\label{1bdry C^s}
    Suppose $\Omega$ is a locally $C^{1,1}$ domain, $0<s<1$, $g\in L^\infty(\Omega\cap B_4)$ and $u$ is a nonnegative classical solution of
    \begin{equation}
        \begin{cases}
            \mbox{$(-\triangle)^s$} u = g,&\text{in } \Omega\cap B_4,\\
            u=0, &\text{in } B_4\backslash \Omega.\\
        \end{cases}
    \end{equation}
    Then $u\in C^{s}(\Omega\cap B_{1/2})$ and
    \[
    \lVertu\rVert_{C^s(\Omega\cap B_{1/2})}\le C(\lVertg\rVert _{L^\infty(\Omega\cap B_4)} + \lVertu\rVert _{L^\infty(\Omega\cap B_4)} ),
    \]
    where $C$ depends on $n,s$ and $C^{1,1}_{loc}$ norm of $\partial\Omega$, and $B_r$ denotes a ball centered at any fixed point on the boundary $\partial\Omega$ with radius $r$.

\begin{theorem}\label{1bdry C^s}
    Suppose $\Omega$ is a locally $C^{1,1}$ domain, $0<s<1$, $g\in L^\infty(\Omega\cap B_4)$ and $u$ is a nonnegative classical solution of
    \begin{equation}
        \begin{cases}
            \flap u = g,&\text{in } \Omega\cap B_4,\\
            u=0, &\text{in } B_4\backslash \Omega.\\
        \end{cases}
    \end{equation}
    Then $u\in C^{s}(\Omega\cap B_{1/2})$ and
    \[
    \norm{u}_{C^s(\Omega\cap B_{1/2})}\le C(\norm{g} _{L^\infty(\Omega\cap B_4)} + \norm{u} _{L^\infty(\Omega\cap B_4)} ),
    \]
    where $C$ depends on $n,s$ and $C^{1,1}_{loc}$ norm of $\partial\Omega$, and $B_r$ denotes a ball centered at any fixed point on the boundary $\partial\Omega$ with radius $r$.
\end{theorem}

\begin{theorem}\label{conv}
    Assume $n\ge 1$, $s\in (0,1)$, $\beta\in (0,1)$. $\Omega$ is an unbounded domain with boundary. Suppose nonnegative functions $\{u_i\}\subset \mathcal{L}_{2s}\cap C^{2s+\beta}_{loc}(\Omega)$ vanishes outside $\Omega$, and $u\in \mathcal{L}_{2s}$ vanishes outside $\Omega$. If $\{u_i\}$
    converges in $C^{2s+\beta}_{loc}(\Omega)$ to a function $u\in \mathcal{L}_{2s}$ , and $\{\flap u_i \}$ converges pointwisely in $\Omega$, then there exists a constant $b\ge 0 $ such that
    \begin{equation}
        \lim _{i\to\infty}\flap u_i(x) = \flap u(x) -b.
    \end{equation}
\end{theorem}

In this paper, we consider the following fractional problem in  unbounded domain $\Omega$
 \begin{equation}\label{main1}
        \begin{cases}
            \mbox{$(-\triangle)^s$} u(x) = g(x),&\text{in } \Omega,\\
            u(x)=0, &\text{in }  \Omega^c.\\
        \end{cases}
    \end{equation}
First, we derive a boundary regularity estimate for  solutions
in terms of only the local $L^{\infty}$ norm of $u$ and $g$.
Then as an important application, we obtain  a priori estimates for solutions to nonlinear equations on unbounded domains with  boundaries \begin{equation}\label{main2}
\begin{cases}
    \mbox{$(-\triangle)^s$} u (x) = f(x,u)  , & x\in \Omega,\\
    u (x)  = 0 , &x\in \Omega ^c.
\end{cases}
\end{equation}
Here, $(-\Delta)^s$  denotes the fractional Laplacian,  defined as
\begin{eqnarray}\label{eq1-1}
\begin{aligned}
(-\Delta)^s u(x)&=C_{n, s}P.V. \int_{\mathbb{R}^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy\\
&=C_{n, s}\lim_{\varepsilon \to 0}\int_{\mathbb{R}^n\backslash B_\varepsilon(0)}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy,
\end{aligned}
\end{eqnarray}
where $P.V.$
stands for the Cauchy principal value, and $C_{n, s}$ is a
dimensional constant.
In order that the integral on the right hand side of \eqref{eq1-1} is well defined in the classical sense, we require that $u\in C_{loc}^{1, 1}(\Omega)\cap \mathcal{L}_{2s}$, where
$$
\mathcal{L}_{2s}=\left\{u\in L_{loc}^1(\mathbb{R}^n) \,\Big| \int_{\mathbb{R}^n}\frac{|u(x)|}{1+|x|^{n+2s}}dx<\infty \right\}
$$
endowed with norm

\[\lVertu\rVert_{{\mathcal{L}}_{2s}}:=\int_{\mathbb{R}^n}\frac{|u(x)|}{1+|x|^{n+2s}}dx.\]

The fractional Laplacian is a nonlocal integro-differential operator that effectively models memory effects and long-range diffusion phenomena (Chaves \cite{Chave1998}; Metzler and Klafter \cite{MK2000}). Specific applications include random walks with memory, Lévy flights \cite{MS1984, SZK1993, N1986}, and kinetic theories of systems exhibiting chaotic \cite{SZ1997, Z2002} and pseudo-chaotic \cite{ZE2001} dynamics. Beyond these, the operator finds significant applications across diverse domains including probability theory and finance, image processing and computer vision, mathematical biology and ecology, as well as geophysics and climate modeling.

 Nonlocal operators, particularly the fractional Laplacian, have attracted considerable attention in the mathematical community owing to their broad applications. For foundational developments in this field, we refer to the seminal work of Caffarelli and Silvestre \cite{CS07}, along with subsequent contributions by  Stinga \cite{Stinga2019}, Dipierro et al. \cite{Dipierro2019}, and the comprehensive monograph by Bucur and Valdinoci \cite{BV2016}. Regarding qualitative properties of solutions, extensive results can be found in \cite{CDQ,CDG,CG,CGLO,CHL17,DQ, FL,FLS}, with specific advances in a priori estimates documented in \cite{CGL,Chen2016directblowup,ouyang2023}.

    The regularity properties of solutions to fractional order equations have also been extensively studied. The foundation was laid by Silvestre \cite{Silvestre2007Regularity}, who established Schauder and H\"older estimates for entire solutions. Then the interior type estimates are studied in \cite{Chen2010frac}.  Later, Li and Wu \cite{Li2021Pointwise} established interior pointwise regularity estimates for fractional equations, a result analogous in spirit to Caffarelli's seminal work \cite{Caffarelli1989Ann} on pointwise estimates for viscosity solutions of fully nonlinear elliptic equations.
    For boundary regularity, Ros-Oton and Serra \cite{Rosoton2014Dirichlet} proved optimal regularity for the Dirichlet problem in domains satisfying an exterior ball condition (with zero exterior data), while their follow-up work \cite{Rosoton2014extremal} treated general exterior data in $C^{1,1}$ domains.
    The regularity theory has been further developed from different perspectives: Servadei and Valdinoci \cite{Servadei2014weak} investigated both weak and viscosity solutions, while Caffarelli-Stinga \cite{CS2016} and Stinga-Torrea \cite{ST2010} provided additional important regularity criteria.
    For more results concerning the nonlocal elliptic equations, please see \cite{Fall2019regulariy,Fall2020reg,Dyda2020reg}, while for fractional parabolic equations, we refer the readers to \cite{CGL,Chen2010Heat,Bogdan2010heat,Fernandezreal2016boundary,Stinga2017reg,Kassmann2024parabolic}.

In particular, in \cite{chen2025refinedregularitynonlocalelliptic}, the authors establish interior H\"older and Schauder regularity estimates for fractional equation \eqref{main1}. This makes it possible to derive a priori estimate for nonnegative solutions of \eqref{main2} when $\Omega=\mathbb{R}^n$, an  unbounded domain {\em without boundary}.

Then for general unbounded domains  with  boundaries, can one derive a priori estimates for the solutions?

This requires boundary H\"older estimates in terms of only the local $L^{\infty}$ norm of the solutions.

However, in the existing results, such as the ones in \cite{BDGQ2018} and \cite{Barrios2019note}, in order to obtain the $C^s$ boundary H\"older continuity of the solutions for \eqref{main1} when $\Omega=\mathbb{R}^n_+$, the authors  require that solution $u$ is globally bounded. Their proof mainly based on the proposition 1.1 in \cite{Rosoton2014Dirichlet}, in which  an optimal boundary H\"older estimate in terms of $\lVertg\rVert_{L^\infty(\Omega)}$ is established for  a {\em bounded domian} $\Omega$ with zero exterior Dirichlet condition. It is well known that in this situation, one can construst a supersolution  to show that the global $L^{\infty}$ norm of $u$ is bounded by $\lVertg\rVert_{L^\infty(\Omega)}$. When $\Omega$ is an {\em unbounded domain}, it is evident   from their proof that  in order to obtain local boundary H\"older continuity, the solution $u$ is required to be globally bounded, as indicated by \cite{BDGQ2018} and \cite{Barrios2019note}. For general unbounded domains, Dipierro, Soave and Valdinoci in \cite{DSV17} obtain $C^{\alpha}$ boundary regularity for some $0<\alpha<s$ under the condition that the solution $u$ of \eqref{main1} be globally bounded.

With these existing boundary  regularity estimates, it is inadequate to carry out blow up and rescaling arguments on unbounded domains aimed at obtaining a priori estimates, because the rescaled functions may be globally unbounded.

One of our main result is to establish the local version of the boundary regularity for nonnegative solutions,  in which  we use {\em only the local $L^{\infty}$  norm of $u$ instead of the global ones} to control its local H\"older norm up to the boundary.

In this paper, we consider the following fractional problem in unbounded domain $\Omega$
 \begin{equation}\label{main1}
 \begin{cases}
 \mbox{$(-\triangle)^s$} u(x) = g(x),&\text{in } \Omega,\\
 u(x)=0, &\text{in } \Omega^c.\\
 \end{cases}
 \end{equation}
First, we derive a boundary regularity estimate for solutions
in terms of only the local $L^{\infty}$ norm of $u$ and $g$.
Then as an important application, we obtain a priori estimates for solutions to nonlinear equations on unbounded domains with boundaries \begin{equation}\label{main2}
\begin{cases}
 \mbox{$(-\triangle)^s$} u (x) = f(x,u) , & x\in \Omega,\\
 u (x) = 0 , &x\in \Omega ^c.
\end{cases}
\end{equation}
Here, $(-\Delta)^s$ denotes the fractional Laplacian, defined as
\begin{eqnarray}\label{eq1-1}
\begin{aligned}
(-\Delta)^s u(x)&=C_{n, s}P.V. \int_{\mathbb{R}^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy\\
&=C_{n, s}\lim_{\varepsilon \to 0}\int_{\mathbb{R}^n\backslash B_\varepsilon(0)}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy,
\end{aligned}
\end{eqnarray}
where $P.V.$
stands for the Cauchy principal value, and $C_{n, s}$ is a
dimensional constant.
In order that the integral on the right hand side of \eqref{eq1-1} is well defined in the classical sense, we require that $u\in C_{loc}^{1, 1}(\Omega)\cap \mathcal{L}_{2s}$, where
$$
\mathcal{L}_{2s}=\left\{u\in L_{loc}^1(\mathbb{R}^n) \,\Big| \int_{\mathbb{R}^n}\frac{|u(x)|}{1+|x|^{n+2s}}dx<\infty \right\}
$$
endowed with norm

The regularity properties of solutions to fractional order equations have also been extensively studied. The foundation was laid by Silvestre \cite{Silvestre2007Regularity}, who established Schauder and H\"older estimates for entire solutions. Then the interior type estimates are studied in \cite{Chen2010frac}. Later, Li and Wu \cite{Li2021Pointwise} established interior pointwise regularity estimates for fractional equations, a result analogous in spirit to Caffarelli's seminal work \cite{Caffarelli1989Ann} on pointwise estimates for viscosity solutions of fully nonlinear elliptic equations.
 For boundary regularity, Ros-Oton and Serra \cite{Rosoton2014Dirichlet} proved optimal regularity for the Dirichlet problem in domains satisfying an exterior ball condition (with zero exterior data), while their follow-up work \cite{Rosoton2014extremal} treated general exterior data in $C^{1,1}$ domains.
 The regularity theory has been further developed from different perspectives: Servadei and Valdinoci \cite{Servadei2014weak} investigated both weak and viscosity solutions, while Caffarelli-Stinga \cite{CS2016} and Stinga-Torrea \cite{ST2010} provided additional important regularity criteria.
 For more results concerning the nonlocal elliptic equations, please see \cite{Fall2019regulariy,Fall2020reg,Dyda2020reg}, while for fractional parabolic equations, we refer the readers to \cite{CGL,Chen2010Heat,Bogdan2010heat,Fernandezreal2016boundary,Stinga2017reg,Kassmann2024parabolic}.

In particular, in \cite{chen2025refinedregularitynonlocalelliptic}, the authors establish interior H\"older and Schauder regularity estimates for fractional equation \eqref{main1}. This makes it possible to derive a priori estimate for nonnegative solutions of \eqref{main2} when $\Omega=\mathbb{R}^n$, an unbounded domain {\em without boundary}.

However, in the existing results, such as the ones in \cite{BDGQ2018} and \cite{Barrios2019note}, in order to obtain the $C^s$ boundary H\"older continuity of the solutions for \eqref{main1} when $\Omega=\mathbb{R}^n_+$, the authors require that solution $u$ is globally bounded. Their proof mainly based on the proposition 1.1 in \cite{Rosoton2014Dirichlet}, in which an optimal boundary H\"older estimate in terms of $\lVertg\rVert_{L^\infty(\Omega)}$ is established for a {\em bounded domian} $\Omega$ with zero exterior Dirichlet condition. It is well known that in this situation, one can construst a supersolution to show that the global $L^{\infty}$ norm of $u$ is bounded by $\lVertg\rVert_{L^\infty(\Omega)}$. When $\Omega$ is an {\em unbounded domain}, it is evident from their proof that in order to obtain local boundary H\"older continuity, the solution $u$ is required to be globally bounded, as indicated by \cite{BDGQ2018} and \cite{Barrios2019note}. For general unbounded domains, Dipierro, Soave and Valdinoci in \cite{DSV17} obtain $C^{\alpha}$ boundary regularity for some $0<\alpha<s$ under the condition that the solution $u$ of \eqref{main1} be globally bounded.

With these existing boundary regularity estimates, it is inadequate to carry out blow up and rescaling arguments on unbounded domains aimed at obtaining a priori estimates, because the rescaled functions may be globally unbounded.

One of our main result is to establish the local version of the boundary regularity for nonnegative solutions, in which we use {\em only the local $L^{\infty}$ norm of $u$ instead of the global ones} to control its local H\"older norm up to the boundary.

\begin{equation}\label{main1}
        \begin{cases}
            \flap u(x) = g(x),&\text{in } \Omega,\\
            u(x)=0, &\text{in }  \Omega^c.\\
        \end{cases}
    \end{equation}

\begin{equation}\label{main2}
\begin{cases}
    \flap u (x) = f(x,u)  , & x\in \Omega,\\
    u (x)  = 0 , &x\in \Omega ^c.
\end{cases}
\end{equation}

As an important application, we establish a priori estimate for the solutions to \eqref{main2}. We assume that $\Omega \subset \R ^n$ is an unbounded domain with uniformly $C^{1,1}$ boundary, and $f$ satisfies the following condition:
\begin{itemize}
 \item $f(x,t):\Omega \times [0,\infty)\to\R$ is uniformly H\"older continuous with respect to $x$ and continuous with respect to $t$.
\end{itemize}
\begin{theorem}\label{1A1}
Assume $1<p<\frac{n+2s}{n-2s}$, $f$ satisfies
\begin{equation}\label{growth f}
 f(x,t)\le C_0(1+t^p)
\end{equation}
uniformly for all $x$ in $\Omega$, and
\begin{equation}\label{lim f}
 \lim _{t\to\infty}\frac{f(x,t)}{t^p} = K(x),
\end{equation}
 where $K(x)\in (0,\infty)$ is uniformly continuous and $K(\infty):=\lim _{|x|\to\infty }K(x) \in (0,\infty)$.
Then there exists a constant $C$, such that
\begin{equation}\label{bd1}
u(x)\le C,\quad\forall \, x\in \Omega
\end{equation}
holds for all nonnegative solutions $u$ of \eqref{main2}.
\end{theorem}

To begin with, let us recall a classical result in
\cite{Rosoton2014Dirichlet}, in which the authors establish an optimal boundary regularity for the fractional equation with zero exterior Dirichlet condition.
\begin{proposition}[\cite{Rosoton2014Dirichlet}]\label{otonserra}
 Suppose $\Omega$ is a bounded Lipschitz domain with exterior ball condition, $g\in L^\infty(\Omega)$, $s\in (0,1)$, $u$ is a classical solution of the following Dirichlet problem:
 \begin{equation}\label{eq1}
 \begin{cases}
 \flap u = g,&\text{in } \Omega,\\
 u=0, &\text{in } \Omega ^c.\\
 \end{cases}
 \end{equation}
 Then $u\in C^{s}(\R^n)$ and
\begin{equation}\label{hd}
 \norm{u}_{C^s(\R^n)}\le C\norm{g}_{L^\infty(\Omega)},
\end{equation}
where $C=C(n,s)$ is a positive constant.
\end{proposition}
Notice that here $\Omega$ is a {\em bounded domian}. In this situation, a supersolution and a subsolution can be constructed to show that the global $L^{\infty}$ norm of $u$ is bounded by $\norm{g}_{L^\infty(\Omega)}$ as the following. Let
\[\psi(x):=(1-|x|^2)_+^s.\]
It is well known that
\[\flap \psi (x)=a>0.\]
Choose $R$ sufficiently large, so that $\Omega\subset B_R(0).$ Denote
\[\bar{u}(x):=\norm{g}_{L^{\infty}(\Omega)}\frac{R^{2s}}{a}\psi_R(x), ~\mbox{with}~ \psi_R(x):={\psi(\frac{x}{R})}.\]
Then it can be easily verified that $\bar{u}$ and $-\bar{u}$ are super-solution and sub-solution of \eqref{eq1}, respectively. Therefore,
\begin{equation*}
 \norm{u}_{L^{\infty}(\Omega)}\le C\norm{g}_{L^\infty(\Omega)},
\end{equation*}
From the above construction, one can see that in order to obtain \eqref{hd}, it is necessary that $\Omega$ be bounded.
When $\Omega$ is an {\em unbounded domain}, it is evident that the solution $u$ is required to be globally bounded. This requirement can not be fulfilled in the process of employing the blow-up and rescaling argument to obtain a priori estimates for solutions to a corresponding family of nonlinear fractional equations on {\em unbounded domains with boundaries}.

This motivates us to establish a local version of the boundary regularity, in which, instesd of global one, only a local ${L^{\infty}}$ norm of the solution is involved.
\begin{theorem}\label{bdry C^s}
 Suppose $\Omega$ is a unbounded domain with locally $C^{1,1}$ boundary, $0<s<1$, $g\in L^\infty(\Omega\cap B_4)$ and $u$ is a nonnegative classical solution of
 \begin{equation}
 \begin{cases}
 \flap u = g,&\text{in } \Omega\cap B_4,\\
 u=0, &\text{in } B_4\backslash \Omega.\\
 \end{cases}
 \end{equation}
 Then $u\in C^{s}(\Omega\cap B_{1/2})$ and
 \[
 \norm{u}_{C^s(\Omega\cap B_{1/2})}\le C(\norm{g} _{L^\infty(\Omega\cap B_4)} + \norm{u} _{L^\infty(\Omega\cap B_4)} ),
 \]
 where $C$ depends on $n,s$ and $C^{1,1}_{loc}$ norm of $\partial\Omega$, and $B_r$ denotes a ball centered at any fixed point on the boundary $\partial\Omega$ with radius $r$.
\end{theorem}
The idea of the proof is that we divide the solution into two parts: the potential part and the harmonic part. The regularity for the potential part is obtained by Theorem \ref{otonserra}. For the harmonic part, we firstly derived an $\alpha$ power order decay near $\partial\Omega$, here $\alpha$ is a certain positive number which may be less than $s$. Combining the decay estimate and the interior regularity result (Theorem \ref{Holder thm}), we derive the $C^\alpha$ boundary regularity. Finally we refine the boundary regularity by an iteration progress.

To continue estimate $\norm{u}_{C^{\alpha_0}(\Omega\cap B_{1/2})}$, we need the interior H\"older estimate from \cite{chen2025refinedregularitynonlocalelliptic}.
\begin{proposition}[Interior H\"older regularity]\label{Holder thm}
 Assume $0<s<1$, and $u\in C^{1,1}_{loc}(\R^n)\cap \mathcal{L}_{2s}$ is a nonnegative solution of
 \begin{equation}
 \flap u(x) = g(x),\quad x\in B_1
 \end{equation}
 where $g\in L^\infty(B_1)$. Then for any $0<\varepsilon <2s$,
 \begin{equation}
 \norm {u}_{C^{[2s-\varepsilon],\{2s-\varepsilon\}}(B_{1/2})}\le C(\norm{g}_{L^\infty(B_1)}+\norm{u}_{L^\infty(B_1)}).
 \end{equation}
\end{proposition}
Based on this, we arrive at
\begin{lemma}\label{global holder lemma}
 The decay estimate \eqref{decay estimate} and Proposition \ref{Holder thm} implies that
 \begin{equation}
 \norm{u} _{C^{\alpha _0}(\Omega\cap B_{1/2})}\le C(\norm{u}_{L^\infty(\Omega\cap B_4)}+\norm{g}_{L^\infty(\Omega\cap B_4)})
 \end{equation}
 where $C$ depends on $s,n$, and $\norm{\partial \Omega}_{C^{1,1}_{loc}}$, $\alpha_0:=\min\{s,1-s\}\in (0,1/2)$.
\end{lemma}

\section{Appendix}
Here is a generalization of \cite[Theorem 1.1]{Du2023blowup}Du-Jin-Xiong-Yang's convergence result.
\begin{theorem}
 Assume $n\ge 1$, $s\in (0,1)$, $\beta\in (0,1)$. $\Omega$ is an unbounded domain with boundary. Suppose nonnegative functions $\{u_i\}\subset \mathcal{L}_{2s}\cap C^{2s+\beta}_{loc}(\Omega)$ vanishes outside $\Omega$, and $u\in \mathcal{L}_{2s}$ vanishes outside $\Omega$. If $\{u_i\}$
 converges to a function $u\in \mathcal{L}_{2s}$ in $C^{2s+\beta}_{loc}(\Omega)$, and $\{\flap u_i \}$ converges pointwisely in $\Omega$, then there exists a constant $b\ge 0 $ such that
 \begin{equation}
 \lim _{i\to\infty}\flap u_i(x) = \flap u(x) -b.
 \end{equation}
\end{theorem}
\begin{proof}
 Fix any $x\in\Omega$, consider $R\gg|x|+1$.
 \begin{align*}
 \flap u(x)-\flap u_i(x) & = c_{n,s}\int _{B_R}\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy \\ &+c_{n,s}\int _{B_R^c}\frac{(u-u_i)(x)-u(y)}{|x-y|^{n+2s}}dy\\
 & + c_{n,s}\int _{B_R^c}\frac{u_i(y)}{|x-y|^{n+2s}}dy\\
 & := A_i(x,R) + E_i(x,R) + F_i(x,R).
 \end{align*}
 For $E_i$, since $u\in \mathcal{L}_{2s}$,
 \begin{equation}
 \lim _{R\to\infty}\lim _{i\to\infty}E_i(x,R) = -c_{n,s}\lim _{R\to\infty}\int _{B_R^c}\frac{u(y)}{|x-y|^{n+2s}}dy =0.
 \end{equation}
 The consideration of $A_i$ is a little bit different with that in \cite{Du2023blowup}. In our cases, $B_R$ is not contained in $\Omega$ for sufficiently large $R$. Fix $\varepsilon >0$ such that $B_\varepsilon(x)\subset \Omega$. Then
 \begin{align*}
 A_i(x,R) & = c_{n,s}\int _{B_R\backslash B_\varepsilon(x)}\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy \\
 &+c_{n,s}\int _{B_\varepsilon(x)}\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy
 \end{align*}
 By dominated convergence theorem, the first term converges to 0 as $i\to\infty$. For the second term,
 \[
 \lim _{i\to\infty} c_{n,s}\int _{B_\varepsilon(x)}\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy\le c_{n,s}\lim _{i\to\infty}[u-u_i]_{C^{2s+\beta}(B_\varepsilon(x))}\varepsilon ^\beta = 0.
 \]
 Hence $\lim _{i\to\infty}A_i(x,R)=0$. Therefore
 \begin{equation}
 \lim _{R\to\infty}\lim _{i\to\infty}A_i(x,R)=0.
 \end{equation}
 Then the same argument in \cite{Du2023blowup} implies $\lim _{i\to\infty}F_i(x,R)$ exists and
 \[
 \lim _{R\to\infty}\lim _{i\to\infty}F_i(x,R) =\lim _{R\to\infty}\lim _{i\to\infty}F_i(0,R) = :b\ge 0.
 \]
\end{proof}

The idea of the proof is that we divide a given solution into two parts: the potential part and the harmonic part. The regularity for the potential part is obtained by Proposition \ref{otonserra}. For the harmonic part $h$, we rewrite it in terms of the Poisson representation formula in balls. Using this explicit expression, we  first carry out a detailed analysis to   derive an $\alpha$ power order decay near $\partial\Omega$,
\begin{equation*}
        |h(x)|\le C\{\lVertu\rVert_{L^\infty(\Omega \cap B_4)}+\lVertu\rVert_{\cL_{2s}}\}\operatorname{dist}(x,\partial \Omega)^{\alpha },
    \end{equation*}
    where $\alpha = \min\{s,1-s\}$. Combining this decay estimate and the interior regularity result (Theorem \ref{Holder thm}), we derive the $C^\alpha$ boundary regularity. Then by an iteration process, we increase the power $\alpha$ successively until it reaches the desired power $s$.

As an important application, we establish a priori estimate for the solutions  to \eqref{main2}.  We assume that $\Omega \subset \mathbb{R} ^n$ is an unbounded domain with uniformly $C^{1,1}$ boundary, and $f$ satisfies the following condition:
\begin{itemize}
    \item $f(x,t):\Omega \times [0,\infty)\to\mathbb{R}$ is uniformly H\"older continuous with respect to $x$ and continuous with respect to $t$.
\end{itemize}
\begin{theorem}\label{1A1}
Assume $1<p<\frac{n+2s}{n-2s}$, $f$ satisfies
\begin{equation}\label{growth f}
    f(x,t)\le C_0(1+t^p)
\end{equation}
uniformly for all $x$ in $\Omega$, and
\begin{equation}\label{lim f}
    \lim _{t\to\infty}\frac{f(x,t)}{t^p} = K(x),
\end{equation}
    where $K(x)\in (0,\infty)$ is uniformly continuous and $K(\infty):=\lim _{|x|\to\infty }K(x) \in (0,\infty)$.
Then there exists a constant $C$, such that
\begin{equation}\label{bd1}
u(x)\le C,\quad\forall \, x\in \Omega
\end{equation}
holds for all nonnegative solutions $u$ of \eqref{main2}.
\end{theorem}

\begin{remark}
    The assumptions on $f$ are essential for the a priori estimate. When $f=0$ and $\Omega = B_1$, Abatangelo, Jarohs and Salda\~na \cite{Abatangelo2018Green} constructed nontrivial nonnegative solutions to \eqref{main2} that are unbounded, see also \cite{Li2023unique}.
\end{remark}

During the process of blow-up and rescaling, it can be shown that a sequence of the rescaled functions (still denoted by $u_i$) converges to a limiting function $u$ in the sense of $C^{2s+\beta}_{loc}$. In the case of a local operator, say $-\Delta$, we already have
\[   \lim _{i\to\infty}(-\Delta) u_i(x) = (-\Delta) u(x).\]
However, this is not the case for nonlocal operators, as shown in \cite[Theorem 1.1]{Du2023blowup} by Du-Jin-Xiong-Yang, in which they work on the whole space $\mathbb{R}^n$. In order to establish a priori estimate on general  unbounded domains with boundaries, we generalize  their convergence result as follows.
\begin{theorem}\label{conv}
    Assume $n\ge 1$, $s\in (0,1)$, $\beta\in (0,1)$. $\Omega$ is an unbounded domain with boundary. Suppose nonnegative functions $\{u_i\}\subset \mathcal{L}_{2s}\cap C^{2s+\beta}_{loc}(\Omega)$ vanishes outside $\Omega$, and $u\in \mathcal{L}_{2s}$ vanishes outside $\Omega$. If $\{u_i\}$
    converges in $C^{2s+\beta}_{loc}(\Omega)$ to a function $u\in \mathcal{L}_{2s}$ , and $\{\mbox{$(-\triangle)^s$} u_i \}$ converges pointwisely in $\Omega$, then there exists a constant $b\ge 0 $ such that
    \begin{equation}
        \lim _{i\to\infty}\mbox{$(-\triangle)^s$} u_i(x) = \mbox{$(-\triangle)^s$} u(x) -b.
    \end{equation}
\end{theorem}

 In Section 2, we establish local boundary H\"older regularity  for \eqref{main1} and prove Theorem \ref{1bdry C^s}. In Section 3, we derive  a priori estimates for \eqref{main2} and validate Theorem \ref{1A1}. Finally, the proof of Theorem \ref{conv} is included in the Appendix.

To prove Theorem~\ref{1bdry C^s}, the proof “divide[s] a given solution into two parts: the potential part and the harmonic part.” The “regularity for the potential part is obtained by Proposition \ref{otonserra}.” For the harmonic part $h$, one “rewrite[s] it in terms of the Poisson representation formula in balls,” and from this explicit expression one derives an $\alpha$-power decay near $\partial\Omega$:
\[
|h(x)|\le C\{\lVertu\rVert_{L^\infty(\Omega \cap B_4)}+\lVertu\rVert_{\cL_{2s}}\}\,\operatorname{dist}(x,\partial \Omega)^{\alpha},
\]
with $\alpha=\min\{s,1-s\}$. “Combining this decay estimate and the interior regularity result (Theorem \ref{Holder thm}),” one gets “the $C^\alpha$ boundary regularity.” Then “by an iteration process,” one “increase[s] the power $\alpha$ successively until it reaches the desired power $s$,” yielding the stated $C^s$ boundary regularity and estimate.

To prove the main theorem, the proof “divide[s] a given solution into two parts: the potential part and the harmonic part.” The “regularity for the potential part is obtained by
\begin{proposition}[\cite{Rosoton2014Dirichlet}]\label{otonserra}
    Suppose $\Omega$ is a bounded Lipschitz domain with exterior ball condition, $g\in L^\infty(\Omega)$, $s\in (0,1)$, $u$ is a classical solution of the following Dirichlet problem:
    \begin{equation}\label{eq1}
        \begin{cases}
            \flap u = g,&\text{in } \Omega,\\
            u=0, &\text{in } \Omega ^c.\\
        \end{cases}
    \end{equation}
    Then $u\in C^{s}(\R^n)$ and
\begin{equation}\label{hd}
    \norm{u}_{C^s(\R^n)}\le C\norm{g}_{L^\infty(\Omega)},
\end{equation}
where $C=C(n,s)$ is a positive constant.
\end{proposition}
For the harmonic part $h$, one “rewrite[s] it in terms of the Poisson representation formula in balls,” and from this explicit expression one derives an $\alpha$-power decay near $\partial\Omega$:
\[
|h(x)|\le C\{\lVertu\rVert_{L^\infty(\Omega \cap B_4)}+\lVertu\rVert_{\cL_{2s}}\}\,\operatorname{dist}(x,\partial \Omega)^{\alpha},
\]
with $\alpha=\min\{s,1-s\}$. Combining this decay estimate and the interior regularity result,
\begin{proposition}[Interior H\"older regularity]\label{Holder thm}
    Assume $0<s<1$, and $u\in C^{1,1}_{loc}(\R^n)\cap \mathcal{L}_{2s}$ is a nonnegative solution of
    \begin{equation}
        \flap u(x) = g(x),\quad x\in B_1
    \end{equation}
    where $g\in L^\infty(B_1)$. Then for any $0<\varepsilon <2s$,
    \begin{equation}
        \norm {u}_{C^{[2s-\varepsilon],\{2s-\varepsilon\}}(B_{1/2})}\le C(\norm{g}_{L^\infty(B_1)}+\norm{u}_{L^\infty(B_1)}).
    \end{equation}
\end{proposition}
one gets “the $C^\alpha$ boundary regularity.” Then “by an iteration process,” one “increase[s] the power $\alpha$ successively until it reaches the desired power $s$,” yielding the stated $C^s$ boundary regularity and estimate.

Question:
Let \(B_r\) denote the ball of radius \(r\) centered at a fixed point of \(\partial\Omega\), and let the fractional Laplacian be defined by \(( -\Delta)^s u(x)=C_{n,s}\,\mathrm{P.V.}\!\int_{\mathbb R^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy\). Suppose \(\Omega\) is a locally \(C^{1,1}\) domain, \(0<s<1\), \(g\in L^\infty(\Omega\cap B_4)\), and \(u\) is a nonnegative classical solution of
\[
\begin{cases}
(-\Delta)^s u=g,&\text{in }\Omega\cap B_4,\\
u=0,&\text{in }B_4\setminus\Omega.
\end{cases}
\]
Which of the following conclusions about \(u\) on \(\Omega\cap B_{1/2}\) is valid under these hypotheses?

The solution satisfies \(u\in C^s(\Omega\cap B_{1/2})\) and
\[
\|u\|_{C^s(\Omega\cap B_{1/2})}\le C\Big(\|g\|_{L^\infty(\Omega\cap B_4)}+\|u\|_{L^\infty(\Omega\cap B_4)}\Big),
\]
where \(C\) depends only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\).

Choices:
B.
The solution satisfies \(u\in C^s(\Omega\cap B_{1/2})\) and
\[
\|u\|_{C^s(\Omega\cap B_{1/2})}\le C\Big(\|g\|_{L^\infty(\Omega\cap B_4)}+\|u\|_{\mathcal L_{2s}}\Big),
\]
where \(C\) depends only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\).
C.
The solution satisfies \(u\in C^{\alpha_0}(\Omega\cap B_{1/2})\), where \(\alpha_0=\min\{s,1-s\}\), and
\[
\|u\|_{C^{\alpha_0}(\Omega\cap B_{1/2})}\le C\Big(\|g\|_{L^\infty(\Omega\cap B_4)}+\|u\|_{L^\infty(\Omega\cap B_4)}\Big),
\]
where \(C\) depends only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\).
D.
The solution satisfies \(u\in C^s(\Omega\cap B_{1/2})\) and
\[
\|u\|_{C^s(\Omega\cap B_{1/2})}\le C\|g\|_{L^\infty(\Omega\cap B_4)},
\]
where \(C\) depends only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\).
E.
The solution satisfies \(u\in C^{2s}(\Omega\cap B_{1/2})\) and
\[
\|u\|_{C^{2s}(\Omega\cap B_{1/2})}\le C\Big(\|g\|_{L^\infty(\Omega\cap B_4)}+\|u\|_{L^\infty(\Omega\cap B_4)}\Big),
\]
where \(C\) depends only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\).

replacement of local \(L^\infty\) control by tail norm \(\mathcal L_{2s}\) in the final estimate

dropped the iteration step upgrading boundary exponent from \(\alpha_0=\min\{s,1-s\}\) to \(s\)

omission of the local \(L^\infty\) dependence on \(u\) needed for the harmonic part

upgrading boundary regularity from \(C^s\) to \(C^{2s}\) by misreading interior estimates as boundary estimates


\begin{theorem}  \label{thm: factor hazard}
Let the hazard function, $h_0:(0,\infty)\to(0,\infty)$, admit the factorization:
\[
h_0(x)=C\,x^{p-1}\,g(x),\qquad C>0,\; p\ge 0,
\]
where $g:(0,\infty)\to(0,\infty)$ is non-decreasing. For $\tau>0$ define:
\[
r(x,\tau)\;=\;\tau\,\frac{h_0(\tau x)}{h_0(x)}.
\]
Then, for every $x>0$,
\[
0\le \tau\le 1
\quad\Longleftrightarrow\quad
0\le r(x,\tau)\le 1.
\]
\end{theorem}

\begin{theorem} \label{thm: ifr hazard}
Let $h_0:(0,\infty)\to(0,\infty)$ be a non-decreasing (i.e., increasing failure rate) hazard function. Define:
\[
r(x,\tau) = \tau\,\frac{h_0(\tau x)}{h_0(x)}, \qquad x>0.
\]
Then:
\[
0 \le r(x,\tau) \le 1 \quad \forall x>0
\quad \Longleftrightarrow \quad
0 \le \tau \le 1.
\]
\end{theorem}

\label{app: thms}

\begin{theorem} \label{thm: ifr hazard}
Let $h_0:(0,\infty)\to(0,\infty)$ be a non-decreasing (i.e., increasing failure rate) hazard function. Define:
\[
r(x,\tau) = \tau\,\frac{h_0

\begin{theorem} \label{thm: cov(x, y)}
Consider an accelerated conditional model of the form:
\[
P(X>x) = \bar{F}_0(x)
\quad \text{and} \quad
P(Y>y \mid X>x) = \bar{F}_1\bigl(\beta(x)y\bigr),
\]
where $\bar{F}_0$ and $\bar{F}_1$ are survival functions. Then:
\[
\beta'(x) \ge 0 \quad \Longrightarrow \quad {Cov}(X,Y) \le 0,
\qquad
\beta'(x) \le 0 \quad \Longrightarrow \quad {Cov}(X,Y) \ge 0.
\]
\end{theorem}

As per \citet{arnold2020bivariate}, the general accelerated failure conditionals model is of the form:
\begin{align}
 \bar{F}_X(x) = P(X>x) = \bar{F}_0(x),\quad x>0, \label{eq: general marginal}
\end{align}
for some survival function $\bar{F}_0(x) \in [0, 1] \ \forall x> 0$, and for each $x>0$:
\begin{equation}
 P(Y>y\mid X>x) = \bar{F}_1\left(\beta(x)y\right), \quad x, y >0,\label{eq: general cond}
\end{equation}
for some survival function $\bar{F}_1(\beta(x)y) \in [0, 1] \ \forall x, y >0$ and a suitable acceleration function $\beta(x)$. As per \citet{arnold2020bivariate}, in the analysis of dependent lifetimes of components in a system, it is more appropriate to consider the conditional density of $Y$ given that the first component, with lifetime $X$, remains operational at time $x$. Hence why, the conditioning event is taken to be $\{X > x\}$, rather than conditioning on the exact value $X = x$. The joint survival function is then:
\begin{align}
 P(X>x, Y>y) = \bar{F}_0(x) \bar{F}_1\left(\beta(x)y\right), \quad x, y>0. \label{eq: general joint}
\end{align}
Assuming differentiability and $X,Y$ are continuous, we obtain the marginal densities:
\[
f_X(x) = -\frac{d}{dx}P(X>x) = -\frac{d}{dx}\bar{F}_0(x) = f_0(x),
\]
and since $\lim_{x\to 0^+}\bar{F}_0(x)=1$, where $\beta(0):=\lim_{x\to 0^+}\beta(x)$:
\[
P(Y>y) = \lim_{x\to 0^+}P(Y>y,X>x) = \bar{F}_1\bigl(\beta(0)y\bigr).
\]
Now, the density of $Y$ is:
\[
f_Y(y) = -\frac{d}{dy}P(Y>y)
= -\frac{d}{dy}\bar{F}_1(\beta(0)y)
= \beta(0)\,f_1(\beta(0)y),
\]
where $f_1(t)=-\bar{F}_1'(t)$ denotes the density associated with $\bar{F}_1$. Hence the marginal distribution of $Y$ belongs to the same family as $\bar{F}_1$, but with its argument scaled by $\beta(0)$. For \eqref{eq: general joint} to be a valid survival function, it must have a non-negative
mixed partial derivative. Denoting $f_0$ and $f_1$ as the densities corresponding to the survival function $\bar{F}_0$ and $\bar{F}_1$, and differentiating, we obtain:
\begin{align}
 \frac{\partial}{\partial x} \frac{\partial}{\partial y} P(X>x, Y>y) &=
f_0(x) f_1(\beta(x) y)\beta(x)
- \bar{F}_0(x) f_1'(\beta(x) y) \beta(x) \beta'(x) y
- \bar{F}_0(x) f_1(\beta(x) y) \beta'(x) \nonumber \\
&= f_0(x) f_1(\beta(x) y)\beta(x)
- \beta'(x) \bar{F}_0(x) \left[ f_1(\beta(x) y) + f_1'(\beta(x) y) \beta(x) y\right]. \nonumber
\end{align}
Denoting $t = \beta(x)y$, we have:
\begin{align}
 \frac{\partial}{\partial x} \frac{\partial}{\partial y} P(X>x, Y>y)
&= f_0(x) f_1(t)\beta(x)
- \beta'(x) \bar{F}_0(x) \left[ f_1(t) + f_1'(t)t\right] \nonumber \\
& = f_0(x) f_1(t)\beta(x)
- \beta'(x) \bar{F}_0(x) \left[ t f_1(t) \right]' .\nonumber
\end{align}
Thus a necessary and sufficient condition for $ \frac{\partial}{\partial x} \frac{\partial}{\partial y} P(X>x, Y>y) \geq 0$ is:
\[
\beta'(x)\;\begin{cases}
\le\; \dfrac{f_0(x)}{\bar F_0(x)}\;\dfrac{f_1(t)}{[t f_1(t)]'}\;\beta(x), & \text{if } [t f_1(t)]'>0,\\[10pt]
\ge\; \dfrac{f_0(x)}{\bar F_0(x)}\;\dfrac{f_1(t)}{[t f_1(t)]'}\;\beta(x), & \text{if } [t f_1(t)]'<0, \\
\text{unconstrained}, & \text{if } [t f_1(t)]'=0.
\end{cases}
\]
Denoting the hazard function $h_0(x) = \frac{f_0(x)}{\bar{F}_0(x)}$ and $S(t) = \frac{f_1(t)}{[t f_1(t)]'}$, we may write the bounds as such:
\begin{align}
\sup_{t: [t f_1(t)]'<0} h_0(x) S(t)\beta(x)
\;\leq\; \beta'(x)
\;\leq\;
\inf_{t: [t f_1(t)]'>0} h_0(x) S(t)\beta(x). \label{eq: B'(x) bounds}
\end{align}
A special case of the upper bound for $B'(x)$ arises for distribution families satisfying $f_1'(t) \leq0$ - that is, for non-increasing densities. In this case, we note that $S(t) = \frac{f_1(t)}{\left[tf_1(t) \right]'} = \frac{f_1(t)}{f_1(t) + tf_1'(t) } \geq1$ for $t \geq 0$, hence $\beta'(x)
\;\leq\;
 h_0(x)\beta(x)$.\\

It should be noted that multiple functional forms for $\beta(x)$ are possible. The question of selecting a form of $\beta(x)$ that provides the best model fit has already been investigated by \citet{arnold2020bivariate}. Accordingly, the present study restricts attention to those choices of $\beta(x)$ that arise from the extreme case in which $\beta'(x)$ attains its theoretical upper bound, that is, given $\inf_{t: [t f_1(t)]'>0} S(t) = c^*$:
\begin{align}
 \beta'(x) &= c^*h_0(x) \beta(x), \nonumber \\
 \therefore \int \frac{1}{\beta(x)} d\beta(x) &= \int c^* h_0(x) dx, \nonumber\\
 \therefore \log\left( \beta(x) \right) &= -c^* \log\left( \bar{F}_0 (x)\right) + K. \nonumber \\
 \therefore \beta(x) &= \gamma \frac{1}{\left[\bar{F}_0(x)\right]^{c^*}}, \label{eq: beta(x) general}
 \end{align}
for $\gamma = e^K>0$. This suggests $\beta(x)$ is non-decreasing for $c^*>0$. Furthermore, $\beta(0): = \lim_{x\to 0^+} \beta(x) = \gamma$ since $\lim_{x\to 0^+}\bar{F}_0(x) = 1$. We also note that since $\bar{F}_0(x)$ approaches zero only as $x\rightarrow \infty$, the expression in \eqref{eq: beta(x) general} remains finite for all finite $x$.\\

The role of the acceleration function $\beta(x)$ in determining the sign of the correlation between $X$ and $Y$ is established in Theorem~\ref{thm: cov(x, y)}. In particular, since the present study considers only non-decreasing specifications of $\beta(x)$, the resulting framework is restricted to modelling data in which $X$ and $Y$ exhibit negative correlation.

\subsection{Moments}
Noting $X \sim Weibull(\alpha, \lambda)$ and $Y \sim Weibull(\gamma, \nu)$, we have:
\begin{align}
 E(X) &= \frac{1}{\alpha} \Gamma\left(1+ \frac{1}{\lambda}\right), \label{eq: Weibull ex} \\
 E(Y) &= \frac{1}{\gamma} \Gamma\left(1+ \frac{1}{\nu}\right), \label{eq: Weibull ey} \\
 {Var}(X) &= \frac{1}{\alpha^2}\left( \Gamma\left( 1+ \frac{2}{\lambda}\right) - \left(\Gamma\left(1+\frac{1}{\lambda}\right)\right)^2 \right),\label{eq: Weibull varx} \\
 {Var}(Y) &= \frac{1}{\gamma^2}\left( \Gamma\left( 1+ \frac{2}{\nu}\right) - \left(\Gamma\left(1+\frac{1}{\nu}\right)\right)^2 \right).\label{eq: Weibull vary}
\end{align}
Accordingly:
\begin{align}
\operatorname{Cov}(X, Y)
&= \int_0^\infty \int_0^\infty
 \big( P(Y>y , X> x) - P(X> x)P(Y> y) \big) \,dy\,dx \nonumber \\
&= \int_0^\infty \int_0^\infty
 \bar{F}_0(x) \left( \bar{F}_1\bigl(\beta(x)y\bigr) - \bar{F}_1\bigl(\beta(0)y\bigr) \right) \,dy\,dx \nonumber \\
&= \int_0^\infty e^{-(\alpha x)^\lambda}
 \left( \int_0^\infty \left( e^{-(\beta(x)y )^\nu}- e^{-(\beta(0)y )^\nu} \right) dy \right) dx \nonumber \\
&= \int_0^\infty e^{-(\alpha x)^\lambda}
 \left( \frac{\Gamma\!\left(1+\frac{1}{\nu}\right)}{\beta(x)} -\frac{\Gamma\!\left(1+\frac{1}{\nu}\right)}{\beta(0)} \right) dx \nonumber \\
&= \frac{1}{\gamma } \Gamma\!\left( 1+ \frac{1}{\nu}\right)
 \left[\int_0^\infty e^{-(\alpha x)^\lambda - \frac{1}{\nu}(\alpha \tau x)^\lambda} \, dx
 - \int_0^\infty e^{-(\alpha x)^\lambda} \, dx \right] \nonumber \\
&= \frac{1}{\alpha \gamma} \Gamma\!\left(1+ \frac{1}{\nu} \right)\Gamma\!\left(1+ \frac{1}{\lambda} \right)
 \left[ \left(\frac{\nu}{\nu + \tau^\lambda} \right)^{\frac{1}{\lambda}} - 1 \right].
\label{eq: Weibull cov}
\end{align}
Now, using \eqref{eq: Weibull varx}, \eqref{eq: Weibull vary} and \eqref{eq: Weibull cov}, the correlation function is:
\begin{align}
 \rho(X,Y)
 &= \frac{\operatorname{Cov}(X,Y)}{\sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}} \nonumber\\
 &= \frac{\Gamma\left(1+ \frac{1}{\nu} \right)\Gamma\left(1+ \frac{1}{\lambda} \right)\left[ \left(\frac{\nu}{\nu + \tau^\lambda} \right)^{\frac{1}{\lambda}} - 1 \right]}{\sqrt{\left( \Gamma\left( 1+ \frac{2}{\lambda}\right) - \left(\Gamma\left(1+\frac{1}{\lambda}\right)\right)^2 \right)\left( \Gamma\left( 1+ \frac{2}{\nu}\right) - \left(\Gamma\left(1+\frac{1}{\nu}\right)\right)^2 \right)}}.
 \label{eq: Weibull rho}
\end{align}
Clearly, \eqref{eq: Weibull rho} is strictly decreasing in $\tau \in [0, 1]$. Hence:
\begin{align}
 \rho_{\text{min}}(\lambda, \nu):=\frac{\Gamma\left(1+ \frac{1}{\nu} \right)\Gamma\left(1+ \frac{1}{\lambda} \right)\left[ \left(\frac{\nu}{\nu + 1} \right)^{\frac{1}{\lambda}} - 1 \right]}{\sqrt{\left( \Gamma\left( 1+ \frac{2}{\lambda}\right) - \left(\Gamma\left(1+\frac{1}{\lambda}\right)\right)^2 \right)\left( \Gamma\left( 1+ \frac{2}{\nu}\right) - \left(\Gamma\left(1+\frac{1}{\nu}\right)\right)^2 \right)}} \leq \rho(X, Y) \leq 0.
\end{align}
Now, $\inf_{\lambda, \nu > 0} \rho_{\text{min}}(\lambda, \nu) = \lim_{\nu \to \infty}\rho_{\text{min}}(\lambda = 1, \nu) = -\frac{\sqrt{6}}{\pi}$. Hence, this Weibull model with acceleration function $\beta(x) = \gamma e^{\frac{1}{\nu}(\alpha \tau x)^\lambda}$ will be able to accommodate correlations in the range $\rho(X,Y)\in\big[-\tfrac{\sqrt{6}}{\pi},\,0\big]$.

\appendix
\section*{Appendices}
\section{Joint Densities and Log-likelihoods} \label{app: joint lik}
\subsection{Exponential}
We obtain the joint density function by differentiating the joint survival function \eqref{eq: exp joint survival}:
\[
f_{X,Y}(x,y)
= \alpha \gamma \,\bigl(\gamma \tau y e^{\alpha \tau x} - \tau + 1 \bigr)
 e^{\!\left(\alpha x(\tau - 1) - \gamma y e^{\alpha \tau x}\right)},
 \qquad x,y>0.
\]
where $\alpha, \gamma > 0 $ and $\tau \in [0, 1]$. The log-likelihood is thus:
\begin{align}
\ell(\alpha,\gamma,\tau)
&= n \log \alpha + n \log \gamma
+ \sum_{i=1}^n \log\!\Big( \gamma \tau y_i e^{\alpha \tau x_i} - \tau + 1 \Big) \nonumber \\
&\quad + \alpha (\tau - 1)\sum_{i=1}^n x_i
- \gamma \sum_{i=1}^n y_i e^{\alpha \tau x_i}. \nonumber
\end{align}
\subsection{Lomax}
We obtain the joint density function by differentiating the joint survival function \eqref{eq: Lomax joint survival}:
\begin{align*}
f_{X,Y}(x,y)
&= \nu \,(1+\alpha x)^{-3\lambda - 1}\,
 \bigl(1+\gamma y (1+\alpha \tau x)^{\lambda}\bigr)^{-3\nu - 4} \\[6pt]
&\quad \times \Biggl[
 \frac{\alpha \gamma^{2} \lambda \tau y (\nu+1)\,(1+\alpha x)^{2\lambda+1}(1+\alpha \tau x)^{2\lambda}
 \,\bigl(1+\gamma y (1+\alpha \tau x)^{\lambda}\bigr)^{2\nu+2}}
 {1+\alpha \tau x} \\[6pt]
&\qquad - \frac{\alpha \gamma \lambda \tau \,(1+\alpha x)^{2\lambda+1}(1+\alpha \tau x)^{\lambda}
 \,\bigl(1+\gamma y (1+\alpha \tau x)^{\lambda}\bigr)^{2\nu+3}}
 {1+\alpha \tau x} \\[6pt]
&\qquad + \alpha \gamma \lambda \,\bigl((1+\alpha x)^{2}(1+\alpha \tau x)\bigr)^{\lambda}
 \,\bigl(1+\gamma y (1+\alpha \tau x)^{\lambda}\bigr)^{2\nu+3}
 \Biggr], \quad x,y >0,
\end{align*}
where $\alpha, \lambda, \nu,\gamma > 0 $ and $\tau \in [0, 1]$. The log-likelihood is thus:
\begin{align}
\ell(\alpha,\lambda,\nu,\gamma,\tau)
&= n \log \nu
 - (3\lambda+1)\sum_{i=1}^n \log(1+\alpha x_i) \nonumber \\
&\quad - (3\nu+4)\sum_{i=1}^n \log\!\bigl(1+\gamma y_i (1+\alpha \tau x_i)^{\lambda}\bigr) \nonumber \\
&\quad + \sum_{i=1}^n
 \log\!\Biggl\{
 \frac{\alpha \gamma^{2} \lambda \tau y_i (\nu+1)\,(1+\alpha x_i)^{2\lambda+1}(1+\alpha \tau x_i)^{2\lambda}}
 {1+\alpha \tau x_i}
 \bigl(1+\gamma y_i (1+\alpha \tau x_i)^{\lambda}\bigr)^{2\nu+2} \nonumber \\
&\qquad\qquad
 - \frac{\alpha \gamma \lambda \tau (1+\alpha x_i)^{2\lambda+1}(1+\alpha \tau x_i)^{\lambda}}
 {1+\alpha \tau x_i}
 \bigl(1+\gamma y_i (1+\alpha \tau x_i)^{\lambda}\bigr)^{2\nu+3} \nonumber \\
&\qquad\qquad
 + \alpha \gamma \lambda \,\bigl((1+\alpha x_i)^{2}(1+\alpha \tau x_i)\bigr)^{\lambda}
 \bigl(1+\gamma y_i (1+\alpha \tau x_i)^{\lambda}\bigr)^{2\nu+3}
 \Biggr\}. \nonumber
\end{align}
\subsection{Weibull}
We obtain the joint density function by differentiating the joint survival function \eqref{eq: Weibull joint survival}:
\[
f_{X, Y}(x,y) = \alpha^{\lambda} \gamma^{\nu} \lambda \nu x^{\lambda - 1} y^{\nu - 1}
\left( \tau^{\lambda} (\gamma y)^{\nu} e^{(\alpha \tau x)^\lambda}
- \tau^{\lambda} + 1 \right)
e^{(\alpha x)^\lambda(\tau^\lambda - 1)
- (\gamma y)^{\nu} e^{(\alpha\tau x)^{\lambda}}}, \quad x, y>0,
\]
where $\alpha, \lambda, \nu,\gamma > 0 $ and $\tau \in [0, 1]$. The log-likelihood is thus:
\begin{align}
\ell(\alpha,\lambda,\nu,\gamma,\tau)
&= n \lambda \log \alpha + n \nu \log \gamma
 + n \log \lambda + n \log \nu \nonumber\\
&\quad + (\lambda - 1) \sum_{i=1}^n \log x_i
 + (\nu - 1) \sum_{i=1}^n \log y_i \nonumber \\
&\quad + \sum_{i=1}^n
 \log\!\Bigl(\tau^{\lambda} (\gamma y_i)^{\nu} e^{(\alpha \tau x_i)^\lambda}
 - \tau^{\lambda} + 1 \Bigr) \nonumber \\
&\quad + (\tau^{\lambda} - 1)\sum_{i=1}^n (\alpha x_i)^\lambda
 - \sum_{i=1}^n (\gamma y_i)^{\nu} e^{(\alpha \tau x_i)^\lambda}. \nonumber
\end{align}
\subsection{Log-logistic}
We obtain the joint density function by differentiating the joint survival function \eqref{eq: Log joint survival}:
\[
\begin{aligned}
f_{X,Y}(x,y)
&= \frac{\lambda \nu\,(\gamma y)^{\nu}}
{x\,y\,\bigl((\alpha x)^{\lambda}+1\bigr)^{2}
\Bigl( (\gamma y)^{\nu}\bigl((\alpha \tau x)^{\lambda}+1\bigr)+1 \Bigr)^{3}}
\\[4pt]
&\quad\times
\Biggl\{
(\alpha x)^{\lambda}\bigl((\alpha \tau x)^{\lambda}+1\bigr)
\Bigl( (\gamma y)^{\nu}\bigl((\alpha \tau x)^{\lambda}+1\bigr)+1 \Bigr)
\\
&\qquad\qquad
+ (\alpha \tau x)^{\lambda}\bigl((\alpha x)^{\lambda}+1\bigr)
\Bigl( (\gamma y)^{\nu}\bigl((\alpha \tau x)^{\lambda}+1\bigr)+1 \Bigr)
\\
&\qquad\qquad
- 2\,(\alpha \tau x)^{\lambda}\bigl((\alpha x)^{\lambda}+1\bigr)
\Biggr\},
\qquad x,y>0,
\end{aligned}
\]

\begin{proof}
Since $P(Y>y , X> x) = \bar{F}_0(x)\bar{F}_1\bigl(\beta(x)y\bigr)$ then $P(Y>y) = \lim_{x\to 0^+}P(Y>y , X> x) = \bar{F}_0(0)\bar{F}_1\bigl(\beta(0)y\bigr) = \bar{F}_1\bigl(\beta(0)y\bigr)$ since $\lim_{x\to 0^+}\bar{F}_0(x) = 1$ and $\beta(0): = \lim_{x\to 0^+} \beta(x)$. We note from Hoeffding's covariance identity:
\begin{align}
    Cov(X, Y) & = \int_0^\infty \int_0^\infty\left( P(Y>y , X> x) - P(X> x)P(Y> y)   \right) \ dx dy \nonumber \\
    & = \int_0^\infty \int_0^\infty  \bar{F}_0(x) \left( \bar{F}_1\bigl(\beta(x)y\bigr) - \bar{F}_1\bigl(\beta(0)y\bigr)  \right) \ dx dy.\nonumber
\end{align}
Now since $\bar{F}_1'(\cdot)\le 0$ (as $\bar{F}_1$ is a survival function), if $\beta'(x) \geq 0$ then $\bar{F}_1\bigl(\beta(x)y\bigr) \leq \bar{F}_1\bigl(\beta(0)y\bigr)$ and $Cov(X, Y) \leq 0$ for $x, y \geq 0$. Conversely, if $\beta'(x) \leq 0$ then $\bar{F}_1\bigl(\beta(x)y\bigr) \geq \bar{F}_1\bigl(\beta(0)y\bigr)$ and $Cov(X, Y) \geq 0$ for $x, y \geq 0$. Finally, if $\beta(x)$ is constant, say $\beta(x)\equiv c$, then $P(X>x,Y>y)=\bar F_0(x)\,\bar F_1(cy)=P(X>x)\,P(Y>y)$, so $X$ and $Y$ are independent and ${Cov}(X,Y)=0$.
\end{proof}
Furthermore, as done in \citet{arnold2020bivariate}, we introduce a dependence parameter $\tau$ which controls the strength of the dependence between $X$ and $Y$. Extending from \eqref{eq: beta(x) general}, we define the acceleration function as such:
\begin{align}
    \beta(x) &= \gamma \frac{1}{\left[\bar{F}_0(\tau x)\right]^{c^*}}. \label{eq: beta(x) general 2}
\end{align}
Given the bounds established in \eqref{eq: B'(x) bounds}, and upon selecting a non-decreasing specification for $\beta(x)$ as defined in \eqref{eq: beta(x) general 2}, and again assuming $\inf_{t: [t f_1(t)]'>0} S(t) = c^* > 0$, we obtain bounds for $\tau$ as such:
\begin{align}
    0 &\leq \beta'(x) \leq c^* h_0(x)\, \beta(x), \nonumber\\
    \therefore
    0 &\leq
    \frac{c^* \tau \gamma f_0(\tau x)}{\big[\bar{F}_0(\tau x)\big]^{c^* + 1}}
    \leq
    c^* \gamma \,
    \frac{f_0(x)}{\big[\bar{F}_0(x)\big]^{c^*}}
    \frac{1}{\big[\bar{F}_0(\tau x)\big]^{c^*}}, \nonumber\\
    \therefore
    0 &\leq \tau \leq \frac{h_0(x)}{h_0(\tau x)}. \nonumber\end{align}
We re-write this inequality as such:
\begin{align}
    0 \le r(x, \tau) \le 1,
    \label{eq: tau bounds}
\end{align}
where $r(x, \tau) = \tau \frac{h_0(\tau x)}{h_0(x)}$, where the endpoint $\tau=0$ is included by continuity, with $r(x,0):=\lim_{\tau\to 0^+}r(x,\tau)=0$. Theorem \ref{thm: ABC} and Theorems \ref{thm: ifr hazard} and \ref{thm: factor hazard} in Appendix \ref{app: thms}, serve as a basis for establishing bounds on the dependence parameter $\tau$ that satisfy \eqref{eq: tau bounds}.

\begin{theorem}
\label{thm: ABC}
Let $h_0:(0,\infty)\to(0,\infty)$ be a hazard function. For $x>0$ and $\tau>0$ define:
\[
r(x,\tau):=\tau\,\frac{h_0(\tau x)}{h_0(x)},\qquad
\phi(x):=x\,h_0(x).
\]
Let
\[
A:\ \ 0\le r(x,\tau)\le 1\ \ \text{for all }x>0,\qquad
B:\ \ \tau\in(0,1],\qquad
C:\ \ \phi \text{ is non-decreasing on }(0,\infty).
\]
Then:
\begin{enumerate}
\item[(1)] $(A \wedge B)\ \Rightarrow\ C$.
\item[(2)] $(C \wedge B)\ \Rightarrow\ A$.
\item[(3)] $(C \wedge A)\ \Rightarrow\ B$, \emph{unless} $\phi$ is identically constant on $(0,\infty)$, in which case $A$ holds for all $\tau>0$ and $B$ cannot be inferred.
\end{enumerate}

\end{theorem}

\begin{proof}
\textbf{(1) $(A \wedge B)\Rightarrow C$.}
Fix $0<y<x$ and set $\tau:=y/x\in(0,1]$ so that $B$ holds. By $A$,
\[
1 \ \ge\ r(x,\tau)=\tau\,\frac{h_0(\tau x)}{h_0(x)}=\frac{y\,h_0(y)}{x\,h_0(x)}.
\]
Hence $y\,h_0(y)\le x\,h_0(x)$ for all $0<y<x$, i.e.\ $\phi$ is non-decreasing. Thus $C$ holds.

\medskip
\textbf{(2) $(C \wedge B)\Rightarrow A$.}
Assume $C$ and $\tau\in(0,1]$. Then $\tau x\le x$ and, by the non-decreasing nature of $\phi$,
\[
\phi(\tau x)\le \phi(x)\ \Longleftrightarrow\ \tau x\,h_0(\tau x)\le x\,h_0(x)
\ \Longleftrightarrow\ r(x,\tau)\le 1.
\]
Since $h_0>0$, we also have $r(x,\tau)\ge 0$. Hence $A$ holds.

\medskip
\textbf{(3) $(C \wedge A)\Rightarrow B$, unless $\phi$ is constant.}
Assume $C$ and $A$ hold. Because $h_0(x)>0$, the lower bound $r(x,\tau)\ge 0$ implies $\tau \ge 0$. Now suppose, toward a contradiction, that $\tau>1$, then $\tau x > x$ and, by assuming $\phi$ is now strictly increasing:
\[
\phi(\tau x) > \phi(x)\ \Longleftrightarrow\ \tau x\,h_0(\tau x) > x\,h_0(x)
\ \Longleftrightarrow\ r(x,\tau) > 1.
\]
contradicting the upper bound. Hence $\tau \le 1$. If instead $\phi$ is identically constant,
say $\phi(x)\equiv K>0$, then $h_0(x)=K/x$ and
\[
r(x,\tau)=\tau\,\frac{K/(\tau x)}{K/x}=1 \quad (\forall x>0,\ \forall \tau>0),
\]
so $A$ holds for all $\tau>0$ and no restriction $B$ follows.
\end{proof}
\begin{remark} When convenient, extend $B$ to $\tau\in[0,1]$ by defining $r(x,0):=\lim_{\tau\to 0^+} r(x,\tau)=0$ (whenever the limit exists). All implications above then carry over with this convention.
\end{remark}
\begin{remark}
If $h_0$ is non-decreasing, then $\phi(x) =  x h_0(x)$ is non-decreasing (product of increasing, positive functions), so Theorem~\ref{thm: ABC} recovers Theorem \ref{thm: ifr hazard}.
If $h_0(x)=C\,x^{p-1}g(x)$ with $C>0$, $p\ge 0$, and $g(x)$ non-decreasing, then $\phi(x) = C x^pg(x)$ is non-decreasing, so Theorem \ref{thm: ABC} recovers Theorem \ref{thm: factor hazard}.
\end{remark}

The following sections present the range of models employed in this study. In each case, the survival function forms are specified according to a single particular rate-shape distributional family such that $X \sim Family (\alpha, \lambda)$ and $Y \sim Family (\beta(0), \nu)$ where $\alpha$ and $\beta(0)$ denote the rate parameters and $\lambda$ and $\nu$ denote the shape parameters (shape parameters are not applicable for exponential and half-Cauchy models, however). Closed-form expressions of the models' moments are derived, and additionally, theoretical correlation bounds are obtained (with the exception of the half-Cauchy and gamma models).

Using $P(Y>y,X>x)=\bar F_0(x)\,\bar F_1(\beta(x)y)$, compute the marginal tail of $Y$ by letting $x\to 0^+$:
\[
P(Y>y)=\lim_{x\to 0^+}P(Y>y,X>x)=\bar F_1(\beta(0)y),\qquad \beta(0):=\lim_{x\to 0^+}\beta(x),
\]
(since $\lim_{x\to 0^+}\bar F_0(x)=1$). Then apply Hoeffding's covariance identity:
\[
\operatorname{Cov}(X,Y)=\int_0^\infty\!\int_0^\infty \Big(P(Y>y,X>x)-P(X>x)P(Y>y)\Big)\,dx\,dy
=\int_0^\infty\!\int_0^\infty \bar F_0(x)\Big(\bar F_1(\beta(x)y)-\bar F_1(\beta(0)y)\Big)\,dx\,dy.
\]
Since $\bar F_1'(\cdot)\le 0$, if $\beta'(x)\ge 0$ then $\bar F_1(\beta(x)y)\le \bar F_1(\beta(0)y)$, making the integrand non-positive for $x,y\ge 0$, hence $\operatorname{Cov}(X,Y)\le 0$; conversely, if $\beta'(x)\le 0$ then $\bar F_1(\beta(x)y)\ge \bar F_1(\beta(0)y)$, so $\operatorname{Cov}(X,Y)\ge 0$. Finally, if $\beta(x)\equiv c$ is constant then $P(X>x,Y>y)=\bar F_0(x)\,\bar F_1(cy)=P(X>x)P(Y>y)$, so $X$ and $Y$ are independent and $\operatorname{Cov}(X,Y)=0$.

\label{thm: cov(x, y)}
Consider an accelerated conditional model of the form:
\[
P(X>x) = \bar{F}_0(x)
\quad \text{and} \quad
P(Y>y \mid X>x) = \bar{F}_1\bigl(\beta(x)y\bigr),
\]
where $\bar{F}_0$ and $\bar{F}_1$ are survival functions. To prove the main theorem, it suffices to show that
\[
\beta'(x) \ge 0 \quad \Longrightarrow \quad {Cov}(X,Y) \le 0,
\qquad
\beta'(x) \le 0 \quad \Longrightarrow \quad {Cov}(X,Y) \ge 0.
\]

Question:
Let \(X\) and \(Y\) be random variables in an accelerated conditional model with survival structure
\[
P(X>x)=\bar F_0(x), \qquad P(Y>y\mid X>x)=\bar F_1\bigl(\beta(x)y\bigr),
\]
where \(\bar F_0\) and \(\bar F_1\) are survival functions and \(\beta(x)\) is the acceleration function. Which statement about the covariance \(\operatorname{Cov}(X,Y)\) holds under these assumptions?

If \(\beta'(x)\ge 0\), then \(\operatorname{Cov}(X,Y)\le 0\); and if \(\beta'(x)\le 0\), then \(\operatorname{Cov}(X,Y)\ge 0\).

Choices:
B. If \(\beta'(x)\ge 0\), then \(\operatorname{Cov}(X,Y)\ge 0\); and if \(\beta'(x)\le 0\), then \(\operatorname{Cov}(X,Y)\le 0\).
C. If \(\beta(x)\equiv c\) is constant for some \(c>0\), then \(X\) and \(Y\) are independent, and hence \(\operatorname{Cov}(X,Y)=0\).
D. If \(\beta'(x)\ge 0\), then \(\operatorname{Cov}(X,Y)< 0\); and if \(\beta'(x)\le 0\), then \(\operatorname{Cov}(X,Y)> 0\).
E. If \(\beta'(x)\ge 0\) for all sufficiently large \(x\), then \(\operatorname{Cov}(X,Y)\le 0\); and if \(\beta'(x)\le 0\) for all sufficiently large \(x\), then \(\operatorname{Cov}(X,Y)\ge 0\).

monotonicity-sign transfer through decreasing survival function

dropped the full sign-of-\(\beta'\) conclusion and retained only the constant-\(\beta\) independence case

equality case \(\beta'(x)\equiv 0\) leading to independence and zero covariance

global monotonicity in \(x\) needed to control the integrand for all \(x,y\)


\begin{theorem}\label{thm:toric-deg}
Let $\pi\colon \mathcal{X}\rightarrow \mathbb{D}$ be a projective fibration.
Let $(\mathcal{X},\mathcal{X}_0+\mathcal{B})$ be a log Calabi--Yau pair over $\mathbb{D}$ for which $(\mathcal{X},\mathcal{X}_0)$ is plt.
If $(\mathcal{X}_t,\mathcal{B}_t)$ is a toric pair\footnote{Meaning that $\mathcal{X}_t$ is a projective toric variety and $\mathcal{B}_t$ is the reduced torus invariant boundary.} for $t \in \mathbb{D}^*$,
then $(\mathcal{X}_0,\mathcal{B}_0)$ is a finite quotient of a toric pair.
\end{theorem}

\begin{proposition}\label{prop:tor-deg}
Let $\mathcal{X}\rightarrow \mathbb{D}$ be a klt Fano degeneration of $\pp(1,1,n)$ with $n\geq 3$, then $\mathcal{X}_0$ is toric if and only if $\mathcal{X}_0\cong \pp(x^2,y^2,n)$ such that $x^2+y^2+n=(n+2)xy$.
\end{proposition}

\begin{definition}\label{def:pt}
{\em
A pair $(X,B)$ is said to be {\em purely terminal} if the following conditions are satisfied:
\begin{enumerate}
\item the pair $(X,B)$ is plt; and
\item for every prime component $S\subset \lfloor B\rfloor$ the pair $(S,B_S)$, obtained from adjunction of $(X,B)$ to $S$, is terminal.
\end{enumerate}
}
\end{definition}

\begin{corollary}\label{cor:markov-no-deg}
Let $(a,b,c)\in \zz_{\geq 2}^3$ be a Markov triple.
Then, the weighted projective plane $\pp(a^2,b^2,c^2)$ has no non-trivial $\qq$-Gorenstein klt degenerations.
\end{corollary}

\begin{example}\label{ex:P^2-toric-model-1}
{\em
We consider the pair
$(\pp^2,B)$ where $B$ is the sum of a quadric and a transversal line.
This example is of cluster type, as described by Gross, Hacking, and Keel~\cite{GHK15}.
The choice of a torus embedding $\mathbb{G}_m^2\hookrightarrow \pp^2\setminus B$ is equivalent to the choice of a {\em toric model}
$\phi \colon (T,B_T)\dashrightarrow (\pp^2,B)$.
More precisely, $\phi$ is a crepant birational map that only extracts log canonical places and $(T,B_T)$ is a toric pair. In this example, we demonstrate
that the choice of such a toric model, or equivalently the choice of an embedding $\mathbb{G}_m^2\hookrightarrow \pp^2\setminus B$ naturally induces a degeneration of $\pp^2$ to a toric pair.
We will consider a toric model
$(\mathbb{F}_2,B_T)$ of $(\pp^2,B)$
where $\mathbb{F}_2$ is the second Hirzebruch surface. This toric model induces an embedding $j_0\colon \mathbb{F}_2\setminus B_T \simeq \mathbb{G}_m^2 \hookrightarrow \pp^2\setminus B$.

We start with the trivial family
\begin{equation}\label{fam1}
(\mathcal{T},B_{\mathcal{T}})\rightarrow \mathbb{D},
\end{equation}
where all the fibers are isomorphic
to the second Hirzebruch surface plus its toric boundary.
In other words, we have
\[
(\mathcal{T},B_{\mathcal{T}}) \simeq
(\mathbb{F}_2,B_T)\times \mathbb{D}.
\]
Let $B_{\mathcal{T}}=S_{\mathcal{T}}+\Delta_{0,\mathcal{T}}+F_{0,\mathcal{T}}+F_{\infty,\mathcal{T}}$ where $S_{\mathcal{T}}\sim \Delta_0+2F\subset \mathbb{F}_2$ and the $f$'s are the fibers of the projection $\mathbb{F}_2\rightarrow \pp^1$. We blow-up the image of a section
$s(\mathbb{D})\subset \mathcal{T}$ whose image is contained in
$F_{0,\mathcal{T}}$ and $s(0)$
is the intersection of
$F_{0,\mathcal{T},0}$ and $\Delta_{0,\mathcal{T}}$.
This leads to a family
\begin{equation}\label{fam2}
(\mathcal{T}',B_{\mathcal{T}'})\rightarrow \mathbb{D}
\end{equation}
where the general fiber is a blow-up of $\mathbb{F}_2$ along a general point of a fiber of $\mathbb{F}_2\rightarrow \pp^1$
and the central fiber is a toric surface.
We may blow-down the strict transform of
$F_{0,\mathcal{T}}$ in $\mathcal{T}'$ to obtain a family
\begin{equation}\label{fam3}
(\mathcal{Y}^+,B_{\mathcal{Y}^+}) \rightarrow \mathbb{D}.
\end{equation}
The general fiber of this family is
isomorphic to $\mathbb{F}_1$ and
$\mathcal{B}_{Y^+,t}$ has three components;
a fiber, a section with negative self-intersection, and a section with positive self-intersection.
The central fiber of~\eqref{fam3} is $\mathbb{F}_3$ with its toric boundary.
By \cite[Corollary 3.23]{HTU17}, the family admits an anti-flip which leads to a degeneration
\begin{equation}\label{fam4}
(\mathcal{Y},B_{\mathcal{Y}})\rightarrow \mathbb{D}.
\end{equation}
The general fiber of~\eqref{fam4}
is isomorphic to the general fiber of~\eqref{fam3}.
The central fiber of~\eqref{fam4} is
isomorphic to the blow-up of
$\mathbb{P}(1,1,4)$ at a torus invariant point (see for example \cite[Section 7]{UZ24}).
The family $\mathcal{Y}\rightarrow \mathbb{D}$ admits a blow-down
$\mathcal{X}\rightarrow \mathbb{D}$.
We obtain a family of Calabi--Yau pairs
\begin{equation}\label{fam5}
(\mathcal{X},\mathcal{B})\rightarrow \mathbb{D}.
\end{equation}

Below, we give an explicit description, modulo change of coordinates, of the family~\eqref{fam5} and the blow-up
$\mathcal{Y}\rightarrow \mathcal{X}$.
The family is given by
\[
\mathcal{X}_t=\{ [x:y:z:w] \mid xy=z^2+tw = 0\} \subset \pp(1,1,1,2)
\]
for $t\in \mathbb{D}$.
It is straightforward to check that $\mathcal{X}_t\cong \pp^2$ for $t\neq 0$ and $\mathcal{X}_0\cong \pp(1,1,4)$.
Indeed, $\mathcal{X}_t$, for $t\neq 0$, is a smooth del Pezzo surface of Picard rank one. The family of curves $\mathcal{B}\to \mathbb{D}$, where
\begin{equation}
\mathcal{B}_t=\{[x:y:z:w] \mid xyw^2+tz^2w^2=0\} \subset \mathcal{X}_t \subset \pp(1,1,1,2).
\end{equation}
When $t=0$, $\mathcal{B}_0$ corresponds to the toric boundary of $\pp(1,1,4)$, while for $t\neq 0$ it follows that
\begin{equation*}
B_t=\{[x:y:z:w] \mid w^2((1+t)z^2+tw)=0\}
\end{equation*}
 which corresponds to the union of the line $L_t=\{w=0\}$ and the irreducible conic
\begin{equation*}
Q_t=\{[x:y:z:w]\mid (1+t)z^2+tw=0\}.
\end{equation*}
The intersection points of $L_t$ and $Q_t$ are $[1:0:0:0]$ and $[0:1:0:0]$ which specialize to the smooth toric invariant points of $\mathcal{X}_0$.
The family~\eqref{fam4} described above is obtained by blowing up the section $s:\mathbb{D}\to \mathcal{X}$ such that $s(t)=[1:0:0:0]$, we obtain a $\qq$-Gorenstein degeneration $\mathbb{F}_1\rightsquigarrow Bl_p\pp(1,1,4)$.

By construction, the family~\eqref{fam5} admits the embedding of a product
\[
\xymatrix{
\mathbb{D} \times \mathbb{G}_m^2 \ar@{^{(}->}[rr]\ar[rd]_-{\pi_1} & &  \mathcal{X}\setminus \mathcal{B} \ar[ld] \\
& \mathbb{D}.&
}
\]
The restriction of the previous embedding to the general fiber recovers $j_0$
while the restriction to the central fiber
is just the embedding of the open torus into the toric variety $\pp(1,1,4)$.
}
\end{example}

In this article, we study degenerations of varieties and pairs.
We focus on degenerations with central fiber having klt singularities.
These are known as {\em klt degenerations}.
The problem is particularly interesting when the general fiber $\mathcal{X}_t$ of the degeneration $\mathcal{X}\rightarrow \mathbb{D}$ is a Fano variety.
In this case, there are many possible degenerations $\mathcal{X}_0$
and these degenerations are interesting even from the combinatorial perspective.
For instance, it is known that the toric degenerations of $\mathbb{P}^2$ are given by
\begin{equation}\label{eq:markov-triple}
\mathbb{P}(a^2,b^2,c^2) \text{ with $a^2+b^2+c^2=3abc$.}
\end{equation}
These triples $(a,b,c)\in \zz_{\geq 1}^3$ are known as {\em Markov triples}.
The topic of degenerations of projective surfaces is classic in algebraic geometry.
In~\cite{HP05}, Hacking and Prokhorov proved that any degeneration of $\mathbb{P}^2$ is indeed a partial smoothing of a toric surface as in~\eqref{eq:markov-triple}.
These surfaces were studied by Manetti and are nowadays known as
{\em Manetti surfaces} (see~\cite{Man91}).
In~\cite{HP10}, Hacking and Prokhorov classify del Pezzo surfaces, with quotient singularities and Picard rank one, admitting $\mathbb{Q}$-Gorenstein smoothings.
In~\cite{Pro19}, Prokhorov studied log canonical degenerations with Picard rank one of del Pezzo surfaces in $\mathbb{Q}$-Gorenstein families.
In~\cite{UZ24}, Urzua and the second author explain how to relate
the degenerations as in~\eqref{eq:markov-triple} via birational transformations.
Given two degenerations $\mathcal{X} \rightarrow \mathbb{D}$
and $\mathcal{Y}\rightarrow \mathbb{D}$ of $\mathbb{P}^2$ into
$\mathcal{X}_0 \simeq \mathbb{P}(a,b,c)$
and $\mathcal{Y}_0\simeq \mathbb{P}(d,e,f)$,
the authors explain how to perform small birational modifications to go from
$\mathcal{X}\rightarrow \mathbb{D}$ to $\mathcal{Y}\rightarrow \mathbb{D}$
and how these small modifications reflect on the combinatorics of the Markov triples.
In higher dimensions, H\"oring and Peternell proved that a klt degeneration $X$ of $\mathbb{P}^n$ is indeed isomorphic to the $n$-dimensional projective space whenever $T_X$ is semistable (see~\cite{HP24}).
They also classify normal degenerations of $\mathbb{P}^3$ with canonical singularities.
In~\cite{HKW25}, Hausen, Kir\'aly, and Wrobel explicitly determine all log terminal,
rational, degenerations of $\mathbb{P}^2$ that admit non-trivial torus actions.
In~\cite{UZ25}, Urzua and the second author classified all Wahl singularities that appear
in the degenerations of del Pezzo surfaces of degree $d$,
extending the work of Manetti and Hacking-Prokhorov in degree $9$.
Recently in~\cite{Pen25}, Peng studied special $\mathbb{G}_m$-degenerations of del Pezzo surfaces $X$
induced by log canonical places of pairs $(X,C)$ where $C$ is a nodal curve.
Peng proved that the space of special valuations of $(X,C)$ is connected and admits a
partition, which is locally finite, and each interval corresponds to a different $\mathbb{G}_m$-degeneration of $X$.

Throughout this work, we focus on understanding the degenerations of cluster type
varieties and cluster type pairs.
A {\em cluster type pair} is a pair $(X,B)$ with mild singularities
for which $K_X+B\sim 0$ and there is an embedding in codimension one
$\mathbb{G}_m^n \dashrightarrow X\setminus B$ (see Definition~\ref{def:ct}).
A {\em cluster type variety} is a variety $X$ that admits a cluster type pair $(X,B)$ structure.
In this case, we call $B$ a cluster type boundary.
Cluster type varieties and pairs can be thought of as a generalization of toric varieties and pairs.
Indeed, any toric pair is a cluster type pair.
However, the realm of cluster type pairs is much broader.
A del Pezzo surface of degree $d\geq 2$ as well as a general del Pezzo
surface of degree $d=1$ is cluster type (see, e.g.,~\cite[Theorem 2.1 and Remark 2.2]{ALP23}).
Thus, all the examples discussed in the introduction are cluster type varieties.
Our first aim is to understand degenerations of cluster type pairs
for which the limit of the cluster type boundary still has reasonable singularities.
In the next subsection, we prove that there are not many such degenerations
for toric pairs, but quite a lot for cluster type pairs.

\subsection{Degenerations of cluster type pairs}

Our first theorem states that degenerations of toric pairs
are finite quotients of toric pairs. We impose some conditions
on the singularities of the central fiber.

\begin{definition}\label{def:ct}
{\em
Let $(X,B)$ be log Calabi--Yau pair of dimension $n$.
We say that $(X,B)$ is of {\em cluster type}
or that $(X,B)$ is a {\em cluster type pair} if there exists an embedding
in codimension one
\[
\mathbb{G}_m^n \dashrightarrow X\setminus B,
\]
i.e., there exists a closed subset of codimension at least two $Z\subset \mathbb{G}_m^n$
and an embedding $\mathbb{G}_m^n\setminus Z \hookrightarrow X\setminus B$.
}
\end{definition}

Our first theorem states that degenerations of toric pairs
are finite quotients of toric pairs. We impose some conditions
on the singularities of the central fiber.

In the previous theorem,
the boundary $\mathcal{B}_0$ on the central fiber $\mathcal{X}_0$
is defined via adjunction.
This means that $\mathcal{B}_0$ is defined via the formula.
\[
K_{\mathcal{X}_0}+\mathcal{B}_0 = (K_{\mathcal{X}}+\mathcal{B}+\mathcal{X}_0)|_{\mathcal{X}_0}.
\]
In Example~\ref{ex:non-toric-central},
we show that in general $(\mathcal{X}_0,\mathcal{B}_0)$ is not a toric pair. Indeed, the central fiber $\mathcal{X}_0$ does not need to be a rational variety under the assumptions of Theorem~\ref{thm:toric-deg}.
Our next theorem gives a similar statement for degenerations of cluster type pairs.
We need to impose stronger conditions on the singularities in this case.
In the proof of Theorem~\ref{thm:toric-deg}, we will see that $(\mathcal{X},\mathcal{B})\rightarrow \mathbb{D}$ is indeed a finite quotient of an isotrivial toric family. Thus, in the toric case, there are not many interesting degenerations for which the degeneration of the toric boundary still has lc singularities.
Now, we turn to discuss the case of cluster type pairs.

\begin{theorem}\label{thm:ct-deg}
Let $\pi\colon \mathcal{X}\rightarrow \mathbb{D}$ be a projective Fano fibration.
Let $(\mathcal{X},\mathcal{X}_0+\mathcal{B})$ be a log Calabi--Yau pair of index one over $\mathbb{D}$.
Assume that $(\mathcal{X},\mathcal{X}_0)$ is plt
and purely terminal on the complement of $\mathcal{B}$.
If $(\mathcal{X}_t,\mathcal{B}_t)$ is of cluster type for $t\in \mathbb{D}^*$, then
$(\mathcal{X}_0,\mathcal{B}_0)$ is a finite quotient of a cluster type pair.
\end{theorem}

\begin{proof}[Proof of Theorem~\ref{thm:toric-deg}]
Let $f\colon \mathcal{X'}\rightarrow \mathcal{X}$ be the finite cover given by Lemma~\ref{lem:finite-cover}.
By Lemma~\ref{lem:finite-cover}.(2), we have an induced
finite cover of log Calabi--Yau pairs $(\mathcal{X}',\mathcal{B}')\rightarrow (\mathcal{X},\mathcal{B})$.
Then, up to shrinking $\mathbb{D}$ near $\{0\}$ we have a commutative diagram
\[
\xymatrix{
(\mathcal{X},\mathcal{B})\ar[d]_-{\pi} & (\mathcal{X}',\mathcal{B}') \ar[l]_-{f}\ar[d]^-{\pi'} \\
\mathbb{D} & \mathbb{D}\ar[l]_-{f_\mathbb{D}}
}
\]
where $f_\mathbb{D}$ is simply given by $t\mapsto t^k$ for some suitable positive integer $k$.
The morphism $f'\colon \mathcal{X}'\rightarrow \mathbb{D}$ is a fibration by Lemma~\ref{lem:finite-cover}.(3).
By assumption $(\mathcal{X},\mathcal{X}_0)$ is plt so by Riemann-Hurwitz we conclude that $(\mathcal{X}',\mathcal{X}_0')$ is plt as well. In particular, the variety $\mathcal{X}_0'$ is irreducible.
By Lemma~\ref{lem:finite-cover}.(4), every component of $\mathcal{B}'_t$
is the restriction to $\mathcal{X}'_t$ of a component of $\mathcal{B}'$.
Note that $\pi$ and $\pi'$ have the same general log fibers; indeed $f$ is induced by a finite cover of $\mathbb{D}$ ramified over $\{0\}$.
Therefore, the general fiber $(\mathcal{X'},\mathcal{B'})$ is a projective toric variety of dimension $n$
and Picard rank $\rho$.
Shrinking $\mathbb{D}$ around $\{0\}$, we may assume that all the fibers over $\pi'$ are irreducible.
Therefore, we conclude that $\rho(\mathcal{X}'/\mathbb{D})\leq \rho$.
On the other hand, as every component of $\mathcal{B}'_t$ is the restriction to $\mathcal{X}'_t$ of a component of $\mathcal{B}$,
we conclude that $\mathcal{B}$ has at least $n+\rho$ components.
Thus, we can compute the relative complexity of
the log Calabi--Yau pair $(\mathcal{X}',\mathcal{B}'+\mathcal{X}_0')$ over $\{0\} \in \mathbb{D}$.
We obtain
\[
c_{\{0\}}(\mathcal{X}'/\mathbb{D},\mathcal{B}'+\mathcal{X}_0') =
\dim \mathcal{X}' + \rho(\mathcal{X}'/\mathbb{D}) - |\mathcal{B}'+\mathcal{X}_0'| \leq
n+1+\rho - (n+\rho+1).
\]
Therefore, by~\cite[Theorem 1]{MS21}, we conclude that
$(\mathcal{X}',\mathcal{B}'+\mathcal{X}'_0)$ is a formally toric morphism near $\{0\}$.
Thus, the pair $(\mathcal{X}'_0,\mathcal{B}'_0)$ obtained from adjunction
of $(\mathcal{X}',\mathcal{B}'+\mathcal{X}'_0)$ must be a projective
toric log Calabi--Yau pair.
Henceforth, we have a finite crepant morpshim
of log Calabi--Yau pairs
\[
f_0\colon (\mathcal{X}'_0,\mathcal{B}'_0)
\rightarrow (\mathcal{X}_0,\mathcal{B}).
\]
We conclude that the pair $(\mathcal{X}_0,\mathcal{B}_0)$
is a finite quotient of a toric pair.
\end{proof}

\begin{theorem}\label{thm:1-comp}
Let $\pi\colon \mathcal{X}\rightarrow \mathbb{D}$ be a Fano projective morphism where $\mathcal{X}_0$ is reduced in $\mathcal{X}$ and
$(\mathcal{X},\mathcal{X}_0)$ plt.
Assume that the following conditions hold for the general fiber $\mathcal{X}_t$ with $t\neq 0$:
\begin{enumerate}
\item $\mathcal{X}_t$ is a toric surface of Picard rank one;
\item ${\rm mld}(\mathcal{X}_t)<\frac{1}{6}$;
\item $\mathcal{X}_t$ does not have singularities in the baskets
$\mathcal{F}_1,\dots,\mathcal{F}_4,\mathcal{D}$.
\end{enumerate}
Then, the pair $(\mathcal{X},\mathcal{X}_0)$ admits a $1$-complement over $\mathbb{D}$.
\end{theorem}
\begin{proof}
We break the proof into five steps.\\

\begin{example}\label{ex:non-toric-central}
{\em
We give an example of a degeneration of a toric pair where the central fiber is not a toric variety.
Fix $n\geq 3$ and consider the variety $\mathcal{Y}:=(\pp^1)^n \times \mathbb{D}$
and let $\pi_2\colon \mathcal{Y}\rightarrow \mathbb{D}$ be the projection onto the second component.
Let $B_T$ be the torus-invariant boundary of $(\pp^1)^n$
and $\mathcal{B}_\mathcal{Y}:=B_T\times \mathbb{D}$.
Therefore, the pair $(\mathcal{Y},\mathbb{B}_{\mathcal{Y}}+\mathcal{Y}_0)$
is a log Calabi--Yau pair for which every fiber of $\pi_2$ is isomorphic to
$((\pp^1)^n,B_T)$.
Consider the group $\zz_2$ acting on $\mathcal{Y}$ via
\begin{align*}
\mu & \colon (\pp^1)^n\times \mathbb{D} \rightarrow
(\pp^1)^n \times \mathbb{D} \\
\mu & \cdot ([x_1:y_1],[x_2:y_2],\dots,[x_n:y_n],t) :=
([y_1:x_1],[y_2:x_2],\dots,[y_n:x_n],-t).
\end{align*}
The log Calabi--Yau pair $(\mathcal{Y},\mathcal{B}_{\mathcal{Y}}+\mathcal{Y}_0)$
is invariant under the $\zz_2$-action.
Let $(\mathcal{X},\mathcal{B}+\mathcal{X}_0)$ be the induced quotient, so
we obtain a commutative diagram:
\[
\xymatrix{
(\mathcal{X},\mathcal{B}+\mathcal{X}_0) \ar[d]^-{\pi} & (\mathcal{Y},\mathcal{B}_{\mathcal{Y}}+\mathcal{Y}_0) \ar[d]^-{\pi_2} \ar[l]^-{/\zz_2} \\
\mathbb{D} & \mathbb{D} \ar[l]^-{/\zz_2}
}
\]
The morphism $\pi$ is a degeneration of the toric log Calabi--Yau pair
$((\pp^1)^n,B_T)$ into $(\mathcal{X}_0,\mathcal{B}_0)=((\pp^1)^n,B_T)/\zz_2$.
We have that $\mathcal{D}(\mathcal{X}_0,\mathcal{B}_0)\simeq_{\rm PL} \mathbb{P}_\rr^{n-1}$.
Thus, the central fiber $(\mathcal{X}_0,\mathcal{B}_0)$ is not a toric pair.
}
\end{example}

In the previous theorem,
the boundary $\mathcal{B}_0$ on the central fiber $\mathcal{X}_0$
is defined via adjunction.
This means that $\mathcal{B}_0$ is defined via the formula.
\[
K_{\mathcal{X}_0}+\mathcal{B}_0 = (K_{\mathcal{X}}+\mathcal{B}+\mathcal{X}_0)|_{\mathcal{X}_0}.
\]
In Example~\ref{ex:non-toric-central},
we show that in general $(\mathcal{X}_0,\mathcal{B}_0)$ is not a toric pair. Indeed, the central fiber $\mathcal{X}_0$ does not need to be a rational variety under the assumptions of Theorem~\ref{thm:toric-deg}.
Our next theorem gives a similar statement for degenerations of cluster type pairs.
We need to impose stronger conditions on the singularities in this case.
In the proof of Theorem~\ref{thm:toric-deg}, we will see that $(\mathcal{X},\mathcal{B})\rightarrow \mathbb{D}$ is indeed a finite quotient of an isotrivial toric family. Thus, in the toric case, there are not many interesting degenerations for which the degeneration of the toric boundary still has lc singularities.
Now, we turn to discuss the case of cluster type pairs.

\begin{theorem}\label{thm:ct-deg}
Let $\pi\colon \mathcal{X}\rightarrow \mathbb{D}$ be a projective Fano fibration.
Let $(\mathcal{X},\mathcal{X}_0+\mathcal{B})$ be a log Calabi--Yau pair of index one over $\mathbb{D}$.
Assume that $(\mathcal{X},\mathcal{X}_0)$ is plt
and purely terminal on the complement of $\mathcal{B}$.
If $(\mathcal{X}_t,\mathcal{B}_t)$ is of cluster type for $t\in \mathbb{D}^*$, then
$(\mathcal{X}_0,\mathcal{B}_0)$ is a finite quotient of a cluster type pair.
\end{theorem}

We refer the reader to Definition~\ref{def:pt} for the concept of purely terminal pairs.
In contrast to the toric case, a cluster type pair may have several interesting degenerations for which the degeneration of the cluster type boundary still has lc singularities. In Example~\ref{ex:P^2-toric-model-1}, we show that some toric degenerations of $\mathbb{P}^2$ can be regarded as cluster type degenerations for different embeddings of algebraic tori
$\mathbb{G}_m^2 \hookrightarrow \mathbb{P}^2\setminus C$ where $C$ is a nodal cubic.

\subsection{Degenerations of singular surfaces}

Now, we restrict ourselves to the study of degenerations of singular toric surfaces.
One of our aims is to understand how the singularities affect the possible degenerations.
Our third theorem states that almost every weighted projective plane has no interesting degenerations.

\begin{theorem}\label{no-deg-almost}
For almost all well-formed triples $(a,b,c)\in \zz_{\geq 1}^3$
the weighted projective plane $\mathbb{P}(a,b,c)$ has no non-trivial
$\mathbb{Q}$-Gorenstein klt degenerations.
\label{weighteddeg}
\end{theorem}

In the previous theorem, when we write {\em almost all}, we mean that the statement holds up to a subset $S\subsetneq \zz_{\geq 1}^3$ that has density zero with the natural density endowed from $\mathbb{Z}^3$.
We say that a degeneration is {\em trivial} if the central fiber is isomorphic to the general fiber.
However, in the setting of Theorem~\ref{no-deg-almost}, we will prove something stronger; the family is a product near the origin of the disk.
In order to prove Theorem~\ref{weighteddeg}, we will prove Theorem~\ref{thm:1-comp}, which is a general statement about
the existence of complements for degenerations of singular toric surfaces.
This means that, under some mild conditions, we prove that given a degeneration
$\mathcal{X}\rightarrow \mathbb{D}$ of a singular toric surface,
there exists some boundary $\mathcal{B}\in |-K_{\mathcal{X}}|$ for which $(\mathcal{X}_t,\mathcal{B}_t)$ has log canonical singularities for $t$ near $\{0\}\in \mathbb{D}$.
In the setting of Theorem~\ref{weighteddeg}, in most cases, we can argue that $(\mathcal{X}_t,\mathcal{B}_t)$ is toric for every $t$ and so
the statement is similar to that of Theorem~\ref{thm:toric-deg}, which states that there are no interesting such toric degenerations.
Theorem~\ref{thm:1-comp} is rather technical and depends on some meticulous analysis of basket of singularities.
The idea of using the theory of complements to understand degenerations of del Pezzo surfaces goes back to Hacking and Prokhorov.
The situation becomes a bit more delicate when we allow the general fiber of the degeneration to have singularities.
We will argue that for a Markov triple $(a,b,c)\in \zz_{\geq 2}^3$ the triple
$(a^2,b^2,c^2)$ belongs to the complement of the subset $S\subsetneq \zz_{\geq 0}^3$ of density zero mentioned above.
Thus, we conclude the following corollary.

\begin{corollary}\label{cor:markov-no-deg}
Let $(a,b,c)\in \zz_{\geq 2}^3$ be a Markov triple.
Then, the weighted projective plane $\mathbb{P}(a^2,b^2,c^2)$ has no non-trivial $\mathbb{Q}$-Gorenstein klt degenerations.
\end{corollary}

We note that Corollary~\ref{cor:markov-no-deg} can also be concluded from the work of Hacking and Prokhorov~\cite{HP10}. Indeed, every iterated degeneration of $\mathbb{P}(a^2,b^2,c^2)$ is indeed a degeneration of $\mathbb{P}^2$.
In upcoming work~\cite{Zun25}, the second author will prove
some structural theorems about $\mathbb{Q}$-Gorenstein klt degenerations
of Hirzebruch surfaces.
This will aim to finish the classification of $\mathbb{Q}$-Gorenstein
klt degenerations of minimal smooth rational surfaces.
This motivates us to pay particular attention to the weighted projective plane
$\mathbb{P}(1,1,n)$.
In this direction, using the tools introduced above, we can give
a complete classification of klt degenrations of $\mathbb{P}(1,1,n)$ with $n\geq 3$.

\begin{theorem}\label{thm:1-1-n}
Let $\mathcal{X}\rightarrow \mathbb{D}$ be a klt Fano degeneration of $\mathbb{P}(1,1,n)$ with $n \geq 3$,
then  for $\mathcal{X}_0$ one of the following holds:
\begin{enumerate}
\item $\mathcal{X}_0$ is a weighted projective plane, or
\item $\mathcal{X}_0$ is a $\mathbb{G}_m$-surface which is not toric.
\end{enumerate}
Furthermore, in the second case $\mathcal{X}_0$ is a $\mathbb{Q}$-Gorenstein deformation of a weighted projective plane.
\end{theorem}

In Proposition~\ref{prop:tor-deg},
we give a explicit classification
of the weighted projective planes
which are $\mathbb{Q}$-Gorenstein klt degenerations of $\mathbb{P}(1,1,n)$ with $n\geq 3$.

The paper is organized as follows.
In Section~\ref{sec:prelims}, we write some preliminary results regarding cluster type pairs, theory of complements, dual complexes, T-singularities, and Wahl singularities.
In Section~\ref{sec:degen-cluster-type}, we prove Theorem~\ref{thm:toric-deg} and Theorem~\ref{thm:ct-deg} regarding degenerations of toric pairs as well as cluster type pairs.
In Section~\ref{sec:complements-degen-klt-surfaces}, we prove some general statements regarding the existence of complements for degenerations of singular toric surfaces of Picard rank one. In this section, we also prove Theorem~\ref{weighteddeg} regarding the degenerations of weighted projective planes $\mathbb{P}(a,b,c)$.
In Section~\ref{sec:degen-wps}, we classify the degenerations of weighted projective planes $\mathbb{P}(1,1,n)$ with $n\geq 3$.
Finally, in Section~\ref{sec:ex-and-quest}, we give some examples and propose some questions for further research.

\subsection*{Acknowledgements}

The authors would like to thank Audric Lebovitz, Tomoki Oda, Giancarlo Urzúa, and Jos\'e Ignacio Y\'a\~nez
for many discussions related to this article.

In the proof of the main theorem, we will see that $(\mathcal{X},\mathcal{B})\rightarrow \mathbb{D}$ is indeed a finite quotient of an isotrivial toric family. Thus, in the toric case, there are not many interesting degenerations for which the degeneration of the toric boundary still has lc singularities.,

Question: Let \(\pi\colon \mathcal{X}\to \mathbb{D}\) be a projective fibration, let \(\mathcal{X}_0=\pi^{-1}(0)\), and let \((\mathcal{X},\mathcal{X}_0+\mathcal{B})\) be a log Calabi--Yau pair over \(\mathbb{D}\) (so \(K_{\mathcal{X}}+\mathcal{X}_0+\mathcal{B}\) is trivial over \(\mathbb{D}\)). Assume that \((\mathcal{X},\mathcal{X}_0)\) is plt. For every \(t\in \mathbb{D}^*:=\mathbb{D}\setminus\{0\}\), suppose the fiber pair \((\mathcal{X}_t,\mathcal{B}_t)\) is a toric pair, meaning that \(\mathcal{X}_t\) is a projective toric variety and \(\mathcal{B}_t\) is its reduced torus-invariant boundary. Which conclusion about the central fiber pair \((\mathcal{X}_0,\mathcal{B}_0)\) holds?

The central fiber pair \((\mathcal{X}_0,\mathcal{B}_0)\) is a finite quotient of a toric pair; equivalently, there exists a toric pair \((Y,B_Y)\), with \(Y\) a projective toric variety and \(B_Y\) its reduced torus-invariant boundary, together with a finite morphism of pairs \((Y,B_Y)\to (\mathcal{X}_0,\mathcal{B}_0)\).

Choices:
B. The central fiber pair \((\mathcal{X}_0,\mathcal{B}_0)\) is itself a toric pair; equivalently, \(\mathcal{X}_0\) is a projective toric variety and \(\mathcal{B}_0\) is its reduced torus-invariant boundary.
C. The central fiber pair \((\mathcal{X}_0,\mathcal{B}_0)\) is birational to a toric pair; that is, there exists a projective toric pair \((Y,B_Y)\) and a birational map of pairs \((Y,B_Y) \dashrightarrow (\mathcal{X}_0,\mathcal{B}_0)\).
D. There exists a single projective toric pair \((Y,B_Y)\) such that, after possibly shrinking \(\mathbb{D}\), every fiber pair \((\mathcal{X}_t,\mathcal{B}_t)\) for \(t\in\mathbb{D}\) is isomorphic to \((Y,B_Y)\); in particular, the family is isotrivial and the central fiber is a toric pair isomorphic to the general fiber.
E. The central fiber pair \((\mathcal{X}_0,\mathcal{B}_0)\) admits a finite crepant morphism to a toric pair; equivalently, there exists a projective toric pair \((Y,B_Y)\) and a finite morphism of pairs \((\mathcal{X}_0,\mathcal{B}_0)\to (Y,B_Y)\).