ in $\Omega$ and \eqref{condition}.
Then any convex solution $u \in C^3 (\Omega) \cap C^2 (\overline \Omega)$
to the Dirichlet problem \eqref{1-1} satisfies the estimates
\begin{equation}
\label{main-estimate1}
\max_{\partial \Omega}|D^2 u| \leq C,
\end{equation}
where the positive constant $C$ depends on $n$, $k$, $\Omega$, $|\varphi|_{C^{3,1} (\partial \Omega)}$ and $|f^{1/(k-1)}|_{C^{1,1} (\overline \Omega)}$
but is independent of the lower bound $\inf_\Omega f$.
\end{theorem}
Theorem \ref{conj2} can be regarded as a generalization of the main results in \cite{GTW99}. The major difficulty to prove \eqref{main-estimate1} is
 the estimates for double normal derivatives which we establish in two
steps. One is Lemma \ref{cor-lemma}, which generalizes Lemma 3.1 of \cite{GTW99}.
We utilize the idea of Trudinger \cite{T}, then we can obtain the same lower bound of tangential-tangential derivatives only assuming that the solutions are $k$-convex for $k$-Hessian equations.
Our method is completely different from \cite{GTW99}.
The other one is an estimate of mixed tangential-normal derivatives in terms of  tangential-tangential derivatives, namely, Lemma \ref{lemma-2}. Similar estimate was
 proved in \cite{GTW99} for Monge-Amp\`{e}re equation,
where the special structure of Monge-Amp\`{e}re equation plays a key role to obtain the bound of the tangential-normal derivatives. Unfortunately, the $k$-Hessian
equations with $2 \leq k \leq n-1$ do not possess such structure. Therefore, we need some new idea to reestablish these estimates without using the affine transformation, which is our novelty of Section 6 and Section 7.

Since the constant in \eqref{main-estimate1} is independent of $\inf_\Omega f$, by approximation and
the $C^1$ estimates, global $C^2$ estimates in section 2, section 3,
we obtain the following $C^{1,1}$ estimates for degenerate $k$-Hessian equations.
\begin{theorem}
\label{conj1}
Suppose $\Omega$ is uniformly convex with $\partial \Omega \in C^{3, 1}$, $\varphi \in C^{3,1} (\partial \Omega)$,
$f \geq 0$ in $\Omega$ and \eqref{condition}.
Then any convex solution $u \in C^4 (\Omega) \cap C^2 (\overline \Omega)$
to the Dirichlet problem \eqref{1-1} satisfies the estimates
\begin{equation}
\label{main-estimate}
|u|_{C^{1,1} (\overline \Omega)} \leq C,
\end{equation}
where the positive constant $C$ depends on $n$, $k$, $\Omega$, $|\varphi|_{C^{3,1} (\partial \Omega)}$ and $|f^{1/(k-1)}|_{C^{1,1} (\overline \Omega)}$.
\end{theorem}

The rest of the paper is organized as follows. In Section 2, we establish the \emph{a priori} $C^1$ estimates. Section 3
is devoted to the maximal principle for second order derivatives. We concern the boundary estimates of the pure tangential derivatives and mixed
second order derivatives  in Section 4. In Section 5, we prove a lower bound of the pure tangential derivatives on the boundary.
In the last two sections, we obtain an upper bound of the tangential-normal derivatives in terms of the pure tangential derivatives for convex solutions.

\section{$C^1$ estimates}

In Section 2 to Section 5, we assume  that $u$ is the $k$-convex solution to \eqref{1-1}.
We establish the $C^1$ estimates for $u$ in this section.
By using the $(k - 1)$-convexity of the domain $\Omega$, we can construct
a subsolution $\underline u$ to \eqref{1-1}  as \cite{CNS},
\begin{equation}
\label{2-1}
\left\{ \begin{aligned}
   \sigma_k \big(\lambda(D^{2} \underline u)\big) & \geq f  \;\;\mbox{ in }~ \Omega, \\
                \underline u &= \varphi  \;\;\mbox{ on }~ \partial \Omega.
\end{aligned} \right.
\end{equation}
Let $h$ be the harmonic function in $\Omega$ with $h = \varphi$ on $\partial \Omega$.
By the comparison principal, we have
\[
\underline u \leq u \leq h \mbox{ in } \Omega \mbox{ and } \underline u = u = h \mbox{ on } \partial \Omega
\]
Thus, we  have
\begin{equation}
\label{C1-1}
\sup_{\overline \Omega} |u| + \sup_{\partial \Omega} |D u| \leq C,
\end{equation}
where the constant $C > 0$ depends only on $|\underline u|_{C^1 (\overline \Omega)}$ and $|h|_{C^1 (\overline \Omega)}$. Define
\[
S_k [r] = \sigma_k (\lambda (r))
\]
for a symmetric matrix $r = \{r_{ij}\}$ with $\lambda (r) \in \Gamma_k$ and
\[
S^{ij}_k = \frac{\partial S_k [D^2u]}{\partial r_{ij}}.
\]
By the Newton-Maclaurin inequality, we have
\begin{equation}
\label{GC2-9}
\sum_i S^{ii}_k = (n - k + 1) S_{k - 1} \geq c_0 S_k^{1 - 1/(k-1)} S_1^{1/(k-1)}
\end{equation}
for some positive constant $c_0$ depending only on $n$ and $k$.
\begin{proposition}
\label{C1-3}
Assume that
\begin{equation}
\label{C1-8}
|D f(x)|\leq A f^{1-\frac{1}{k-1}}(x)
\end{equation}
holds for  some positive constant  $A$ and any $x\in \Omega$.
Then there exists positive constant $C$ depending only on $n$, $k$, $A$
and $\Omega$ such that
\begin{equation}
\label{C1-4}
\sup_{\overline \Omega} |D u| \leq C (1 + \sup_{\partial \Omega} |D u|).
\end{equation}
\end{proposition}
\begin{proof}
We may assume $0 \notin \overline \Omega$ and $|x|^2 \geq \delta_0 > 0$ for all $x \in \overline \Omega$.
Suppose
\[
W := \max_{x \in \overline \Omega, \xi \in \mathbb{S}^n} \{u_\xi + B |x|^2\}
\]
is attained at an interior point $x_0 \in \Omega$ and $\xi_0 \in \mathbb{S}^n$, where $B$
is a positive constant sufficiently large to be determined. We may assume $\xi_0$ is in the
direction $x_1$ by rotating the coordinates.
We have, at $x_0$,
\begin{equation}
\label{C1-9}
\sum_{l=1}^n u^2_{1l} = 4 B^2 \sum_l x^2_l = 4 B^2 |x|^2 \geq 4 B^2 \delta_0.
\end{equation}
By \eqref{C1-8} and \eqref{GC2-9}, at $x_0$, we have
\begin{equation}
\label{C1-5}
\begin{aligned}
0 \geq & S^{ij}_k (u_1 + B |x|^2)_{ij} = f_1 + 2 B \sum_i S^{ii}_k\\
  \geq & - A f^{1 - 1/(k-1)} + 2 c_0 B S_1^{1/(k-1)} S_k^{1 - 1/(k-1)}\\
     = & - A f^{1 - 1/(k-1)} + 2 c_0 B (\Delta u)^{1/(k-1)} f^{1 - 1/(k-1)}.
\end{aligned}
\end{equation}
Since $\lambda (D^2 u) \in \Gamma_k \subset \Gamma_2$, we have
\[
\begin{aligned}
0 < 2 \sigma_2 (\lambda (D^2 u)) = \,& 2 \sum_{1 \leq i < j \leq n} (u_{ii} u_{jj} - u_{ij}^2)\\
   = \,& (\Delta u)^2 - \sum_i u_{ii}^2 - \sum_{i \neq j} u_{ij}^2.
\end{aligned}
\]
We have, by \eqref{C1-9},
\[
(\Delta u)^2 \geq \sum_l u^2_{1l} \geq 4 B^2 \delta_0.
\]
Thus, in view of  \eqref{C1-5}, we derive
\[
0 \geq - A f^{1 - 1/(k-1)} + 2 c_0 B (4 B^2 \delta_0)^{1/(k-1)} f^{1 - 1/(k-1)} > 0
\]
provided $B$ is sufficiently large which is a contradiction.
Then we conclude that $W$ is attained on the boundary when choosing
$B$ sufficiently large and
\eqref{C1-4} holds.
\end{proof}
The $C^1$ estimate has been established in terms of \eqref{C1-1} and \eqref{C1-4}.

\section{The maximal principle for second order derivatives}

In this section, we prove
\begin{proposition}
\label{GC2-2}
Suppose \eqref{C1-8} and
\begin{equation}
\label{GC2-10}
\inf\{\Delta f^{\frac{1}{k-1}}(x)\}\geq -\frac{A}{k-1}
\end{equation}
hold for  some positive constant  $A$ and any $x\in \Omega$.
Then there exists a positive constant $C$ depending on $n$, $k$, $A$ and $\Omega$ such that
\begin{equation}
\label{GC2-3}
\sup_{\overline \Omega} |D^2 u| \leq C (1 + \sup_{\partial \Omega} |D^2 u|).
\end{equation}
\end{proposition}
To prove Theorem \ref{GC2-2}, we need the following lemma which is Lemma 3.2 in \cite{GLL12}.
\begin{lemma}
\label{GLL}
Let $\alpha = \frac{1}{k-1}$. If $\lambda (D^2 u) \in \Gamma_k$, then
\begin{equation}
\label{GLL-1}
\sum_i S_k^{pq, rs} u_{pqi} u_{rsi} \leq - S_k\sum_h\left[\frac{(S_k)_h}{S_k}-\frac{(S_1)_h}{S_1}\right]
\left[(\alpha-1)\frac{(S_k)_h}{S_k}-(\alpha+1)\frac{(S_1)_h}{S_1}\right],
\end{equation}
where
\[
S^{pq, rs}_k = \frac{\partial^2 S_k [D^2u]}{\partial r_{pq} \partial r_{rs}}.
\]
\end{lemma}
Lemma \ref{GLL} comes from the concavity of $\big(\frac{\sigma_k}{\sigma_1}\big)^{1/(k-1)}$
in $\Gamma_k$.
\begin{proof}[Proof of Propsotion \ref{GC2-2}]
We consider the test function
\[
H=\Delta u+\frac{B}{2}|x|^2,
\]
where $B$ is a positive undetermined  constant.
First, by differentiating the equation \eqref{1-1}, we have
\begin{equation}
\label{GC2-11}
S_k^{ij}(\Delta u)_{ij} + \sum_i S_k^{pq,rs}u_{pqi}u_{rsi} = \Delta f,
\end{equation}
which implies
\begin{equation}
\label{GC2-12}
S_k^{ij} H_{ij} = -\sum_i S_k^{pq,rs}u_{pqi}u_{rsi} + B (n - k + 1) S_{k-1} + \Delta f.
\end{equation}
Suppose $H$ attains its maximum at an interior point $x_0 \in \Omega$.
Write $\alpha := \frac{1}{k-1}$. We have, at $x_0$,
\begin{equation}
\label{GC2-14}
0 = H_i = (\Delta u)_i + B x_i
\end{equation}
for each $1 \leq i \leq n$ and
therefore, by \eqref{GLL-1}, \eqref{GC2-9}, \eqref{GC2-10} and \eqref{C1-8}, we get
\begin{equation}
\label{GC2-13}
\begin{aligned}
0 \geq & - \sum_i S_k^{pq,rs} u_{pqi}u_{rsi} + B (n - k + 1) S_{k-1} + \Delta f\\
  \geq & S_k\sum_h\left(\frac{(S_k)_h}{S_k}-\frac{(S_1)_h}{S_1}\right)
\left((\alpha-1)\frac{(S_k)_h}{S_k}-(\alpha+1)\frac{(S_1)_h}{S_1}\right)\\
       & +(c_0 B S_1^{\alpha}-A) f^{1-\alpha} + (1-\alpha)\frac{|\nabla f|^2}{f}\\
     = & (\alpha - 1)\frac{|\nabla f|^2}{f} - 2\alpha \frac{\nabla \Delta u}{\Delta u} \cdot \nabla f
         + (1 + \alpha) f \frac{|\nabla \Delta u|^2}{(\Delta u)^2}\\
       & + (c_0 B (\Delta u)^{\alpha} - A) f^{1-\alpha} + (1-\alpha)\frac{|\nabla f|^2}{f}\\
     = & \frac{2 \alpha B x \cdot \nabla f}{\Delta u} +(1 + \alpha) f \frac{B^2 |x|^2}{(\Delta u)^2}
         + (c_0 B (\Delta u)^{\alpha} - A) f^{1-\alpha}\\
  \geq & \left(c_0 B (\Delta u)^{\alpha} - A - \frac{C B}{\Delta u}\right)f^{1-\alpha},
\end{aligned}
\end{equation}
where $C$ is some constant only depending on $\Omega,n,k$ and $A$.
Thus, for sufficient large $B$ and $\Delta u$, we have the desired estimates.
\end{proof}
If there is no degenerate point on $\partial\Omega$, we have the usual boundary estimate as \cite{CNS}. Therefore, using Proposition \ref{C1-3}  and Proposition \ref{GC2-2},  we can prove
\begin{theorem}
\label{cor1}
Suppose $\Omega$ is $(k-1)$-convex with $\partial \Omega \in C^{3, 1}$, $\varphi \in C^{3,1} (\overline \Omega)$,
$f \geq 0$ in $\Omega$, $f > 0$ on $\partial \Omega$ and \eqref{C1-8}, \eqref{GC2-10} hold for some constant $A>0$.
Then the estimate \eqref{main-estimate} holds for any
 $k$-convex  $C^{1,1}$ solution $u$ of \eqref{1-1}.
\end{theorem}

\section{Estimates for mixed tangential-normal derivatives}

In this and the following sections, we derive the second order boundary estimates.
For any point $x_0 \in \partial \Omega$, we may assume that $x_0$ is the origin and that the
positive $x_n$-axis is in the interior normal direction to $\partial \Omega$ at the origin.
Suppose near the origin, the boundary $\partial \Omega$ is given by
\begin{equation}
\label{BC2-1}
x_n = \rho (x') = \frac{1}{2} \sum_{\alpha, \beta < n} B_{\alpha\beta} x_\alpha x_\beta + O (|x'|^3),
\end{equation}
where $x' = (x_1, \ldots, x_{n-1})$ and $B_{\alpha\beta}$ is the second fundamental form of $\partial \Omega$ at $x_0$. Differentiating the
boundary condition $u = \varphi$ on $\partial \Omega$ twice, we can find a constant $C$ depending on
$|\varphi|_{C^2 (\partial \Omega)}$ and $|u|_{C^1 (\overline \Omega)}$ such that
\begin{equation}
\label{BC2-2}
|u_{\alpha \beta} (0)| \leq C \mbox{  for } \alpha, \beta \leq n - 1.
\end{equation}
Next, we establish the estimate
\begin{equation}
\label{BC2-3}
|u_{\alpha n} (0)| \leq C \mbox{  for } \alpha \leq n - 1.
\end{equation}
For $x\in\partial\Omega$ near the origin, let
\[
T_\alpha = \partial_{\alpha}
  + \sum_{\beta < n} B_{\alpha \beta} (x_{\beta}\partial_{n}-x_{n}\partial_{\beta}), \;\;
  \mbox{ for } \alpha < n,
\]
$T_n = \partial_n$ and $\omega_\delta = \{x \in \Omega: \rho (x') < x_n < \rho (x') + \delta^2 , |x'| < \delta\}$.
We have
\[
S_k^{ij} (T_\alpha u)_{ij} = T_\alpha f.
\]
It follows that
\begin{equation}
\label{bd-2}
|S_k^{ij} (T_\alpha (u - \varphi))_{ij}| \leq C \Big(S_{k - 1} + f^{1 - 1/ (k - 1)}\Big)
\end{equation}
and
\begin{equation}
\label{bd-3}
|T_\alpha (u - \varphi)| \leq C |x'|^2 \mbox{ on } \partial \Omega \cap \overline{\omega}_\delta
    \mbox{ for } \alpha < n
\end{equation}
when $\delta$ is sufficiently small since $u = \varphi$ on $\partial \Omega$. Because $\Omega$ is
uniformly $(k - 1)$- convex, there exist positive constants $\theta$ and $K$ such that
\[
(\kappa_1 - 2 \theta, \ldots, \kappa_{n-1} - 2 \theta, 2 K) \in \Gamma_k.
\]
Define
\begin{equation}
\label{BC2-6}
\Psi = \rho (x') - x_n - \theta |x'|^2 + K x_n^2.
\end{equation}
Note that the boundary $\partial \omega_\delta$ consists three parts: $\partial \omega_\delta
= \partial_1 \omega_\delta \cup \partial_2 \omega_\delta \cup \partial_3 \omega_\delta$, where
$\partial_1 \omega_\delta$, $\partial_2 \omega_\delta$ are  defined by $\{x_n=\rho\} \cap\overline{\omega}_{\delta}$, $\{ x_n=\rho+\delta^2\}\cap\overline{\omega}_{\delta}$
respectively, and $\partial_3 \omega_\delta$ is defined by $\{|x'| = \delta\}\cap\overline{\omega}_{\delta}$.

We see that when $\delta$ is sufficiently small (depending on $\theta$ and $K$),
$\Psi \leq 0$ in $\omega_\delta$ and furthermore,
\begin{equation}
\label{BC2-12}
\begin{aligned}
\Psi \leq & - \frac{\theta}{2} |x'|^2, \mbox{ on } \partial_1 \omega_\delta\\
\Psi \leq & - \frac{\delta^2}{2}, \mbox{ on } \partial_2 \omega_\delta\\
\Psi \leq & - \frac{\theta \delta^2}{2}, \mbox{ on } \partial_3 \omega_\delta.
\end{aligned}
\end{equation}
We derive from \eqref{BC2-1} that $\lambda (D^2 \Psi) \in \Gamma_k$ and
\begin{equation}
\label{BC2-4}
S_k^{ij} \Psi_{ij} \geq \eta_0 S_{k - 1} \mbox{ on } \overline{\omega}_\delta
\end{equation}
for some uniform constant $\eta_0 > 0$ by further requiring $\delta$ small.

We claim that there exist uniform positive constants $A$ and $\delta$ such
that $A \Psi \pm T_\alpha (u - \varphi) \leq 0$ on $\overline{\omega}_\delta$. It is easy to
get \eqref{BC2-3} from the claim since $A \Psi (0) \pm T_\alpha (u - \varphi) (0) = 0$.

Now we prove the claim. We first note that $A \Psi \pm T_\alpha (u - \varphi) \leq 0$
on $\partial \omega_\delta$ by \eqref{BC2-12} when $A$ is sufficiently large.
So we may suppose
\[
W := \max_{\overline{\omega}_\delta} (A \Psi
\pm T_\alpha (u - \varphi))
\]
is attained at an interior point $x_0 \in \omega_\delta$
and $A$ is sufficiently large to be chosen. We may assume, at $x_0$, $A \Psi
\pm T_\alpha (u - \varphi) \geq 0$ for otherwise we are done. Now we consider two cases:
(i) $\Delta u (x_0) \leq 1$ and (ii) $\Delta u (x_0) > 1$.

\textbf{Case (i).} As in the gradient estimates, we see $|u_{ij} (x_0)| \leq C_0 \Delta u (x_0) \leq C_0$
for some uniform positive constant $C_0$ and each $1 \leq i, j \leq n$. We note that, at $x_0$,
\[
\begin{aligned}
0 = \,& (A \Psi \pm T_\alpha (u - \varphi))_n\\
  = \,& - A + 2 A K x_n \nonumber\\
      & \pm \left[(u - \varphi)_{\alpha n}+\sum_{\beta<n}B_{\alpha\beta}(x_{\beta}(u-\varphi)_{nn}-x_n(u-\varphi)_{\beta n})-\sum_{\beta<n}B_{\alpha\beta}(u-\varphi)_{\beta}\right]\\
  < \,& 0,
\end{aligned}
\]
if we choose $\delta$ is sufficiently small and $A$ is sufficiently large using the bound of $|D^2u|$ and $|Du|$ at $x_0$. Then we have a contradiction.

\textbf{Case (ii).} By \eqref{GC2-9} and \eqref{bd-2}, we see at $x_0$ where $W$ is attained,
\[
\begin{aligned}
0 \geq \,& S_k^{ij} (A \Psi \pm T_\alpha (u - \varphi))_{ij}
  \geq A \eta_0 S_{k - 1} - C \Big(S_{k - 1} + f^{1 - 1/ (k - 1)}\Big)\\
  \geq \,& \frac{A}{2} \eta_0 S_{k-1} + \frac{A}{2} \eta_0 c_0 f^{1 - 1/ (k - 1)}
     - C \Big(S_{k - 1} + f^{1 - 1/ (k - 1)}\Big) > 0
\end{aligned}
\]
provided $A$ is sufficiently large, which is a contradiction.
The claim follows and \eqref{BC2-3} is proved.

For the homogenous problem, combining \eqref{BC2-2},\eqref{BC2-3} with the estimates for double normal derivatives in \cite{Dong06}, the conditions \eqref{C1-8} and \eqref{GC2-10} are sufficient to obtain the existence of $C^{1,1}$ solutions, which is a slight different from Dong's theorem \cite{Dong06}.
\begin{theorem}
\label{main-result-2}
Suppose $\Omega$ is uniformly $(k - 1)$-convex with $\partial \Omega \in C^{3,1}$, $f \geq 0$ and
 there exists a positive constant $A$ such that \eqref{C1-8} and \eqref{GC2-10} hold.
Then the equation \eqref{1-1} with $\varphi \equiv 0$ has a unique $k$-convex solution $u \in C^{1,1} (\overline \Omega)$
satisfying the estimates \eqref{main-estimate}, where the constant $C$ depends on $\Omega$, $A$, $(\sup_\Omega f)^{-1}$
and modulus continuity of $f$ in $\Omega$.
\end{theorem}

\section{An inequality on the boundary}

Suppose  $W$  is a $(0,2)$ tensor on $\overline{\Omega}$, namely $W\in C^2(T^*\overline{\Omega}\otimes T^*\overline{\Omega})$,
where $T^*\overline{\Omega}$ is the co-tangent bundle of $\overline{\Omega}$.
Let $W'$
be the projection of $W|_{\partial\Omega}$ in the bundle $T^* \partial \Omega\otimes T^*\partial\Omega$, where $T^*\partial \Omega$ is the co-tangent bundle of $\partial \Omega$.  $\lambda' (W')$ denotes
the eigenvalue vector of $W'$ with respect to the induced metric on $\partial \Omega$.
In this section, we prove
\begin{lemma}
\label{cor-lemma}
There exists a uniform positive constant $\delta_0$ such that
\begin{equation}
\label{cor-1}
\sigma_{k-1} (\lambda' [(D^2 u)']) \geq \delta_0 f
\end{equation}
on $\partial \Omega$.
\end{lemma}
\begin{proof}
We use an idea being closed to \cite{T}.
Define $G (r) = \sigma_{k-1}^{1/(k-1)} (\lambda' (r))$ for a $(n-1) \times (n-1)$ matrix
$r$ with its eigenvalues $\lambda' (r) \in \Gamma_{k-1}$. Let
\[
m := \inf_{\partial \Omega} \frac{G ((D^2 u)')}{\widetilde{f}},
\]
where $\widetilde{f} = f^{1/(k-1)}$. It suffices
to prove $m \geq \delta_0$ for some positive constant $\delta_0$.
Suppose $m$ is attained at $x_0 \in \partial \Omega$. We may assume that $x_0$ is the origin and the
positive $x_n$-axis is in the interior normal direction to $\partial \Omega$ at the origin as before.
Choose local orthonormal frames
$\{e_1, \ldots, e_n\}$ around $x_0$ such that $e_n$ is the interior normal to $\partial \Omega$.
$\nabla$ denotes the standard connection of $\mathbb{R}^n$. Write $e_i (x) = e^j_i (x)\partial_j$ for $i = 1, \ldots n$, where $\partial_1,\cdots,\partial_n$ is the  rectangular coordinate system. Thus, we have
 \[
\nabla_i u:=\nabla_{e_i}u= e_i^j \partial_ju=e_i^ju_j
\]
and
\[
\nabla_{ij} u:=\nabla_{e_i}\nabla_{e_j}u =e_i^ke_j^l\partial_{k}\partial_l u = e_i^k e_j^l u_{kl}.
\]
We may also assume that $e_i^j (x_0) = \delta_{ij}$
for $1 \leq i, j \leq n$ and $\{\nabla_{\alpha \beta} u (x_0)\}_{1 \leq \alpha, \beta \leq n-1}$ is diagonal.
Let $\underline u$ be a $k$-convex subsolution satisfying \eqref{2-1}.
Since $u - \underline u = 0$ on $\partial \Omega$, we find
\begin{equation}
\label{cor-2}
\nabla_{\alpha \beta} u = \nabla_{\alpha \beta} \underline u - \nabla_n (u - \underline u) \sigma_{\alpha \beta}, \ 1 \leq \alpha, \beta \leq n - 1
\end{equation}
on $\partial \Omega$ near $x_0$, where $\sigma_{\alpha \beta} = \langle D_{e_\alpha} e_\beta, e_n\rangle$ is the second fundamental form of $\partial \Omega$.
Let
\[
G^{\alpha \beta}_0 = \frac{\partial G}{\partial r_{\alpha \beta}} (\nabla_{\alpha \beta} u (x_0)), 1 \leq \alpha, \beta \leq n-1.
\]
By the concavity of $G$, we have
\begin{equation}
\label{cor-3}
G^{\alpha \beta}_0 (r_{\alpha \beta} - \nabla_{\alpha \beta} u (x_0))
  \geq G (r) - G (\nabla_{\alpha \beta} u (x_0))
\end{equation}
for any matrix $r$ satisfying $\lambda' (r) \in \Gamma_{k-1}$. It follows from \eqref{cor-2}, \eqref{cor-3}
and the definition of $m$ that
\begin{equation}
\label{cor-4}
\begin{aligned}
- G^{\alpha \beta}_0 \sigma_{\alpha \beta} \nabla_n (u - \underline u)
   = \,& G^{\alpha \beta}_0 (\nabla_{\alpha \beta} u - \nabla_{\alpha \beta} \underline u)\\
   = \,& G^{\alpha \beta}_0 (\nabla_{\alpha \beta} u - \nabla_{\alpha \beta} u (x_0))\\ & + G^{\alpha \beta}_0 \nabla_{\alpha \beta} u (x_0)
      - G^{\alpha \beta}_0 \nabla_{\alpha \beta} \underline u\\
\geq \,& G (\nabla_{\alpha \beta} u) - G (\nabla_{\alpha \beta} u (x_0))\\ & + G^{\alpha \beta}_0 \nabla_{\alpha \beta} u (x_0)
      - G^{\alpha \beta}_0 \nabla_{\alpha \beta} \underline u\\
\geq \,& m \widetilde{f}
     - G^{\alpha \beta}_0 \nabla_{\alpha \beta} \underline u
\end{aligned}
\end{equation}
on $\partial \Omega$ near $x_0$. Since $\Omega$ is uniformly $(k-1)$-convex and $\sigma_{\alpha \beta}$ is the second
fundamental form of $\partial \Omega$, we have
\[
\lambda' (\{\sigma_{\alpha \beta} - \theta \delta_{\alpha \beta}\}) \in \Gamma_{k-1}
\]
on $\partial \Omega$ near $x_0$ for some positive constant $\theta$ depending only on the geometry of $\partial \Omega$.
Thus, by the concavity of $G$, we get
\begin{equation}
\label{useful}
\begin{aligned}
G^{\alpha \beta}_0 \sigma_{\alpha \beta} \geq \,& G (\sigma_{\alpha \beta} - \theta \delta_{\alpha \beta}) + \theta \sum_{\alpha =1}^{n-1} G^{\alpha\alpha}_0\\
   \geq \,& \gamma + \theta \sum_{\alpha =1}^{n-1} G^{\alpha\alpha}_0
\end{aligned}
\end{equation}
for some positive constant $\gamma$.
Let
\[
\Omega_\delta := \{x \in \Omega: |x - x_0| < \delta\}
\]
for $\delta$ sufficiently small and
$\eta := G^{\alpha \beta}_0 \sigma_{\alpha \beta}$. Define the following barrier in $\Omega_\delta$,
\[
\tilde{\Phi} = - \nabla_n(u - \underline u) + \frac{1}{\eta} \Big
      G^{\alpha \beta}_0 \nabla_{\alpha \beta} \underline u - m \widetilde{f} \Big).
\]
It follows that $\tilde{\Phi} \geq 0$ on $\partial \Omega \cap \partial \Omega_\delta$ and $\tilde{\Phi} (x_0) = 0$.
Since $e_j^i = \delta_{ij}$ at $x_0 = 0$, we have
\begin{equation}
\label{boundary1}
|e_\alpha^\alpha (u - \underline u)_\alpha| \equiv |\sum_{j\neq \alpha} e_\alpha^j (u - \underline u)_j| \leq C\sum_{j\neq \alpha}|e_{\alpha}^j|\leq C_0 |x|
\end{equation}
on $\partial \Omega \cap \partial \Omega_\delta$ for $1 \leq \alpha \leq n-1$ and the constant $C_0$ depends only on
the bound of $|D (u - \underline u)|$ and $|D e_\alpha^j|$ for $j \neq \alpha$. We may also assume that $\inf_{\Omega_\delta} e_\alpha^\alpha e_n^n \geq c_0 > 0$
for any $1 \leq \alpha \leq n-1$ and some positive constant $c_0$ by choosing $\delta$ sufficiently small.
By \eqref{boundary1} and that $e_j^i = \delta_{ij}$ at $x_0 = 0$ again, we have
\[
\begin{aligned}
\nabla_n (u - \underline u) = \,& e_n^n (u - \underline u)_n + \sum_{\alpha = 1}^{n-1} e_n^{\alpha} (u - \underline u)_\alpha\\
  = \,& e_n^n (u - \underline u)_n + \sum_{\alpha = 1}^{n-1} \frac{e_n^{\alpha} e^\alpha_\alpha (u - \underline u)_\alpha}{e^\alpha_\alpha}
  \geq e_n^n (u - \underline u)_n - CC_0 |x|^2
\end{aligned}
\]
on $\partial \Omega \cap \partial \Omega_\delta$, where the constant $C$ depends only on $\inf_{\Omega_\delta} e_\alpha^{\alpha}$
and the bound of $|D e_n^\alpha|$, $1 \leq \alpha \leq n-1$.
Since $\tilde{\Phi} \geq 0$ on $\partial \Omega \cap \partial \Omega_\delta$ and $\tilde{\Phi} (x_0) = 0$, we have
\[
\Phi := - (u - \underline u)_n + \frac{1}{\eta e_n^n} \Big(
      G^{\alpha \beta}_0 \nabla_{\alpha \beta} \underline u - m \widetilde{f} \Big) + M |x|^2 \geq 0
\]
on $\partial \Omega \cap \partial \Omega_\delta$ and $\Phi (x_0) = 0$,
where $M = CC_0/c_0$.
Let $\Psi$ be the function defined in \eqref{BC2-6}. Note that $\Psi$ satisfies \eqref{BC2-12}
and \eqref{BC2-4}. Consider the function $B \Psi - \Phi$ on $\overline{\omega}_\delta$, where
$B$ is a positive sufficiently large constant to be chosen later.

Because of \eqref{bd-2} and $\widetilde{f} \in C^{1,1} (\overline \Omega)$, we have
\[
S^{ij}_k \Phi_{ij} \leq C \Big(S_{k - 1} + f^{1 - 1/ (k - 1)}\Big) \mbox{ on } \overline{\omega}_\delta.
\]
As in Section 4, we can take sufficiently large $B$ to staisfy $B \Psi - \Phi \leq 0$ on $\partial \omega_\delta$.
Suppose the maximum of $B \Psi - \Phi$ on $\overline{\omega}_\delta$ is achieved at an interior point $x_1 \in \omega_\delta$.
We see, at $x_1$,
\[
(B \Psi - \Phi)_n = - B + 2 BK x_n - \Phi_n = 0,
\]
which implies that $u_{nn} (x_1) \geq B/2$  provided $B$ is sufficiently large
and $\delta$ is sufficiently small. Therefore, we get $\Delta u (x_1) \geq B/2$.
Using \eqref{GC2-9} again, we have,
at $x_1$,
\[
0 \geq S^{ij}_k (B \Psi - \Phi)_{ij} > 0
\]
if $B$ is sufficiently large, which is a contradiction. Therefore, $B \Psi - \Phi \leq 0$ on $\overline{\omega}_\delta$.
It follows that
$\Phi_n (x_0) \geq B \Psi_n (x_0) \geq -C$.
Thus, we get
\[
u_{nn} (x_0) \leq C (1 + m),
\]
where $C$ depends on $|u|_{C^1 (\overline{\Omega})}$, $\sup_{\partial \Omega} |(D^2 u)'|$, $|\underline u|_{C^2 (\overline{\Omega})}$
and $|\widetilde{f}|_{C^{1,1} (\overline{\Omega})}$. Without loss of generality, we may assume $m \leq 1$. Combining with $\Delta u\geq 0$, we then obtain a bound
\begin{equation}
\label{B-3}
|u_{nn} (x_0)| \leq C.
\end{equation}
Recall that $\nabla_{\alpha \beta} u (x_0) = u_{\alpha \alpha} (x_0) \delta_{\alpha \beta}$ for $1 \leq \alpha, \beta \leq n-1$.
Let $b_\alpha = u_{\alpha \alpha} (x_0)$ for $1 \leq \alpha \leq n-1$ and
$b = (b_1, \ldots, b_{n-1})$.
By the equation \eqref{1-1}, we see, at $x_0$,
\begin{equation}
\label{fromeqn}
u_{nn} \sigma_{k - 1} (b) - \sum_{\alpha = 1}^{n-1} u_{\alpha n}^2 \sigma_{k - 2; \alpha} (b) + \sigma_k (b) = f.
\end{equation}
Since we have
\[
\begin{aligned}
\frac{\sigma_k (b)}{\sigma_{k-1} (b)} = \,& \frac{\sum_{\alpha = 1}^{n - 1} b_\alpha \sigma_{k - 1; \alpha} (b)}{k \sigma_{k-1} (b)}\\
        = \,& \frac{\sum_{\alpha = 1}^{n - 1} b_\alpha (\sigma_{k-1} (b) - b_\alpha \sigma_{k - 2; \alpha} (b))}{k \sigma_{k-1} (b)}\\
        = \,& \frac{\sum_{\alpha = 1}^{n - 1} b_\alpha}{k}
           - \frac{\sum_{\alpha = 1}^{n - 1} b_\alpha^2 \sigma_{k - 2; \alpha} (b)}{k \sigma_{k-1} (b)}
        \leq \frac{\sum_{\alpha = 1}^{n - 1} b_\alpha}{k} \leq C,
\end{aligned}
\]
then we get
\[
\frac{f (x_0)}{\sigma_{k - 1} (b)} \leq u_{nn} (x_0) + \frac{\sigma_k (b)}{\sigma_{k-1} (b)} \leq C.
\]
We are done.
\end{proof}

\section{Estimates for convex solutions, the case $k=2$}

From now on, we assume that $u \in C^3 (\Omega) \cap C^2 (\overline \Omega)$ is a convex
solution of \eqref{1-1} and that $\Omega$ is uniformly convex. We consider an arbitrary point $x_0 \in \partial \Omega$.
As in Section 4, we assume $x_0$ is the origin, the
positive $x_n$-axis is in the interior normal direction to $\partial \Omega$ at the origin,
and near the origin, the boundary $\partial \Omega$ is given by \eqref{BC2-1}.

In this and the following sections we always use the notation $x' = (x_1, \ldots, x_{n-1}) \in \mathbb{R}^{n-1}$ to denote the
projection of $x = (x_1, \ldots, x_n) \in \mathbb{R}^n$ in $\mathbb{R}^{n-1}$ by its first $n-1$ coordinates.

We may assume $\{u_{\alpha \beta} (0)\}_{1 \leq \alpha, \beta \leq n-1}$ is diagonal and
\[
\{u_{\alpha \beta} (0)\}_{1 \leq \alpha, \beta \leq n-1} = \mathrm{diag} \{b_1, \ldots, b_{n-1}\}.
\]
We may further assume $b_1 \leq \cdots \leq b_{n-1}$.

Let  $\nabla_{\zeta\eta}$ denote the second order covariant derivative
with respect to the standard connection $\nabla$ for any local vector fields $\zeta$, $\eta$ in $\Omega$ near the origin.
In this and the next section, we shall prove
\begin{lemma}
\label{lemma-2}
Let $\nu$ denote the unit interior normal to $\partial \Omega$ and $\tau_\beta = \partial_\beta + \rho_\beta \partial_n$ be the tangential vector field of $\partial \Omega$ for $\beta \leq n-1$ near the origin.
For $\alpha = n - (k-1), \ldots, n-1$, there exist two positive constants $\theta_\alpha$ and $C$ depending only on $\Omega$,
$|\varphi|_{C^{3,1} (\overline \Omega)}$ and $|f^{1/(k-1)}|_{C^{1,1} (\overline \Omega)}$ such that if any point $x = (x_1, \ldots, x_n) \in \partial \Omega$
satisfies $|x_\beta| \leq \theta_\alpha \sqrt{b_\alpha}$ for any $1 \leq \beta \leq \alpha$
and $|x_\beta| \leq \theta_\alpha \frac{b_\alpha}{\sqrt{b_\beta}}$ for any $\alpha + 1 \leq \beta \leq n-1$, we have the estimates
\begin{equation}
\label{jw-18}
|\nabla_{\nu \tau_\beta} u (x)| \leq C \sqrt{b_\alpha}
\end{equation}
for all $\beta = 1, \ldots, \alpha$.
\end{lemma}
Let $\omega_1 := \{x \in \Omega: x_n < \epsilon_1 b_{n-1}\}$ with a positive constant $\epsilon_1$ sufficiently small to be chosen later. In this section, we prove
\begin{equation}
\label{jw-12}
|\nabla_{\nu \xi}u| \leq C \sqrt{b_{n-1}} \mbox{ on } \partial \omega_1 \cap \partial \Omega,
\end{equation}
for any convex solution $u$ of \eqref{1-1}, where $\xi$ is any unit tangent vector of $\partial \Omega$.
Note that \eqref{jw-12} implies Lemma \ref{lemma-2} for $\alpha = n-1$ and that Lemma \ref{lemma-2} follows
immediately for the case $k=2$.

Since $f^{1/(k-1)} \in C^{1,1} (\overline \Omega)$, we have
\begin{equation}
\label{jw-1}
|\tilde{f}_\alpha (0)| \leq C \sqrt{\tilde{f} (0)}, \mbox{ for } \alpha \leq n-1,
\end{equation}
where $\tilde{f} := f^{1/(k-1)}$ and the constant $C$ depends only on $|\tilde{f}|_{C^{1,1} (\overline \Omega)}$.
By \eqref{cor-1}, \eqref{jw-1} and Taylor's expansion of $\tilde{f}$, we have
\[
\tilde{f} (x) \leq \tilde{f} (0) + \sum_{i=1}^n \tilde{f}_i (0) x_i + C |x|^2
   \leq C \left(\sigma_{k-1}^{1/(k-1)} (b) + \sigma_{k-1}^{1/2(k-1)} (b) |x'| + x_n + |x|^2\right)
\]
in $\Omega$ near the origin, where $x'=(x_1,\cdots, x_{n-1})$. It follows that
\begin{equation}
\label{taylor}
f (x) \leq C \left(\sigma_{k-1} (b) + \sqrt{\sigma_{k-1} (b)} |x'|^{k-1} + x_n^{k-1} + |x|^{2 (k-1)}\right)
\end{equation}
in $\Omega$ near the origin.
Define
$\omega := \{x \in \Omega: x_n < 2 b_{n-1}\}$. Therefore, we have
\begin{eqnarray}\label{add1}
|x'|\leq C\sqrt{b_{n-1}}, \text{ for } x\in\overline{\omega}
\end{eqnarray}
by \eqref{BC2-1}.
In the following, we always suppose $b_{n-1}$ is sufficiently small, otherwise by \eqref{BC2-3}, we have got \eqref{jw-12}.
Thus, by \eqref{BC2-1} and \eqref{taylor} we get
\begin{equation}
\label{taylor-1}
f \leq C b_{n-1}^{k-1} \mbox{ in } \omega.
\end{equation}
Subtracting $D u (0) \cdot x + u (0)$, we may assume $\inf_\Omega u = u (0) = 0$ and $D u (0) = 0$.
By Taylor's expansion, we have
\begin{equation}
\label{jw-2}
\varphi (x', \rho (x')) = \frac{1}{2} \sum_{\alpha \leq n-1} b_\alpha x_\alpha^2 +
  \frac{1}{6} \sum_{\alpha, \beta, \gamma \leq n-1} \varphi_{\alpha \beta \gamma} (0) x_\alpha x_\beta x_\gamma
    + O (|x'|^4),
\end{equation}
near the origin, where $\varphi_{\alpha \beta \gamma} (x') = \frac{\partial^3 \varphi (x', \rho (x'))}{\partial x_\alpha \partial x_\beta \partial x_\gamma}$.
Since $u = \varphi \geq 0$ on $\partial \Omega$ and \eqref{add1}, we find
\begin{equation}
\label{jw-3}
|\varphi_{\alpha \beta \gamma} (0)| \leq C \sqrt{b_{n-1}}.
\end{equation}
We extend the boundary function $\varphi$ to be the right hand side of \eqref{jw-2} near the origin in $\Omega$ and still denote it by $\varphi$.
By the convexity of $u$, the maximum value  of $u$ on $ \{x_n = 2 b_{n-1}\}$ achieves on $\partial\Omega$.
Therefore, again by convexity, we have
\begin{equation}
\label{jw-4}
|u - \varphi| \leq \sup_{\partial\Omega\cap\partial \overline{\omega}}(|u|+|\varphi|)\leq C b_{n-1}^2 \mbox{ on } \overline \omega.
\end{equation}
Let $d (x) := \mathrm{dist} (x, \partial \Omega)$ and
\[
v := - d + \frac{1}{8 b_{n-1}} d^2.
\]
We have $d \leq x_n$ near the origin so that $v \leq 0$ in $\omega$. Moreover, at $x=0$, we have
\begin{eqnarray}\label{d2v}
D^2v\geq \mathrm{diag}\left\{\frac{\kappa_1}{2},\cdots,\frac{\kappa_{n-1}}{2},\frac{1}{4b_{n-1}}\right\},
\end{eqnarray}
where $\kappa_1,\cdots, \kappa_{n-1}$ are the principal curvatures of $\partial \Omega$ at origin.
Thus  $v$ is convex near origin and
\[
\sigma_k (D^2 v) \geq \frac{\delta_0}{b_{n-1}} \mbox{ in } \omega
\]
for some positive constant $\delta_0$ because $\Omega$ is uniformly convex.
Let
\[
F (r) = \sigma_k^{1/k} (\lambda(r)) \mbox{ with } \lambda(r) \in \Gamma_k
\]
where $r$ is a symmetric matrix  and $\lambda(r)$ is the eigenvalue vector of $\lambda$,
and
\[
F^{ij} = \frac{\partial F(D^2 u)}{\partial u_{ij}}.
\]
By using \eqref{d2v}, $D^2v-\delta_1I$ is a positive definite matrix for some sufficiently small constant $\delta_1$, where $I$ is the identical matrix. By the concavity of $F$ and the uniformly convexity of $\Omega$, we find
\begin{equation}
\label{jw-5}
F^{ij} v_{ij} \geq \delta_1 \left(\frac{1}{b_{n-1}^{1/k}} + \sum_{i=1}^n F^{ii}\right)
  \mbox{ in } \omega.
\end{equation}
Let $\omega_0 := \{x \in \Omega: x_n < \epsilon b_{n-1}\}$, where $\epsilon$ is a positive constant sufficiently small to be determined.
For any fixed $y = (y_1, \cdots, y_n)
\in \partial \omega_0 \cap \partial \Omega$, we have
\begin{equation}
\label{add-2}
x_n - y_n \geq (2 - \epsilon) b_{n-1} \mbox{ on } \partial \omega \cap \{x_n = 2 b_{n-1}\}.
\end{equation}
Next, for any $x = (x_1, \cdots, x_n) \in \partial \omega \cap \partial \Omega$, we have
\[
\begin{aligned}
x_n - y_n = \,& \sum_{\beta = 1}^{n-1}\rho_\beta (y') (x_\beta - y_\beta) + O (|x'-y'|^2)\\
   \geq \,& \sum_{\beta = 1}^{n-1}\rho_\beta (y') (x_\beta - y_\beta) - \kappa |x'-y'|^2
\end{aligned}
\]
by Taylor's expansion of $\rho$ at $y'$, where $\kappa$ is a positive constant depending only on the principal curvatures of $\partial \Omega$.
Let
\[
L (x) := \sum_{\beta = 1}^{n-1}\rho_\beta (y') (x_\beta - y_\beta).
\]
It follows that
\begin{equation}
\label{add-1}
w (x) := x_n - y_n - L (x) + \kappa |x'-y'|^2 \geq 0 \mbox{ on } \partial \omega \cap \partial \Omega.
\end{equation}
Note that $\rho_\beta (0) = 0$ for each $1 \leq \beta \leq n-1$. We find, by \eqref{BC2-1},
\[
|L (x)| \leq |D \rho (y')| \cdot |x'-y'| \leq C |y'| \cdot |x'-y'| \leq C \sqrt{\epsilon} b_{n-1}
\]
for any $x \in \overline \omega$. In particular, by \eqref{add-2}, we have
\begin{equation}
\label{add-3}
w (x) \geq (2 - \epsilon - C \sqrt{\epsilon}) b_{n-1} \geq b_{n-1}
\end{equation}
on $\partial \omega \cap \{x_n = 2 b_{n-1}\}$ by fixing $\epsilon$ small enough.
Thus, by \eqref{jw-4}, \eqref{add-1} and \eqref{add-3}, we have
\begin{equation}
\label{jw-17}
|u - \varphi| \leq C b_{n-1} w \mbox{ on } \partial \omega.
\end{equation}
From \eqref{taylor-1}, \eqref{jw-17}, \eqref{jw-5} and using Lemma \ref{cor-lemma}, we derive
\begin{equation}
\label{jw-7}
\begin{aligned}
F^{ij} (b_{n-1} (A_1 v - A_2 w) \pm (u - \varphi))_{ij} \geq \,& 0 \mbox{ in } \omega\\
  b_{n-1} (A_1 v - A_2 w) \pm (u - \varphi) \leq \,& 0 \mbox{ on } \partial \omega
\end{aligned}
\end{equation}
by choosing $A_1 \gg A_2 \gg 1$. It follows from the maximal principle that
\begin{equation}
\label{plus2}
|u_{\nu} (y)| \leq C b_{n-1},
\end{equation}
for $y\in \partial\Omega\cap\partial \omega_0$.
By \eqref{BC2-1}, \eqref{jw-2}, \eqref{plus2} and the convexity of $u$, we have
\begin{equation}
\label{plus1}
\sup_{\overline \omega_0} |u_\alpha| \leq \sup_{\partial \omega_0 \cap \partial \Omega} |\varphi_\alpha - u_n \rho_\alpha| \leq C b_{n-1}^{3/2}, \mbox{ for each } 1 \leq \alpha \leq n-1.
\end{equation}
Let  $\epsilon_1 = \epsilon/2$. For any $x = (x', \epsilon_1 b_{n-1}) \in \partial \omega_1 \cap
\{x_n = \epsilon_1 b_{n-1}\}$, by the convexity of $u$, \eqref{jw-4} and \eqref{plus1},
\begin{equation}
\label{add-6}
- \epsilon_1 b_{n-1} u_n (x) \leq u(0) - u(x) + \sum_{\alpha = 1}^{n-1} x_\alpha u_\alpha (x) \leq C b_{n-1}^2.
\end{equation}
It follows that
\begin{equation}
\label{plus3}
u_n (x) \geq - \frac{C}{\epsilon_1} b_{n-1}.
\end{equation}
On the other hand, we fix a point $y = (y', \epsilon b_{n-1}) \in \partial \omega_0 \cap \partial \Omega$.
By the convexity of $u$, \eqref{jw-4} and \eqref{plus1}, we find
\begin{equation}
\label{add-7}
\epsilon_1 b_{n-1} u_n (x) \leq u(y) - u(x) + \sum_{\alpha = 1}^{n-1} (y_\alpha - x_\alpha) u_\alpha (x) \leq C b_{n-1}^2
\end{equation}
and we obtain
\begin{equation}
\label{plus4}
u_n (x) \leq \frac{C}{\epsilon_1} b_{n-1}.
\end{equation}
Combining \eqref{plus1}, \eqref{plus3} with \eqref{plus4}, we get
\[
|Du| \leq C b_{n-1} \mbox{ on } \partial \omega_1.
\]
By the convexity of $u$ again, we have
\begin{equation}
\label{jw-8}
|Du| \leq C b_{n-1} \mbox{ in } \omega_1.
\end{equation}
In the following, we always denote $\nabla_{\alpha}=\nabla_{\tau_{\alpha}}$.
Note that $|\rho_\alpha| \leq C |x'| \leq C b_{n-1}^{1/2}$ on $\overline \omega_1$.
By \eqref{plus1} and \eqref{jw-8}, we see
$|\nabla_\alpha u| \leq C b_{n-1}^{3/2}$ on $\overline{\omega}_1$. It follows that
\begin{equation}
\label{jw-6}
|\nabla_\alpha (u - \varphi)| \leq C \sqrt{b_{n-1}} w \mbox{ on } \partial \omega_1
\end{equation}
by using \eqref{add-3}.
Since $\tau_\alpha$ is a tangential vector field of $\partial \Omega$ near the origin, we have
\begin{equation}
\label{smooth}
|\nabla_\alpha \tilde{f}| \leq C\sqrt{\tilde{f}}.
\end{equation}
near the origin. The proof of \eqref{smooth} can be found in Lemma 3.1 of \cite{Blocki03}.
Differentiating the equation $F (D^2 u) = f^{1/k}$ and using Lemma \ref{cor-lemma}, we get
\begin{equation}
\label{jw-10}
\begin{aligned}
|F^{ij} (\nabla_\alpha u)_{ij}| \leq \,& |\nabla_\alpha f^{1/k}| + C \sum_{i,j=1}^n F^{ij} u_{ij}
   + C |Du| \sum_{i=1}^n F^{ii}\\
    \leq \,& Cf^{\frac{1}{k}-\frac{1}{2(k-1)}} + C b_{n-1} \sum_{i=1}^n F^{ii}\\
    \leq\, &C b_{n-1}^{1/2-1/k}+ C b_{n-1} \sum_{i=1}^n F^{ii}
    \end{aligned}
\end{equation}
in $\omega_1$. By \eqref{jw-3}, we have
\begin{equation}
\label{jw-11}
|F^{ij} (\nabla_\alpha \varphi)_{ij}| \leq C \sqrt{b_{n-1}} \sum_{i=1}^n F^{ii} \mbox{ in } \omega_1.
\end{equation}
Combining \eqref{jw-6}, \eqref{jw-10}, \eqref{jw-11} and \eqref{jw-5}, we can choose positive constants
$A_1 \gg A_2 \gg 1$ again such that
\[
\begin{aligned}
F^{ij} \left(\sqrt{b_{n-1}} (A_1 v - A_2 w) \pm \nabla_\alpha (u - \varphi)\right)_{ij} \geq \,& 0 \mbox{ in } \omega_1\\
  A \sqrt{b_{n-1}} (A_1 v - A_2 w) \pm \nabla_\alpha (u - \varphi) \leq \,& 0 \mbox{ on } \partial \omega_1
\end{aligned}
\]
By the maximal principle again we get
\[
\sqrt{b_{n-1}} (A_1 v - A_2 w) \pm \nabla_\alpha (u - \varphi) \leq 0 \mbox{ in } \omega_1
\]
and
\begin{equation}
\label{jw-13}
|\nu(\nabla_{\alpha}u) (y)| \leq C \sqrt{b_{n-1}} \mbox{ for } \alpha \leq n-1,
\end{equation}
for any $y\in\partial\Omega\cap\partial\omega_1$, where $\nu$ is the unit interior normal of $\partial \Omega$ at $y$. Therefore, in view of \eqref{jw-8}, we get \eqref{jw-12}.

Now we consider the case $k = 2$.
We have, at the origin,
\[
u_{nn} \sigma_1 (b) - \sum_{\alpha \leq n-1} u_{n \alpha}^2 + \sigma_2 (b) = f.
\]
Thus, by \eqref{cor-1} and \eqref{jw-12}, we obtain
\begin{equation}
\label{jw-14}
u_{nn} (0) \leq C.
\end{equation}

\section{Estimates for convex solutions, the case $k \geq 3$}

In this section, we continue to prove Lemma \ref{lemma-2} for the case $k \geq 3$.

As \cite{GTW99}, we will prove Lemma \ref{lemma-2} by induction. Our induction hypothesis is that for some given $\alpha = n - (k-1), \ldots, n-2$ and any index
$\alpha + 1\leq \gamma\leq  n-1$, there exist two positive constants $\theta_\gamma$ and $C$ depending only on $\Omega$,
$|\varphi|_{C^{3,1} (\overline \Omega)}$ and $|\tilde{f}|_{C^{1,1} (\overline \Omega)}$ such that if any point $(x', \rho (x')) \in \partial \Omega$
satisfies $|x_\beta| \leq \theta_\gamma \sqrt{b_\gamma}$ for any $1 \leq \beta \leq \gamma$
and $|x_\beta| \leq \theta_\gamma \frac{b_\gamma}{\sqrt{b_\beta}}$ for any $\gamma + 1 \leq \beta \leq n-1$, we have the estimates
\begin{equation}
\label{jw-19}
|\nabla_{\nu \tau_\beta} u (x', \rho (x'))| \leq C \sqrt{b_\gamma}
\end{equation}
for $1\leq \beta \leq \gamma$.

From \eqref{jw-12}, we see that \eqref{jw-19} holds for $\alpha = n-2$.
In the following, we will prove there exist positive constants $\theta_\alpha$ and $C$ depending only on $\Omega$,
$|\varphi|_{C^{3,1} (\overline \Omega)}$ and $|\tilde{f}|_{C^{1,1} (\overline \Omega)}$ such that if any point $(x', \rho (x')) \in \partial \Omega$
satisfies  $|x_\beta| \leq \theta_\alpha \sqrt{b_\alpha}$ for any $1 \leq \beta \leq \alpha$
and $|x_\beta| \leq \theta_\alpha \frac{b_\alpha}{\sqrt{b_\beta}}$ for any $\alpha + 1 \leq \beta \leq n-1$, we have
\begin{equation}
\label{jw-21}
|\nabla_{\nu \tau_\beta} u (x', \rho (x'))| \leq C \sqrt{b_\alpha}
\end{equation}
for $1\leq \beta \leq \alpha$.

Let
\[
\omega := \{x \in \Omega: |x_\beta| < \delta \frac{b_\alpha}{\sqrt{b_\beta}}, \beta = \alpha + 1, \ldots, n-1,
   x_n < \delta^2 b_\alpha\},
\]
where $\delta$ is a positive  sufficiently small constant independent of $b_\alpha$ to be determined later.
By \eqref{jw-2} and that $u \geq 0$ in $\Omega$, we have
\begin{equation}
\label{jw-15}
|\varphi_{\mu \beta \gamma} (0)| \leq \left\{\begin{aligned}
& C \frac{\sqrt{b_\mu b_\beta b_\gamma}}{b_\alpha}, & \ \mbox{ if } \mu, \beta, \gamma > \alpha  \\
& C \frac{\sqrt{b_\beta b_\gamma}}{\sqrt{b_\alpha}}, & \ \mbox{ if } \mu \leq \alpha; \beta, \gamma > \alpha\\
& C \sqrt{b_\gamma}, & \ \mbox{ if } \mu, \beta \leq \alpha; \gamma > \alpha\\
& C \sqrt{b_\alpha}, & \ \mbox{ if } \mu, \beta, \gamma \leq \alpha
\end{aligned}\right.
\end{equation}
provided $b_\alpha$ is sufficiently small.
By \eqref{taylor} and the definition of $\omega$, we have
\begin{equation}
\label{jw-16}
f (x) \leq C b_{n-1} \cdots b_{\alpha+1} b_\alpha^{k+\alpha-n} \mbox{ in } \omega.
\end{equation}
We then note that
\begin{equation}
\label{jw-47}
u \leq C b_\alpha^2 \mbox{ in } \omega
\end{equation}
by the convexity of $u$.

Let
\[
\underline u (x) = \frac{\sigma}{2} \left(b_\alpha |\hat{x}|^2 + \sum_{\beta = \alpha + 1}^{n-1} b_\beta x_\beta^2\right)
    + \frac{1}{2} K x_n^2 - K^2 b_\alpha x_n,
\]
where $\hat{x} := (x_1, \ldots, x_\alpha)$, $\sigma$ and $K$ are sufficiently small and sufficiently large positive constants
to be chosen later. We then have
\begin{equation}
\label{jw-22}
\sigma_k (\lambda (D^2 \underline u)) \geq \sigma^{k-1} K b_{n-1} \cdots b_{\alpha+1} b_\alpha^{k+\alpha-n}.
\end{equation}

Next, we prove
\begin{equation}
\label{jw-23}
\underline u \leq \frac{1}{2} u
\end{equation}
on $\partial \omega$ by choosing suitable $\delta$, $\sigma$ and $K$. Note that
\[
\partial \omega = \partial_1 \omega \cup \partial_2 \omega \cup \partial_3 \omega,
\]
where $\partial_1 \omega := \partial \omega \cap \partial \Omega$, $\partial_2 \omega := \partial \omega \cap \{x_n = \delta^2 b_\alpha\}$
and $\partial_3 \omega := \partial \omega \cap \{|x_\beta| = \frac{\delta b_\alpha}{\sqrt{b_\beta}}$ for some $\beta = \alpha + 1, \ldots, n-1\}$.

We first consider any point $x \in \partial_1 \omega$. By \eqref{jw-2} and \eqref{jw-15},
\[
u (x) \geq \left(\frac{1}{2} - C \delta\right) \sum_{\beta = \alpha + 1}^{n-1} b_\beta x_\beta^2 - C b_\alpha |\hat{x}|^2
   \geq \frac{1}{4} \sum_{\beta = \alpha + 1}^{n-1} b_\beta x_\beta^2 - C b_\alpha |\hat{x}|^2
\]
provided $\delta$ is sufficiently small. By the uniformly convexity of $\Omega$, we have
\[
\underline u (x) \leq \frac{\sigma}{2} \sum_{\beta = \alpha + 1}^{n-1} b_\beta x_\beta^2 - b_\alpha \left(\frac{1}{2} K^2 x_n -
    \frac{\sigma}{2} |\hat{x}|^2\right) \leq \frac{\sigma}{2} \sum_{\beta = \alpha + 1}^{n-1} b_\beta x_\beta^2 - \frac{K^2}{4} b_\alpha |\hat{x}|^2
\]
if $\sigma$ is sufficiently small. Thus, \eqref{jw-23} holds on $\partial_1 \omega$ provided $\sigma$ is sufficiently small and $K$ is sufficiently large.

Next, for $x \in \partial_2 \omega$
\[
\underline u (x) \leq C \delta^2 b_\alpha^2 \sigma - \frac{K^2 \delta^2 b_\alpha^2}{2} \leq 0 \leq \frac{1}{2} u(x)
\]
provided $\sigma$ is sufficiently small or $K$ is sufficiently large.

For $\partial_3 \omega$, as \cite{GTW99}, we only consider the piece $\partial_3' \omega := \partial \omega \cap \left\{x_{n-1} = \frac{\delta b_\alpha}{\sqrt{b_{n-1}}}\right\}$
and other cases can be handled similarly. We first prove that
\begin{equation}
\label{jw-24}
u (x) \geq \frac{1}{4} \delta^2 b_\alpha^2 \mbox{ on } \partial_3' \omega \cap \{x_n < \epsilon_0 \delta^2 b_\alpha\}
\end{equation}
provided $\epsilon_0$ is sufficiently small. On $\partial_3' \omega \cap \{x_n \geq \epsilon_0 \delta^2 b_\alpha\}$, \eqref{jw-23} holds as on $\partial_2 \omega$. Therefore, we only consider the set $\partial_3' \omega \cap \{x_n < \epsilon_0 \delta^2 b_\alpha\} $.
Fix a point $x = (\hat{x}, \tilde{x}, x_n) \in \partial_3' \omega \cap \{x_n < \epsilon_0 \delta^2 b_\alpha\}$, where $\hat{x} = (x_1, \ldots, x_\alpha)$
and $\tilde{x} = (x_{\alpha+1}, \ldots, x_{n-1})$. We consider two cases.

\textbf{Case 1.} Suppose $\hat{x} = 0$. Let $x_0 = (0, \tilde{x}, \rho (0, \tilde{x})) \in \partial \Omega$. We may assume
$\delta \ll \min_{\alpha+1 \leq \gamma \leq n-1}\theta_{\gamma}$. Let $\tilde{\nu} := \tilde{\nu} (x') =
(- D\rho (x'), 1) \in \mathbb{R}^n$ for $x' \in \mathbb{R}^n$ near the origin.
Note that $|u_\gamma| = |\varphi_\gamma - \rho_\gamma u_n| \leq \sqrt{b_\alpha}$ on $\partial \omega \cap \partial \Omega$
for $1 \leq \gamma \leq n - 1$.
Therefore, $\nu = \frac{\tilde{\nu}}{|\tilde{\nu}|}$ and
for any point $y = (0, \tilde{y}, y_n) \in \partial \omega \cap \partial \Omega$, by our induction hypothesis,
we have
\begin{equation}
\label{jw-25}
\begin{aligned}
|u_{\tilde{\nu} (y')} (y)| \,& \leq |u_{\tilde{\nu} (0)}(0)| \\
    & + \sum_{\beta = \alpha+1}^{n-1} \sup_{\xi \in \partial \omega \cap \partial \Omega} \left(|\nabla_{\tilde{\nu}\tau_\beta} u (\xi)| + \sum_{\gamma=1}^{n-1}|\rho_{\gamma\beta} (\xi) u_\gamma (\xi)|\right) \cdot |y_\beta|
   \leq C b_\alpha
\end{aligned}
\end{equation}
by \eqref{jw-19} and that $D u (0) = 0$. It follows that $|u_\nu (y)| \leq C b_\alpha$. Therefore $|Du(y)|\leq Cb_{\alpha}$ since
$|u_{\tau_\gamma} (y)| = |\varphi_\gamma (y)| \leq C b_\alpha$ for each $1 \leq \gamma \leq n-1$.
Thus, by the convexity of $u$, $x_n<\epsilon_0\delta^2 b_{\alpha}$ and $\hat{x}=0$, we have
\begin{equation}
\label{jw-26}
\begin{aligned}
u (x) \geq u (x_0) - C b_\alpha |x - x_0| = \,& u (x_0) - C b_\alpha (x_n - \rho (0, \tilde{x}))\\
   \geq \,& u (x_0) - C \epsilon_0 \delta^2 b_\alpha^2.
\end{aligned}
\end{equation}
By \eqref{jw-15} and that $x_{n-1} = \frac{\delta b_\alpha}{\sqrt{b_{n-1}}}$, we have
\begin{equation}
\label{jw-29}
\begin{aligned}
u (x_0)& =\varphi(x_0)= \frac{1}{2} \sum_{\beta=\alpha+1}^{n-1} b_\beta x_\beta^2 + \frac{1}{6} \sum_{\xi, \beta, \gamma \geq \alpha+1} \varphi_{\xi \beta \gamma} (0) x_\xi x_\beta x_\gamma
    + O (|\tilde{x}|^4)\\
&\geq \frac{1}{2} \delta^2 b_\alpha^2 - C \delta^3 b_\alpha^2 \geq \frac{7}{16} \delta^2 b_\alpha^2
\end{aligned}
\end{equation}
provided $\delta$ is sufficiently small. Combining with \eqref{jw-26}, we have
\begin{equation}
\label{jw-27}
u (x) \geq \frac{3}{8} \delta^2 b_\alpha^2
\end{equation}
when $\epsilon_0$ is sufficiently small.

\textbf{Case 2.} Suppose $\hat{x} \neq 0$. We may assume $x \notin \partial \Omega$, otherwise \eqref{jw-27} can be derived
similar as \eqref{jw-29}. Let $x_0 = (0, \tilde{x}, x_n^0) = (0, \tilde{x}, \rho (0, \tilde{x})) \in \partial \Omega$
and $P$ be the 2-dimensional plane spanned by $x$ and the straight line through $x_0$ and parallel to the $x_n$-axis.
Note that $P \subset \{x_{n-1} = \frac{\delta b_\alpha}{\sqrt{b_{n-1}}}\}$.
Let $\gamma$ be the intersection of $\partial \Omega$ and $P$. It is clear that $x^* = (0, \tilde{x}, x_n + \varepsilon_1) \in P$,
where $\varepsilon_1 := - \sum_{\beta=1}^\alpha x_\beta \rho_\beta (0, \tilde{x})$. Note that
\[
\begin{aligned}
x_n^* - x_n^0 = \,& x_n - \sum_{\beta=1}^\alpha x_\beta \rho_\beta (0, \tilde{x}) - \rho (0, \tilde{x}) \\
 > \,& \rho (\hat{x}, \tilde{x}) - \sum_{\beta=1}^\alpha x_\beta \rho_\beta (0, \tilde{x}) - \rho (0, \tilde{x}) \geq 0
\end{aligned}
\]
by the convexity of $\rho$.
Suppose $$e_1=\frac{(\hat{x}, 0, - \varepsilon_1)}{\sqrt{|\hat{x}|^2+\varepsilon_1^2}}, \ \ e_2=\frac{(\varepsilon_1 \hat{x}, 0, |\hat{x}|^2)}{|\hat{x}|\sqrt{|\hat{x}|^2+\varepsilon_1^2}}.$$ It is easy to see that $e_1$ and $e_2$ are the unit tangent vector and unit normal vector of the curve $\gamma$
at $x_0$ respectively. By using the coordinate system $\{x_0; e_1, e_2\}$, the curve $\gamma$ can be written by
\begin{equation}
\label{proj}
\xi_2 = \kappa_0 \xi_1^2 + O (|\xi_1|^3) \mbox{ as } \xi_1 \rightarrow 0
\end{equation}
for some positive constant $\kappa_0$ depending only on $\partial \Omega$.
Thus, the straight line from $x$ to $x^*$ meets $\gamma$ at a point $\bar{x}$ satisfying
\begin{equation}
\label{jw-28}
c_0 |x - x^*| \leq |\bar{x} - x^*| \leq |\bar{x} - x|
\end{equation}
for some $0 < c_0 < 1$ depending only on $\partial \Omega$.

Let $\bar{x} = \bar{\xi}_1e_1+\bar{\xi}_2e_2$. Suppose  $\cos\theta = \langle e_2, E_n \rangle$, where
$E_n = (0, \ldots, 0, 1)$ is the direction of the $x_n$-axis.
Thus, we find $$\bar{\xi}_2 = (x_n^* - x_n^0) \cos \theta \text{ and } |\bar{\xi}_1| \leq C \sqrt{\bar{\xi}_2}$$ for some $C$ depending only on $\kappa_0$ by \eqref{proj},
since $|\varepsilon_1| \leq C \epsilon_0 \delta^2 b_\alpha$ and $x_n^* - x_n^0 \leq C \epsilon_0 \delta^2 b_\alpha$ which can be sufficiently  small. Next, we see
\[
E_n = \frac{- \varepsilon_1}{\sqrt{\varepsilon_1^2 + |\hat{x}|^2}} e_1 + \frac{|\hat{x}|}{\sqrt{\varepsilon_1^2 + |\hat{x}|^2}} e_2.
\]
It follows that, using the definition of $\varepsilon_1$,
\[
\bar{x}_n - x_n^0 = \frac{- \varepsilon_1 \bar{\xi}_1 + |\hat{x}| \bar{\xi}_2}{\sqrt{\varepsilon_1^2 + |\hat{x}|^2}}
   \leq |D \rho (0, \tilde{x})| |\bar{\xi}_1| + |\bar{\xi}_2| \leq C \epsilon_0 \delta^2 b_\alpha
\]
and therefore we get
\[
\bar{x}_n \leq C \epsilon_0 \delta^2 b_\alpha + x_n^0 \leq C \epsilon_0 \delta^2 b_\alpha.
\]
Thus, $\bar{x}\in\omega$ if $\epsilon_0$ is sufficiently small.

By \eqref{jw-2} and $\bar{x}\in\omega$, we have
\begin{equation}
\label{jw-30}
\begin{aligned}
u (\bar{x}) &= \frac{1}{2} \sum_{\beta=1}^{n-1} b_\beta \bar{x}_\beta^2 + + \frac{1}{6} \sum_{\xi, \beta, \gamma } \varphi_{\xi \beta \gamma} (0) \bar{x}_\xi \bar{x}_\beta \bar{x}_\gamma
    + O (|\bar{x}'|^4)\\
& \leq u (x_0) + \frac{1}{2} \sum_{\beta=1}^{\alpha} b_\beta \bar{x}_\beta^2
   + C \delta^3 b_\alpha^2.
  \end{aligned}
\end{equation}
By \eqref{BC2-1}, we have
\begin{equation}
\label{jw-31}
\begin{aligned}
\frac{1}{2} \sum_{\beta=1}^{\alpha} b_\beta \bar{x}_\beta^2 \leq C b_\alpha \bar{x}_n + b_\alpha O (|\bar{x}'|^3)
  \leq C b_\alpha \bar{x}_n + C \delta^3 b_\alpha^2.
\end{aligned}
\end{equation}
Since $|\varepsilon_1| \leq C \epsilon_0 \delta^2 b_\alpha$, using the same argument as Case 1, \eqref{jw-26} and \eqref{jw-27}
hold for $x = x^*$ if $\epsilon_0$ is small enough, namely
$$u (x^*) \geq u (x_0) - C \epsilon_0 \delta^2 b_\alpha^2, \ \  \  u (x^*) \geq \frac{3}{8} \delta^2 b_\alpha^2.$$
Combining with \eqref{jw-30}, \eqref{jw-31}, we get
\begin{equation}
\label{jw-32}
\begin{aligned}
u (\bar{x}) \leq \,& u (x^*) +C\epsilon_0\delta^2b_{\alpha}^2+ Cb_{\alpha}\bar{x}_n
   + C \delta^3 b_\alpha^2\\
\leq \,& u (x^*) + C \left(\epsilon_0 + \delta\right) \delta^2 b_\alpha^2 \leq \left(1+\frac{c_0}{3}\right) u (x^*)
\end{aligned}
\end{equation}
provided $\epsilon_0$ and $\delta$ are sufficiently small.
Therefore, by \eqref{jw-28} and the convexity of $u$, we have
\[
u (x) \geq u (x^*) + \frac{1}{c_0} \min\{0, u(x^*) - u (\bar{x})\} \geq \frac{2}{3} u (x^*) \geq \frac{1}{4} \delta^2 b_\alpha^2
\]
and \eqref{jw-24} is proved. Note that
\[
\underline u (x) \leq C \sigma \delta^2 b_\alpha^2 - \frac{K^2}{2} b_\alpha x_n \mbox{ in } \omega.
\]
Thus, \eqref{jw-23} holds when $\sigma$ is sufficiently small and $K$ is sufficiently large. Then
\eqref{jw-23} is valid on $\partial \omega$.

By \eqref{jw-16} and \eqref{jw-22}, we can further fix $K$ sufficiently large such that
\begin{equation}
\label{jw-46}
\sigma_k (\lambda (D^2 \underline u)) \geq \sup_\omega f \mbox{ in } \omega.
\end{equation}
We have constructed a lower barrier $\underline u$ vanishing at the origin with $\underline u_\nu (0) = - K^2 b_\alpha$.

Now we construct a lower barrier for an arbitrary point $x_0 = (x_1^0, \ldots, x_n^0)
\in \partial \Omega \cap \partial \omega_\epsilon$, where
\[
\omega_\epsilon := \{x \in \Omega: |x_\beta| < \epsilon \delta \frac{b_\alpha}{\sqrt{b_\beta}},
    \beta = \alpha + 1, \ldots, n-1, x_n < \epsilon^2 \delta^2 b_\alpha\}
\]
and $\epsilon$ is a positive constant sufficiently small to be determined later. For $x \in \overline \omega$, write $y = x - x_0$ and define
\begin{equation}
\label{add-8}
\underline u (x) = \underline u (y) = \frac{\sigma}{2} \left(b_\alpha |\hat{y}|^2
    + \sum_{\beta = \alpha + 1}^{n-1} b_\beta y_\beta^2\right) + \frac{1}{2} K y_n^2 - K^2 b_\alpha \bar{y}_n
\end{equation}
as above, where
\[
\bar{y}_n := y_n - \sum_{\beta=1}^{n-1} \rho_\beta (x'_0) y_\beta.
\]
Define $l (y')$ to be the linear function
\[
l (y') := u (x_0) + \sum_{\beta = 1}^{n-1} \nabla_\beta u (x_0) y_\beta.
\]
Now we prove $\underline u \leq u - l (y')$ on $\partial \omega$ by choosing suitable positive constants $\sigma$, $K$
and $\epsilon$.

First we consider $x \in \partial_1 \omega$ as before. Let
$\tilde{b}_\beta := \max\{b_\alpha, b_\beta\}$ for $1 \leq \beta \leq n-1$. Thus, by \eqref{jw-2} and \eqref{jw-15},
we have
\[
\frac{\partial^2 \varphi}{\partial x_\beta^2} (x', \rho (x')) \geq b_\beta -  C \delta \tilde{b}_\beta,
  \  \beta = 1, \ldots, n-1
\]
and
\[
\left|\frac{\partial^2 \varphi}{\partial x_\beta \partial x_\mu} (x', \rho (x'))\right| \leq C \delta
  \sqrt{\tilde{b}_\beta \tilde{b}_\mu},\  \beta, \mu = 1, \ldots, n-1; \beta \neq \mu
\]
for any $(x', \rho (x')) \in \partial_1 \omega$. Hence by Taylor's expansion of $u (x', \rho(x'))
= \varphi (x', \rho (x'))$ at $x_0$, we obtain
\begin{equation}
\label{jw-40}
u (x) \geq l (y') + \frac{1}{4} \sum_{\beta=1}^{n-1} b_\beta y_\beta^2 - C b_\alpha |\hat{y}|^2
\end{equation}
provided $\delta$ is sufficiently small.
By Taylor's expansion of $\rho$ at $x'_0$ and the uniform convexity of $\Omega$, we get
\[
y_n = \rho (x) - \rho (x_0) \geq \sum_{\beta=1}^{n-1} \rho_\beta (x'_0) y_\beta + \kappa |y'|^2,
\]
where $\kappa$ is a positive constant depending only on the principal curvatures of $\partial \Omega$.
It follows that
\begin{equation}
\label{jw-41}
\bar{y}_n \geq \kappa |y'|^2 \mbox{ on } \partial_1 \omega.
\end{equation}
Next, since $|\rho_\beta (x'_0)| \leq C |x'_0| \leq C \epsilon \delta \sqrt{b_\alpha}$
for any $\beta = 1, \ldots, n-1$, we have
\begin{equation}
\label{jw-45}
y_n^2 = \left(\bar{y}_n + \sum_{\beta = 1}^{n-1} \rho_\beta (x'_0) y_\beta\right)^2
  \leq 2 \bar{y}_n^2 + C \epsilon^2 \delta^2 b_\alpha |y'|^2.
\end{equation}
Therefore, by \eqref{jw-41}, \eqref{jw-45} and $|\bar{y}_n| \leq C b_\alpha$, we have
\begin{equation}
\label{jw-42}
\begin{aligned}
\underline u (x) \leq \,& \frac{\sigma}{2} \sum_{\beta = \alpha + 1}^{n-1} b_\beta y_\beta^2
   - b_\alpha \left(\frac{1}{2} K^2 \bar{y}_n - \frac{\sigma}{2} |\hat{y}|^2 - C \epsilon^2 \delta^2 |y'|^2\right)\\
     \leq \,& \frac{\sigma}{2} \sum_{\beta = \alpha + 1}^{n-1} b_\beta y_\beta^2 - \frac{K^2}{4} b_\alpha |y'|^2
\end{aligned}
\end{equation}
provided $\sigma$ is sufficiently small and $K$ is sufficiently large.
Combing \eqref{jw-40} and \eqref{jw-42} we obtain $u (x) - l (y') \geq \underline u (x)$ if $K$ is large enough.

For $x \in \partial_2 \omega$, we note that $\bar{y}_n \geq (1 - \epsilon^2 - C \epsilon) \delta^2 b_\alpha$ since
$x_0 \in \partial \Omega \cap \partial \omega_\epsilon$.
By \eqref{jw-2} and \eqref{jw-15} again, we find
\begin{equation}
\label{jw-44}
|l (y')| \leq C (\epsilon^2 + \epsilon) \delta^2 b_\alpha^2.
\end{equation}
It follows that
\[
\begin{aligned}
\underline u (x) + l (y') \leq \,& C \sigma \delta^2 b_\alpha^2 + CK \delta^2 b_\alpha^2
   - K^2 (1 - \epsilon^2 - C \epsilon) \delta^2 b_\alpha^2
     + C (\epsilon^2 + \epsilon) \delta^2 b_\alpha^2\\
     < \,& 0 \leq u (x)
\end{aligned}
\]
if $K$ is sufficiently large and $\epsilon$ is sufficiently small.

Now we consider $x \in \partial_3 \omega$. We only need to consider the case $x \in \partial'_3 \omega$
as before. First we note that if $x_n \geq \epsilon_0 \delta^2 b_\alpha$, where $\epsilon_0$ is the constant defined by \eqref{jw-24},
\[
\bar{y}_n \geq (\epsilon_0 - \epsilon^2 - C \epsilon) \delta^2 b_\alpha.
\]
Thus, we have $\underline u (x) + l (y') < 0 < u (x)$ if $\epsilon \ll \epsilon_0$ and $K$ is large enough as in the case $x \in \partial_2 \omega$.

Now we consider $x \in \partial_3' \omega \cap \{x_n < \epsilon_0 \delta^2 b_\alpha\}$. By \eqref{jw-24} and \eqref{jw-44}, we have
\[
u (x) - l(y') \geq \left(\frac{1}{4} - C (\epsilon + \epsilon^2)\right) \delta^2 b_\alpha^2
    \geq \frac{1}{8} \delta^2 b_\alpha^2
\]
provided $\epsilon$ is sufficiently small. By \eqref{jw-45}, we have
\[
\underline u (x) \leq C (\sigma + \epsilon) \delta^2 b_\alpha^2 - \frac{K^2}{2} b_\alpha \bar{y}_n.
\]
It follows that $\underline u (x) \leq u (x) - l(y')$ if $\sigma$ and $\epsilon$ are small enough and $K$ is large enough.
We can further fix $K$ sufficiently large such that \eqref{jw-46} holds.

To construct an upper barrier for $x_0 \in \partial \Omega \cap \partial \omega_\epsilon$, we define
\[
\overline u (x) = \overline u (y) := M \left(\frac{1}{2} b_\alpha |\hat{y}|^2
   + \frac{1}{2} \sum_{\beta = \alpha+1}^{n-1} b_\beta y_\beta^2 + b_\alpha \bar{y}_n\right),
\]
where $M$ is a sufficiently large constant to be determined later.
By \eqref{jw-47}, we can fix $\epsilon$ sufficiently small and $M$ sufficiently large such that
\[
u - l (y') \leq \overline u \mbox{ on } \partial \omega.
\]
Furthermore, we find
\[
\det (D^2 \overline u) = 0 \mbox{ in } \omega.
\]
By the convexity of $u$ and the maximal principle, we have
\[
u - l (y') \leq \overline u \mbox{ in } \omega.
\]
Thus, we have
\begin{equation}
\label{add-4}
|u_\nu| \leq C b_\alpha \mbox{ on } \partial_1 \omega_\epsilon.
\end{equation}
Similar to \eqref{plus1}, we find, by \eqref{BC2-1}, \eqref{jw-2}, \eqref{add-4}
and the convexity of $u$,
\begin{equation}
\label{add-5}
\begin{aligned}
\sup_{\overline \omega} |u_\beta| \leq \,& \sup_{\partial \omega \cap \partial \Omega} |\varphi_\beta - u_n \rho_\beta| \leq C b_{\alpha}^{3/2}, \mbox{ for } 1 \leq \beta \leq \alpha,\\
\sup_{\overline \omega} |u_\beta| \leq \,& \sup_{\partial \omega \cap \partial \Omega} |\varphi_\beta - u_n \rho_\beta| \leq C \sqrt{b_\beta} b_{\alpha}, \mbox{ for } \alpha+ 1 \leq \beta \leq n-1.
\end{aligned}
\end{equation}
As \eqref{add-6} and \eqref{add-7}, for any $x = (x', \epsilon^2 \delta^2 b_\alpha) \in \partial_2 \omega_{\epsilon}$,
we have
\[
- \epsilon^2 \delta^2 b_{\alpha} u_n (x) \leq u(0) - u(x) + \sum_{\beta = 1}^{n-1} x_\beta u_\beta (x) \leq C b_\alpha^2
\]
and
\[
\delta^2 (1 - \epsilon^2) b_{\alpha} u_n (x) \leq u(y) - u(x) \leq C b_{\alpha}^2, \mbox{ with } y = (x', \delta^2 b_\alpha) \in \partial_2 \omega
\]
by \eqref{jw-47} and \eqref{add-5}. It follows that $|u_n| \leq C b_\alpha$ on $\partial_2 \omega_{\epsilon}$. Therefore,
by the convexity of $u$ again, we can also get a bound $|u_n| \leq C b_\alpha$ on $\partial_3 \omega_{\epsilon}$.
We then obtain
\begin{equation}
\label{jw-33}
|D u| \leq C b_\alpha \mbox{ on } \overline{\omega}_\epsilon.
\end{equation}
To proceed we consider $\nabla_\beta (u - \varphi)$ for $1 \leq \beta \leq \alpha$.
By \eqref{jw-15}, \eqref{BC2-1}, \eqref{add-5} and \eqref{jw-33}, we have
\begin{equation}
\label{jw-34}
|\nabla_\beta (u-\varphi)| \leq C b_\alpha^{3/2} \mbox{ on } \overline \omega_\epsilon.
\end{equation}
By \eqref{jw-15} again, we find
\begin{equation}
\label{jw-35}
|F^{ij} (\nabla_\beta \varphi)_{ij}| \leq C \sqrt{b_\alpha} \left(\sum_{i=1}^{\alpha} F^{ii} + \sum_{i=\alpha+1}^{n-1} \frac{b_i}{b_\alpha} F^{ii}\right)
   \mbox{ in } \omega_\epsilon.
\end{equation}
Let
\[
m := b_{n-1} \cdots b_{\alpha+1} b_\alpha^{k+\alpha-n}.
\]
We find
\[
m^{-1/2(k-1)} \leq (b_\alpha^{k-1})^{-1/2(k-1)} = b_\alpha^{-1/2}.
\]
By \eqref{smooth}, \eqref{jw-16} and \eqref{jw-33}, we have
\begin{equation}
\label{jw-36}
\begin{aligned}
|F^{ij} (\nabla_\beta u)_{ij}| \leq \,& |\nabla_\beta f^{1/k}| + C \left(f^{1/k} + b_\alpha \sum_{i=1}^n F^{ii}\right)\\
  \leq \,& C m^{1/k} m^{-1/2(k-1)} + C b_\alpha \sum_{i=1}^n F^{ii}\\
   \leq \,& C b_\alpha^{-1/2} m^{1/k} + C b_\alpha \sum_{i=1}^n F^{ii}
\end{aligned}
\end{equation}
in $\omega_\epsilon$. Now we consider an arbitrary point $x_0 = (x_1^0, \ldots, x_n^0)
\in \partial \Omega \cap \partial \omega_{\epsilon_1}$ with $\epsilon_1 \ll \epsilon$ to be
determined later. Let $\underline u$ be the function defined in \eqref{add-8}.
By the concavity of $F$, we have
\begin{equation}
\label{jw-37}
\frac{1}{\sqrt{b_\alpha}} F^{ij} (\underline u - u)_{ij} \geq c_0 \left(\sqrt{b_\alpha}\sum_{i=1}^{\alpha} F^{ii} +
  \sqrt{b_\alpha} \sum_{i=\alpha+1}^{n-1} \frac{b_i}{b_\alpha} F^{ii} + F^{nn}
   + \frac{m^{1/k}}{\sqrt{b_\alpha}}\right)
\end{equation}
in $\omega_\epsilon$ for some positive constant $c_0$ which may depend on $\sigma$ and $K$. Let
\[
v = \frac{1}{\sqrt{b_\alpha}} \sum_{\beta=\alpha+1}^{n-1} b_\beta y_\beta^2 + \sqrt{b_\alpha} \bar{y}_n.
\]
Thus, for any $x \in \partial_2 \omega_\epsilon$, we have
\[
v (x) \geq \sqrt{b_\alpha} \bar{y}_n \geq \sqrt{b_\alpha} (\epsilon^2 - \epsilon_1^2 - C \epsilon_1) \delta^2 b_\alpha
  \geq \frac{\epsilon^2}{2} \delta^2 b_\alpha^{3/2}
\]
if we let $\epsilon_1$ be small enough such that $\epsilon_1^2 + C \epsilon_1 \leq \epsilon/2$.
Next, for any $x \in \partial_3 \omega_\epsilon$, which means $|x_{\beta_0}| = \epsilon \delta b_\alpha/\sqrt{b_{\beta_0}}$
for some $\alpha + 1 \leq \beta_0 \leq n-1$, we find
\[
\begin{aligned}
v (x) \geq \,& \frac{1}{\sqrt{b_\alpha}} \sum_{\beta=\alpha+1}^{n-1} b_\beta y_\beta^2
   \geq \frac{1}{\sqrt{b_\alpha}} b_{\beta_0} y_{\beta_0}^2
      \geq \frac{b_{\beta_0}}{\sqrt{b_\alpha}} \left(\frac{1}{2} x_{\beta_0}^2 - (x_{\beta_0}^0)^2\right)\\
      \geq \,& \left(\frac{\epsilon}{2} - \epsilon_1^2\right) \delta^2 b_\alpha^{3/2}
      \geq \frac{\epsilon}{2} \delta^2 b_\alpha^{3/2}
\end{aligned}
\]
provided $\epsilon_1^2 \leq \epsilon/2$. Combining the above two inequalities and \eqref{jw-34}, we have
\[
|\nabla_\beta (u-\varphi)| \leq C v \mbox{ on } \partial \omega_\epsilon.
\]
Thus, by \eqref{jw-35}, \eqref{jw-36} and \eqref{jw-37}, there exist positive constants $A \gg B \gg 1$ such that
\[
\begin{aligned}
F^{ij} \left(\frac{A}{\sqrt{b_{\alpha}}} (\underline u - u) - B v \pm \nabla_\beta (u - \varphi)\right)_{ij} \geq \,& 0 \mbox{ in } \omega_\epsilon\\
  \frac{A}{\sqrt{b_{\alpha}}} (\underline u - u) - B v \pm \nabla_\beta (u - \varphi) \leq \,& 0 \mbox{ on } \partial \omega_\epsilon
\end{aligned}
\]
By the maximal principle we have
\[
\frac{A}{\sqrt{b_{\alpha}}} (\underline u - u) - B v \pm \nabla_\beta (u - \varphi) \leq 0 \mbox{ in } \omega_\epsilon
\]
and
\[
|\nu (\nabla_\beta u) (x_0)| \leq C \sqrt{b_{\alpha}}, \mbox{ for } 1\leq \beta \leq \alpha.
\]
We thus have proved
\[
|\nabla_{\nu \tau_\beta} u| \leq C \sqrt{b_{\alpha}} \mbox{ on } \partial \omega_{\epsilon_1} \cap \partial \Omega,
  \mbox{ for } 1\leq \beta \leq \alpha.
\]
By induction, \eqref{jw-18} is proved.

At the origin, we have
\begin{equation}
\label{jw-39}
\sigma_{k-1} (b) u_{nn} (0) - \sum_{\alpha \leq n-1} u_{n\alpha}^2 (0) \sigma_{k-2;\alpha} (b) + \sigma_k (b) = f(0).
\end{equation}
Note that, by \eqref{jw-18},
\begin{equation}
\label{jw-38}
\begin{aligned}
& \frac{\sum_{\alpha \leq n-1} u_{n\alpha}^2 \sigma_{k-2;\alpha} (b)}{\sigma_{k-1} (b)}\\
  \leq \,& C\frac{\sum_{\alpha =1}^{n-k} u_{n\alpha}^2 b_{n-k+2}\cdots b_{n-1}}{b_{n-k+1}\cdots b_{n-1}}
    + C\frac{\sum_{\alpha =n-k+1}^{n-1} u_{n\alpha}^2 b_{n-k+1}\cdots \hat{b}_\alpha \cdots b_{n-1}}{b_{n-k+1}\cdots b_{n-1}}\\
  \leq \,& C \frac{b_{n-k+1}\cdots b_{n-1}}{b_{n-k+1}\cdots b_{n-1}} = C.
\end{aligned}
\end{equation}
Combining \eqref{cor-1}, \eqref{jw-39} and \eqref{jw-38}, we obtain
\[
u_{nn} (0) \leq C.
\]

\bigskip

{\bf Acknowledgement.} The authors wish to thank Professor Bo Guan for many helpful discussions.

redpajama_set_name: RedPajamaArXiv


Bollywood: The high profile and buzz-making wedding of the year saw desi girl Priyanka Chopra and international music sensation Nick Jonas getting hitched. The stunning couple got married at the majestic Umaid Bhawan Palace in Jodhpur on December 2, 2018.
The wedding festivities continued for days. But looks like the party is still on! The Jonas' hosted a cool reception party for the newlyweds where PeeCee's mother Dr Madhu Chopra and brother Siddharth were also present. Looks like it took place in Belmont.
Check out the beautiful picture shared by PeeCee on Instagram.
The couple had two ceremonies—a white wedding followed by a traditional Hindu one.
Nickyanka hosted a grand reception on December 4 in Delhi which was largely for VVIPs, relatives and the media. Prime Minister Narendra Modi was amongst the first guests to arrive at the reception and blessed the couple.
Much like Deepika Padukone and Ranveer Singh, Nickyanka too hosted two receptions—one in Delhi and another in Mumbai. Both the receptions were a gala affair with celebrities and several high profile dignitaries making their presence felt at the do.
Meanwhile, PeeCee is gearing up for the release of her Hollywood flick 'Isn't It Romantic?' which releases on February 14.


Plant Parents well trained, licensed, uniformed professional staff emphasizes client satisfaction! Service excellence is never accidental. We take the time and have the commitment to do the job right. Plant Parents has earned a reputation for exceeding horticultural expectations through efficiency, integrity and quality service.
Our Horticultural Technicians are reviewed in their accounts with quality control visits. Regular positive reinforcement assures our clients of consistent quality and helps us build a superior reputation for service excellence.
Plant Parents maintains all necessary insurance as required by the State of Florida. Our employees are covered under a workman's compensation policy. We also carry a comprehensive commercial policy. Plant Parents is a State of Florida licensed pest control company for ornamental plant.
Plant Parents staff is dedicated to continuing education and refining their horticultural skills. Our horticultural technicians are licensed and received on-going training and testing with industry recognized curriculum. Plant Parents is a State of Florida licensed pest control company for ornamental plants.
In our commercial accounts we perform weekly service without interrupting the client's workflow. We are in the business of helping companies provide a healthier more productive work environment for their employees and clients. Residential accounts are serviced with dedication and extra attention to personal respect and trust.
Plant Parents maintains all necessary insurance as required by the State of Florida. Our employees are covered under a workman's compensation policy. We also carry a comprehensive commercial policy.
Plant Parents belongs to the prestigious National Interiorscape Network. The national interiorscape industry has recognized Plant Parents with several Best Project Design Awards. Plant Parents maintains the "Certified Pest Control Operator" designation in ornamental plants from the State of Florida. We also are members of the Florida Nursery Growers and Landscape Association, (FNGLA) and have earned their "Florida Certified Horticultural Professional" designation.
Plant Parents believes in giving back to the community. We have been recognized for our community work with numerous non-profit civic organizations and schools. Plant Parents believes in the positive affirmation of bringing plants into peoples' lives.


text:
\section{Introduction}
The three-flavor neutrino picture is coming into view with oscillation data from many experiments.
Some key tensions remain, many of which can be addressed by the presence of a new light sterile neutrino with a mass $\sim1$ eV and relatively large mixing \cite{Diaz:2019fwt,Boser:2019rta,Giunti:2019aiy,Acero:2022wqg}.
This explanation, however, is in tension with a number of data sets, notably cosmological data  from the early universe \cite{Hannestad:2010kz,Wong:2011ip,Lattanzi:2017ubx,Hagstotz:2020ukm}, as well as those from terrestrial and solar neutrino experiments \cite{Dentler:2018sju}.

In this letter, we propose a novel solution to some of the constraints on light sterile neutrinos by dynamically changing the mass of the sterile neutrino depending on its environment -- an ultralight depth-dependent scalar non-standard interaction in the sterile the sector, see e.g.~\cite{Ge:2018uhz,Wise:2018rnb,Denton:2018dqq,Babu:2019iml,Smirnov:2019cae,Venzor:2020ova,Medhi:2021wxj,Medhi:2022qmu,Denton:2022pxt} for other papers with components of this model.
This is accomplished by coupling the sterile neutrino to an ultralight scalar which in turn is coupled to baryons.
This model evades cosmological constraints as well as solar and atmospheric constraints and the scalar can be a viable candidate for dark matter (DM).

Some aspects of our scenario have been discussed before with key differences such as the lack of a DM candidate \cite{Dasgupta:2013zpn,Escudero:2022gez}, and the lack of an explanation for the $\sim1$ eV sterile neutrino hints \cite{Gonzalez-Garcia:2006vic,Dev:2022bae}.
In addition, none address the constraint from solar or atmospheric neutrinos.
Other models looking to reconcile $\sim1$ eV sterile neutrinos with cosmology often turn to neutrino self-interactions \cite{Hannestad:2013ana,Dasgupta:2013zpn}, although it is known that this introduces more problems \cite{Song:2018zyl,Chu:2018gxk}; such a scenario could partially improve \cite{Kreisch:2019yzn} the Hubble tension \cite{Planck:2018vyg,Riess:2019cxk,Abdalla:2022yfr}.
Other studies have investigated the connection between ultralight particles and neutrino oscillation experiments \cite{Berlin:2016woy,Brdar:2017kbt,Krnjaic:2017zlz,Davoudiasl:2018hjw,Liao:2018byh,Capozzi:2018bps,Huang:2018cwo,Farzan:2019yvo,Cline:2019seo,Dev:2020kgz,Losada:2021bxx,Huang:2021kam,Chun:2021ief} or DM \cite{Davoudiasl:2022ubh}.

\section{Light Sterile Neutrinos}
A new sterile neutrino is an economical way to develop rich neutrino oscillation phenomenology and has been a standard benchmark for many interesting theoretical and experimental questions and may well be related to the nature of neutrino mass.
The anomalous data sets regularly suggest $m_4\sim1$ eV which typically mixes either with $\nu_e$ or both $\nu_e$ and $\nu_\mu$ at the $\sim1\%$ to $\sim10\%$ level.

In the case where $\nu_4$ primarily mixes with $\nu_e$,  the most significant hint comes from gallium data \cite{Giunti:2010zu,SAGE:2009eeu,Kaether:2010ag}.
While the theory involved is somewhat complicated \cite{Kostensalo:2019vmv}, a recent experiment claimed a high significance result at the $\gtrsim4\sigma$ level at $m_4\gtrsim1$ eV and $\sin^22\theta_{ee}\sim0.4$ \cite{Barinov:2021asz}.
These parameters are in some tension with solar data using theoretical predictions for the solar neutrino flux \cite{Goldhagen:2021kxe}.

Additional weak hints come from short-baseline accelerator data and reactor neutrinos compared to theory or reactor spectral data \cite{Mention:2011rk,T2K:2014xvp,Berryman:2020agd,Denton:2021czb} which also point towards a similar picture.
For these data sets either other explanations exist or the significance is quite low \cite{DayaBay:2017jkb,RENO:2018pwo,Arguelles:2021meu,MicroBooNE:2022wdf}.
Reactor spectral data also disfavors the large mixing preferred by some anomalies \cite{Berryman:2020agd}.

The additional inclusion of $\nu_\mu$ mixing has the strengths (and weaknesses) of mixing with just $\nu_e$ and is impacted by several new important data sets.
Notably LSND \cite{LSND:2001aii} and MiniBooNE \cite{MiniBooNE:2020pnu} have each seen evidence for $\nu_\mu\to\nu_e$ mixing at $3.8\sigma$ and $4.8\sigma$ respectively that, when interpreted with other constraints, leads to a preferred region of parameter space of $m_4\sim0.7$ eV and $\sin^22\theta_{\mu e}\sim0.01$ \cite{Dentler:2018sju}.
These parameters are also tested elsewhere notably MINOS+ \cite{MINOS:2017cae} and IceCube \cite{IceCube:2020phf} which have derived constraints that put the high significance results from LSND and MiniBooNE into confusing tension.
Simple extensions to the $\sim1$ eV sterile neutrino picture such as multiple sterile neutrinos in this range \cite{Kopp:2013vaa,Diaz:2019fwt} or new interactions \cite{Liao:2016reh,Denton:2018dqq} do not significantly clarify this picture.

In any case, strong constraints from cosmology exist which are largely flavor independent and strongly disfavor all of the interesting parameter spaces discussed above \cite{Hagstotz:2020ukm}.

\section{An Ultralight Boson Model}

We will consider the Lagrangian for the interactions of the ultralight scalar $\phi$ with the sterile neutrino $\nu_s$  and Earth, {\it i.e.}  electrons $e$ and nucleons $n$
\begin{equation}
{\cal L} \supset -(m_0+ g_s \phi)\, \bar \nu_s \nu_s - g_e \phi\, \bar e e - g_n \phi\,\bar n n\,,
\label{L}
\end{equation}
where $m_0$ is a possible bare mass term, for now taken to be zero or small compared to 1 eV.
A long range force acting on electrons and nucleons is constrained to have strength $g_e \lesssim 1.4 \times 10^{-25}$ and $g_n \lesssim 8.0\times 10^{-25}$ at $2\sigma$, respectively  \cite{Fayet:2017pdp,MICROSCOPE:2022doy}.  For simplicity, we will examine the case where  $g_e\ll g_n$ and thus electron densities can be ignored.

The potential for $\phi$ is assumed to be given by
\begin{equation}
V(\phi) = \frac{1}{2} m_\phi^2 \phi^2 +
\frac{\lambda_\phi}{4!} \phi^4\,,
\label{Vphi}
\end{equation}
where $m_\phi$ is the mass of $\phi$ and $\lambda_\phi$ denotes its self-coupling strength.
We will posit  that the mass\footnote{We define $g_sg_n<0$ to give the sign for a conventional mass term.} implied by laboratory experiments for $\nu_4$,  which is mostly composed of $\nu_s$, is sourced coherently by the terrestrial nucleons (baryons): $m_s \approx g_s \phi$  for $m_0\ll m_s$.
We will consider $m_\phi$ smaller than the inverse of Earth's radius $1/R_\oplus \approx (6400~\text{km})^{-1} \approx 3 \times 10^{-14}$~eV, in order to maximize the source  contribution.
Henceforth, we will take $m_\phi =  5\e{-15}$~eV  as our reference value (see the appendices for the impact of different values of $m_\phi$, including possible annual modulation of terrestrial signals).

Then, at the surface of the Earth, we have
\begin{equation}
\phi_\oplus \approx -\frac{g_n\, N_n^\oplus}{4\pi R_\oplus}e^{-m_\phi R_\oplus}\,.
\label{phi}
\end{equation}
The above expression is valid as long as we can ignore the $\phi^4$ term in $V(\phi)$.  We will justify this assumption below.

Let us take $g_n \approx 7\times 10^{-25}$, consistent with current bounds.  We then have
$\phi_\oplus \approx 6 \times 10^{12}$~eV.  Hence, as a reference value, to get a Dirac mass $m_s \approx1$~eV for $\nu_s$ at the Earth's surface, we need to assume $g_s \approx 2\e{-13}$; see Ref.~\cite{Denton:KG} for the details of calculating $\phi${\footnote{Note that $\phi_\oplus$ is accidentally close to the weak scale, and hence $g_s$ ends up near the typical value of Yukawa couplings inferred for the SM neutrinos.}.
In our scenario, this is the largest fermion coupling of $\phi$, which typically induces a quartic coupling at 1-loop order given by
\begin{equation}
\delta \lambda_\phi \sim \frac{g_s^4}{16 \pi^2}\sim 10^{-53} \left(\frac{g_s}{2\e{-13}}\right)^4\,.
\label{delta-lphi}
\end{equation}
We mention in passing that for the assumed values of $(g_s, g_n)$, finite 1-loop corrections quadratic in fermion bare masses do not destabilize $m_\phi^2$ (for  $m_0\ll$~eV).

\section{Neutrino Mass Generation}
In order to address the anomalies ascribed to sterile neutrinos, we need to induce a mixing between active SM neutrinos and $\nu_s$.
This may be achieved in a number of ways, but here we focus on Dirac neutrinos, for concreteness.
Another case with  Majorana right-handed neutrinos is sketched in the appendix.

\subsection{The Dirac Case}
As a simple example of the above, let us take the case of Dirac neutrinos and lepton number conservation.
Lepton number can be stabilized in a number of ways via $U(1)$ or $Z_n$ type symmetries, see e.g.~\cite{CentellesChulia:2018bkz}.
We consider the three active flavors $\nu_a$ coupled to three right-handed neutrinos $\nu_R$, as well as an additional neutrino\footnote{While we refer to this new fermion as a neutrino for convenience, it does not carry weak charge.} $\nu_s$ which has both left and right components.
We consider the Yukawa couplings
\begin{equation}
-\mathcal L \supset
y_a H \bar \nu_R L_a +
y_s H \bar \nu_s L_a + {\rm H.C.}\,,
\label{Yukawa}
\end{equation}
where $y_{a,s}$ are Yukwa coupling constants, $H$ is the SM Higgs doublet, and $L_a$ is the lepton doublet containing $\nu_a$.  The above interactions generate Dirac mass terms $m_\nu = y_a \vev{H}$ and $m_D = y_s \vev{H}$, with $v\equiv \sqrt{2}\vev{H}\approx 246$~GeV the Higgs vacuum expectation value.
We then also have a contribution to the mass of the $\nu_s$ state from Eq.~\eqref{L} $m_s=m_0+g_s\phi$ which changes with the local baryon density.
One can show that mass eigenvalues are
\begin{equation}
m_{1,2,3} = m_\nu\frac{m_s}{\sqrt{m_s^2 + m_D^2}}\,,
\label{m1}
\end{equation}
and
\begin{equation}
m_4 = \sqrt{m_s^2 + m_D^2 + \frac{m_\nu^2 m_D^2}{m_s^2 + m_D^2}}\,.
\label{m4}
\end{equation}
The mixing between the active flavors $\nu_a$ and $\nu_4$ is governed by the effective two-flavor angle $\theta_{i4}$, where
\begin{equation}
\tan (2 \theta_{i4}) = \frac{2 m_D m_s}{m_s^2-m_D^2-m_\nu^2}\,.
\label{theta-a}
\end{equation}
Generally, $\theta_{i4}$ is required to be $\ord{0.1}$ near the Earth's surface, and implies
$m_D\gtrsim 0.1$~eV consistent with direct mass searches by KATRIN which finds $m_a\lesssim1$ eV at the Earth's surface \cite{KATRIN:2019yun}.

As mentioned before, the above parameters only describe neutrinos near the surface of the Earth.
However, in environments with much higher or lower matter densities those parameters can be very different.  Note that since the postulated long range force is mediated by a scalar, it does not distinguish between particles and antiparticles.

\section{Cosmology}
A primary motivation for our model is to avoid the constraints on sterile neutrinos that disfavor new light degrees of freedom in the early universe.  However, we will illustrate below how one may also obtain a viable DM candidate in our setup, making our proposal significantly  more compelling.

\subsection{Constraints from Big Bang Nucleosynthesis}

Here, we discuss how our proposal  can lead to significant suppression of sterile neutrino production in the early Universe.
Big Bang Nucleosynthesis (BBN) requires a reheat temperature $T_{\rm rh}\gtrsim 5$~MeV \cite{Hannestad:2004px}.
Hence, we assume that it suffices to show that the model yields consistent phenomenology for $T_{\rm rh}\sim \ord{10\rm~MeV}$, for definiteness and as a minimal proof of principle.  It is straightforward to extend our analysis to higher temperratures.  We take  $n_B/s\sim 10^{-10}$, where $n_B$ is the baryon number density and $s\sim g_* T^3$ is the entropy of the  Universe; $g_*\sim 10$ is the  number of relativistic degrees of freedom at temperatures of interest here.  Typical baryogenesis scenarios are  completed by $T\sim T_{\rm rh}$ adopted here and hence we may assume $n_B\sim 10^{-9} T^3$, however the $e^+e^-$ number density $n_e \sim T^3$, until after their annihilation at $T \ll$~MeV, below which $n_e\sim n_B$.

In order to evade severe constraints from thermalization of $\nu_s$ through neutrino oscillation, we will take the induced mass of $m_s \sim g_s \phi$ to be large compared to its terrestrial value $\sim 1$~eV, which in our framework also means a suppressed mixing angle $\theta_{i4}$.
Standard cosmology with  $N_{\rm eff}\approx3$ effective neutrinos after BBN consistent with the data is achieved for $m_s\sim$ keV and $\theta_{i4}\sim10^{-3}$ \cite{Hannestad:2015tea,Acero:2022wqg}.
Let us take $T_{\rm rh} \approx 10$~MeV as a concrete example. We need the right initial scalar value to obtain the above allowed sterile neutrino parameters at
BBN -- corresponding to $T_{\rm BBN}\sim$~MeV --  for $\phi_{\rm BBN} \sim  10^{16}$~eV.   Allowing for some redshift between $T_{\rm rh}$ and $T_{\rm BBN}$, we require $\phi_i >10^{16}$~eV.

As discussed earlier, the 1-loop induced quartic coupling in \eq{delta-lphi} could be naturally given by $\lambda_\phi \sim 10^{-53}$, for our reference choice $g_s = 2\times 10^{-13}$, which would imply the initial  dominance of the quartic potential over the mass term.  Note that the  mass term starts to dominate at $\phi_*\sim 10^{13}$~eV.
A scalar dominated by its quartic coupling redshifts like radiation \cite{Turner:1983he} and hence this corresponds to  $T_*\sim (\phi_*/\phi_{\rm BBN})T_{\rm BBN} \sim$~keV.
At this temperature, the energy density in the scalar field would be given by $m_\phi^2 \phi_*^2 \sim 10^{-3}$~eV$^4$, which is far too small to lead to the correct DM cosmic energy budget today.

The above ``1-loop" choice for $\lambda_\phi$ can be consistent with cosmological data, but would not explain DM.
 We, therefore, consider another regime of  parameters that allows us to identify $\phi$ as the dominant DM in the Universe.
As we will outline below, in this case, the  renormalized value of the quartic coupling is small compared to the 1-loop estimate;  $\lambda_\phi \ll 10^{-53}$.

\subsection{Ultralight Scalar Dark Matter}

Let us now consider what parameters can lead to $\phi$ as viable DM.  Arguments based on the Milky Way satellite population suggest that the behavior of $\phi$ should transition to  matter-like, dominated by its mass term,  by the time the Universe has cooled to $T\sim$~keV \cite{Das:2020nwc}.   Henceforth, we will assume this to  be the  transition temperature $T_{\rm tr}\approx$~keV.  After this point, the scalar energy density would be dominated by $m_\phi^2 \phi^2$ and would redshift like matter.  Thus, by $T_{\rm eq}\approx 1$~eV associated with standard matter-radiation equality era, the energy density in $\phi$ is given by $\sim (T_{\rm eq}/T_{\rm tr})^3\, m_\phi^2 \phi^2/2$.  Hence, at $T_{\rm tr}$, one  needs $\phi_{\rm tr}\sim 10^{19}$~eV.  Since prior to $T_{\rm tr}$ the radiation-like quartic term dominates by assumption, we end up with $\phi_i\sim 10^{23}$~eV at $T_{\rm rh} \sim 10$~MeV.

The preceding considerations  imply that
\begin{equation}
\lambda_\phi \approx \frac{12 \,m_\phi^2}{\phi_{\rm tr}^2}\sim 3\times 10^{-66}\,,\quad \text{($\phi$ viable DM)}
\label{DMcond}
\end{equation}
in order for $\phi$ to be viable as the dominant form of DM.
We will take the above value as a  reference for the rest of this work.  Note that for larger   $T_{\rm tr}$, one would need a smaller value of $\lambda_\phi$ than given in \eq{DMcond}\footnote{In either case considered above, the quartic dominates and hence the value of $\phi$ induced by the nucleon plasma is roughly given by $(g_n n_B/\lambda_\phi)^{1/3}\ll \phi_i$, which can, therefore, be neglected compared to the assumed initial value $\phi_i$.}.
One could also consider a slightly different scenario with $\lambda_\phi=0$ where $\phi$ is still the dark matter;  see the appendix.

\subsection{Absolute Mass Constraints}
The tightest constraint on the absolute neutrino mass scale comes from combining cosmological data \cite{Planck:2018vyg,DiValentino:2021hoh} and is $\sum_{i=1}^3m_i<0.09$ eV.
Notably, however, this constraint is dominated by data in the $10<z<100$ range \cite{Lorenz:2021alz} when the density was low\footnote{In addition the cosmic neutrino background in vacuum is mostly in a sterile state, but near the Earth would return to the typical state.} and thus $m_s\to0$.
In that limit we find that the mass states are mostly active and also $m_{1,2,3}\to0$.
If $\phi$ is not DM and its field value is nearly zero in empty space, we predict that $\sum_{i=1}^3m_i$ would be  vanishingly small away from astrophysical bodies.
In the ULDM case, the presence of DM will source a small value for $m_s$ and thus a small value for $m_{1,2,3}$; see the appendix for estimates.  We find that the empty space neutrino masses would be below the sensitivity of future surveys.  Generally, we predict typical values ($>0.06$ or $>0.1$ eV for normal or inverted ordering respectively) in the early universe.
Future surveys will have resolution at the $0.02$ eV level and will be able to robustly test this scenario \cite{Ferraro:2022cmj}.

\section{Atmospheric Neutrinos}
Atmospheric neutrinos at $E\sim1$ TeV are sensitive to sterile neutrino oscillations at $m_4\sim1$ eV through the Earth's core \cite{akhmedov1988neutrino,Krastev:1989ix,Chizhov:1998ug,Chizhov:1999az,Akhmedov:1999va,Petcov:2016iiu}.
IceCube has strong constraints on $\Delta m^2_{41}\sim0.1-1$~eV$^2$ and a weak hint for a sterile neutrino at $\sin^22\theta_{24}\sim0.1$ and $\Delta m^2_{41}\sim4.5$ eV$^2$ \cite{IceCube:2020phf}.

We solve the Klein-Gordon equation with both $m^2$ and $\lambda_\phi$ terms in the Earth using the PREM distribution of matter in the Earth \cite{Dziewonski:1981xy}.
We find that, for the parameters of interest, the sterile neutrino mixing angle and mass in the Earth's core agree with IceCube's best fit point, as shown in Fig.~\ref{fig:earth}.

We also point that at the surface of the Earth, various gallium experiments have found high significance $>5\sigma$ evidence for sterile neutrino mixing preferring $\Delta m^2_{41}=1.25$ eV$^2$ and $\sin^22\theta_{14}=0.34$ \cite{Barinov:2021asz,Giunti:2022xat} which is also shown in Fig.~\ref{fig:earth}.
Our model shows excellent agreement with each scenario.
We also point out that minor modifications of the parameters in table \ref{tab:parameters} can alternatively lead to a cosmologically safe explanation of the LSND \cite{LSND:2001aii} and MiniBooNE \cite{MiniBooNE:2020pnu} anomalies.
In particular, we have taken $m_\nu=0.03$ eV and $m_D=0.37$ eV, but $m_\nu$ could be larger up to the KATRIN limit $\sim1$ eV \cite{KATRIN:2019yun} and could potentially be detectable by KATRIN in the future.

\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Earth}
\caption{\textbf{Top}: The density of the Earth as a function of radius from the PREM \cite{Dziewonski:1981xy}.
\textbf{Middle}: The sterile neutrino mixing angle as a function of radius for the two models, see table \ref{tab:parameters}.
The best fit values from IceCube \cite{IceCube:2020phf} and gallium experiments \cite{Barinov:2021asz} are shown in red and purple respectively.
\textbf{Bottom}: The sterile neutrino mass as a function of radius. The brown and blue shaded areas identify the Earth's core and surface regions, respectively (the latter is expanded for visual reasons only).}
\label{fig:earth}
\end{figure}

There is flexibility in the scale of the mass and mixing angles due to the couplings $g_n$ and $g_s$, and $m_D$ affects the  mixing angle allowing one to adjust the magnitude of each mixing angle and mass.
In addition, $m_\phi$ affects the shape of $\phi$ through the Earth.
Nonetheless, there is not complete freedom to vary the behavior of the sterile neutrino in the core versus the surface, assuming that the contributions from $\lambda_\phi$ are small in the Earth as they are in both of our cases, and that the masses remain hierarchical.
In fact, we find that the ratio of $\phi$ in the Earth's core and at the surface is constrained to be within the range $[0.13,1]$ as $m_\phi$ is varied; see the appendix.
This leads to a one-dimensional allowed region for the ratio of $\Delta m^2_{41}$'s and $\theta_{i4}$'s in the surface and in the core as shown in Fig.~\ref{fig:sterile allowed}, up to the mild caveats mentioned above.
Note that the ratios implied by the IceCube and gallium data lie exactly on the curve when we take the mixing between $\nu_4$ and each flavor to be equal.

\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Sterile_Allowed}
\caption{The allowed ratios of the sterile parameters at the Earth's surface (superscript $s$) to the Earth's core (superscript $c$) using the PREM.
The location of the IceCube and gallium data are shown as an orange star assuming equal sterile mixing.}
\label{fig:sterile allowed}
\end{figure}

\section{Solar Neutrinos}
In the Sun -- particularly in its core -- the properties of $\nu_s$ would be significantly affected by the high densities.  Assuming a mean mass density of $\rho_c^\odot\sim 100$~g/cm$^3$ and a core radius of $R_c^\odot \sim 10 R_\oplus$, the number of nucleons in the core volume is estimated to be $N_c^\odot\sim 7 \times 10^{55}$.  Since $1/R_c^\odot \sim 3 \times 10^{-15}$~eV, the core size can be considered sufficiently large compared to the range of the scalar set by $m_\phi=5\e{-15}$~eV.  The core nucleon number density is then $n_c^\odot \sim 5 \times 10^{11}$~eV$^3$.
Assuming the dominance of the scalar mass term we would then expect $\phi_c^\odot \sim g_n n_c^\odot/ m_\phi^2 \sim 10^{16}$~eV.
For $\lambda_\phi\gtrsim 10^{-60}$, this field value implies that the quartic term would be larger than the mass term in the potential of $\phi$.
Hence, one could ignore the mass term and approximate the potential by the quartic term.  For a roughly constant core density, we then find
\begin{equation}
\phi_c^\odot \sim -\left(\frac{6 \, g_n \,n_c^\odot }{\lambda_\phi}\right)^{1/3} \sim  -6\times 10^{13}~\text{eV}\left(\frac{10^{-53}}{\lambda_\phi}\right)^{1/3}\,,
\label{phisun}
\end{equation}
which yields $m_s^\odot\sim 12$~eV and $\theta^\odot\sim  3\times 10^{-2}$ at the Solar core.
The small induced mixing angle allows our scenario to evade solar neutrino constraints quite easily which are independent of mass and disfavor mixing angles $>0.17$ \cite{Goldhagen:2021kxe}.

Using the results in Ref.~\cite{Denton:KG,S5}, we then performed a detailed numerical calculation by solving the Klein-Gordon equation with both $m^2\phi^2$ and $\lambda_\phi\phi^4$ terms and the Sun's density profile \cite{Bahcall:2004pz} using the initial conditions $\phi(\infty)=0$ and $d\phi/dr|_{r=0}=0$.
We then computed the average mixing angle and sterile mass over the production region of the four most relevant processes: $^8$B, $^7$Be, $pp$, and $hep$.
Our results are shown in table \ref{tab:solar} which agree well with our rough  estimates.

Here, we would like to provide an estimate for the value of the coupling  $\lambda_*$ which marks the onset of quartic interaction domination.  Assuming a large region with a  constant density of charges $n$ coupled to $\phi$ with strength $g$, we find
\begin{equation}
\lambda_* \sim \frac{m_\phi^6}
{g^2 n^2}\,.
\label{lambda*}
\end{equation}
If $\lambda\ll \lambda_*$, we expect the mass term to dominate.
\begin{table}[h]
\centering
\caption{Reference parameters of the two models considered.
The first is where $\phi$ is the DM and the second is where $\lambda$ is at the 1-loop induced value with no additional tuning.}
\begin{tabular}{|l|c|c|}
\hline
& ULDM & 1-loop induced $\lambda_\phi$ \\\hline
$\lambda_\phi$ & $3\e{-66}$ & $10^{-53}$\\
$g_n$ & $7\times 10^{-25}$ & $7\times 10^{-25}$\\
$g_s$ & $2\e{-13}$ & $2\e{-13}$ \\
$m_\phi$ (eV) & $5\e{-15}$ & $5\e{-15}$\\
$m_D$ (eV) & 0.37 & 0.37\\
$m_\nu$ (eV) & 0.03 & 0.03\\\hline
\end{tabular}
\label{tab:parameters}
\end{table}

\begin{table}
\centering
\caption{The effective mixing angle and sterile neutrino mass in the Sun for the four main production regions assuming the benchmark parameters in table \ref{tab:parameters}.
In this mass range the existing constraint is independent of mass and is at $\theta_{14}<0.17$ at $2\sigma$ \cite{Goldhagen:2021kxe}.}
\begin{tabular}{|c|c|c|c|c|}
\hline
& \multicolumn{2}{c|}{ULDM} & \multicolumn{2}{c|}{1-loop induced $\lambda_\phi$}\\\cline{2-5}
& $\theta_{14}$ & $m_4$ (eV) & $\theta_{14}$ & $m_4$ (eV) \\\hline
$^8$B & $1.6\e{-4}$ & $2.2\e3$ & $2.8\e{-2}$ & 13\\
$^7$Be & $1.8\e{-4}$ & $2.1\e3$ & $3.0\e{-2}$ & 12\\
pp & $2.5\e{-4}$ & $1.7\e3$ & $3.8\e{-2}$ & 9.7\\
hep & $3.0\e{-4}$ & $1.4\e3$ & $4.4\e{-2}$ & 8.4 \\\hline
\end{tabular}
\label{tab:solar}
\end{table}

One could also investigate the impact of sterile neutrinos on the day-night effect of solar neutrinos \cite{Dooling:1999sg,Palazzo:2011rj,Long:2013ota}, however, an early analysis of solar neutrino data indicated the role of the Earth on the night time neutrinos seems to apply to a different region of parameter space than is viable given KamLAND data \cite{Giunti:2000wt}.

\section{Conclusions}
In this letter,  we have shown that it is possible to have a sterile neutrino that acts as suggested by terrestrial experiments on the Earth's surface, but evades the strong bounds from cosmology.
The required input is extremely minimal: a coupling to an ultralight scalar boson that can be dark matter.
This scenario is  compatible with all known cosmological data.
Moreover, our  scenario causes sterile neutrinos to act differently depending on their environment.
For example, the sterile mixing angle is extremely suppressed in the Sun so solar neutrino constraints on sterile neutrinos are no longer competitive.
The model also predicts a relationship between the mass and mixing angle at the Earth's surface and that of the Earth's core that is quite  constrained.
If one sets the parameters to describe the gallium anomaly as measured on the Earth's surface, it agrees excellently with parameters IceCube might be measuring in the Earth's core.
This model also predicts that the cosmological measurement of $\sum_{i=1}^3m_i$ will be well below the normal ordering limit. For our reference parameters, assuming that the sterile mass scale $\sim$~eV is dominantly sourced by the Earth, we expect that long range nucleon interactions mediated by $\phi$ would be uncovered just below current limits.

As the sterile neutrino picture continues to clarify itself in coming years, we point out that this economical scenario predicts different, yet correlated, mixing parameters depending on the environment.
Thus, it can be definitively confirmed or refuted by a careful consideration of surface level and atmospheric experiments.

\begin{acknowledgments}
The authors acknowledge support by the United States Department of Energy under Grant Contract No.~DE-SC0012704.
\end{acknowledgments}