Is there any hope to prove this conjecture (or a similar one)?

Conjecture Let $\Omega_k$ be a family of convex (smooth) domains, and let $u_k$ be convex Alexandrov solution of $$ \begin{cases} det(D^2u_k)=f_k&\mbox{ in }\Omega_k\\ u_k=0 &\mbox{ on }\partial\Omega_k \end{cases} $$ with $0<\lambda\leq f_k\leq\Lambda,$ $f_k\in C^{n,\beta}.$ Assume that $\Omega_k$ converges to some domain $\Omega$ in ``some appropriate distance'' (the Hausdorff distance?) and $f_k\chi_{\Omega_k}\to f$ in $C_{loc}^{n,\beta},$ $f\in C^{n,\beta}.$ Then, if $u$ denotes the unique Alexandrov solution of $$ \begin{cases} det(D^2u)=f&\mbox{ in }\Omega\\ u=0 &\mbox{ on }\partial\Omega \end{cases} $$ for any $\Omega'\subset\subset \Omega,$ we have that $u_k\to u$ in $C^{r,\beta}(\Omega')$ as $k\to\infty,$ where $r>n.$

In Second order stability for the Monge-Ampere equation and strong Sobolev convergence of optima transport maps it is proved that: if $f_k\chi_{\Omega_k}\to f$ in $L^1_{loc}(\Omega),$ then $\|u_k-u\|_{W^{2,1}(\Omega')}\to 0$ as $k\to\infty.$

Is there any hope to prove this conjecture (or a similar one)?

Conjecture Let $\Omega_k$ be a family of convex (smooth) domains, and let $u_k$ be convex Alexandrov solution of $$ \begin{cases} det(D^2u_k)=f_k&\mbox{ in }\Omega_k\\ u_k=0 &\mbox{ on }\partial\Omega_k \end{cases} $$ with $0<\lambda\leq f_k\leq\Lambda,$ $f_k\in C^{n,\beta}.$ Assume that $\Omega_k$ converges to some domain $\Omega$ in ``some appropriate distance'' (the Hausdorff distance?) and $f_k\chi_{\Omega_k}\to f$ in $C_{loc}^{n,\beta},$ $f\in C^{n,\beta}.$ Then, if $u$ denotes the unique Alexandrov solution of $$ \begin{cases} det(D^2u)=f&\mbox{ in }\Omega\\ u=0 &\mbox{ on }\partial\Omega \end{cases} $$ for any $\Omega'\subset\subset \Omega,$ we have that $u_k\to u$ in $C^{r,\beta}(\Omega')$ as $k\to\infty,$ where $r>n.$

In Second order stability for the Monge-Ampère equation and strong Sobolev convergence of optima transport maps it is proved that: if $f_k\chi_{\Omega_k}\to f$ in $L^1_{loc}(\Omega),$ then $\|u_k-u\|_{W^{2,1}(\Omega')}\to 0$ as $k\to\infty.$

I am far from being an expert and I apologise if my question is too easy to answer.

Conjecture Let $\Omega_k$ be a family of convex (smooth) domains, and let $u_k$ be convex Alexandrov solution of $$ \begin{cases} det(D^2u_k)=f_k&\mbox{ in }\Omega_k\\ u_k=0 &\mbox{ on }\partial\Omega_k \end{cases} $$ with $0<\lambda\leq f_k\leq\Lambda,$ $f_k\in C^{n,\beta}.$ Assume that $\Omega_k$ converges to some domain $\Omega$ in ``some appropriate distance'' (the Hausdorff distance?) and $f_k\chi_{\Omega_k}\to f$ in $C_{loc}^{n,\beta},$ $f\in C^{n,\beta}.$ Then, if $u$ denotes the unique Alexandrov solution of $$ \begin{cases} det(D^2u)=f&\mbox{ in }\Omega\\ u=0 &\mbox{ on }\partial\Omega \end{cases} $$ for any $\Omega'\subset\subset \Omega,$ we have that $u_k\to u$ in $C^{r,\beta}(\Omega')$ as $k\to\infty,$ where $r>n.$

In Second order stability for the Monge-Ampere equation and strong Sobolev convergence of optima transport maps it is proved that: if $f_k\chi_{\Omega_k}\to f$ in $L^1_{loc}(\Omega),$ then $\|u_k-u\|_{W^{2,1}(\Omega')}\to 0$ as $k\to\infty.$

Is there any hope to prove this conjecture (or a similar one)?

Is there any hope to prove this conjecture (or a similar one)?

Conjecture Let $\Omega_k$ be a family of convex (smooth) domains, and let $u_k$ be convex Alexandrov solution of $$ \begin{cases} det(D^2u_k)=f_k&\mbox{ in }\Omega_k\\ u_k=0 &\mbox{ on }\partial\Omega_k \end{cases} $$ with $0<\lambda\leq f_k\leq\Lambda,$ $f_k\in C^{n,\beta}.$ Assume that $\Omega_k$ converges to some domain $\Omega$ in ``some appropriate distance'' (the Hausdorff distance?) and $f_k\chi_{\Omega_k}\to f$ in $C_{loc}^{n,\beta},$ $f\in C^{n,\beta}.$ Then, if $u$ denotes the unique Alexandrov solution of $$ \begin{cases} det(D^2u)=f&\mbox{ in }\Omega\\ u=0 &\mbox{ on }\partial\Omega \end{cases} $$ for any $\Omega'\subset\subset \Omega,$ we have that $u_k\to u$ in $C^{r,\beta}(\Omega')$ as $k\to\infty,$ where $r>n.$

In Second order stability for the Monge-Ampere equation and strong Sobolev convergence of optima transport maps it is proved that: if $f_k\chi_{\Omega_k}\to f$ in $L^1_{loc}(\Omega),$ then $\|u_k-u\|_{W^{2,1}(\Omega')}\to 0$ as $k\to\infty.$

I am far from being an expert and I apologise if my question is too easy to answer.

Conjecture Let $\Omega_k$ be a family of convex (smooth) domains, and let $u_k$ be convex Alexandrov solution of $$ \begin{cases} det(D^2u_k)=f_k&\mbox{ in }\Omega_k\\ u_k=0 &\mbox{ on }\partial\Omega_k \end{cases} $$ with $0<\lambda\leq f_k\leq\Lambda$ and $f_k\in C^{n,\beta}.$ Assume that $\Omega_k$ converges to some domain $\Omega$ in ``some appropriate distance'' (the Hausdorff distance?) and $f_k\chi_{\Omega_k}\to f$ in $C_{loc}^{n,\beta}$ and $f\in C^{n,\beta}.$ Then, if $u$ denotes the unique Alexandrov solution of $$ \begin{cases} det(D^2u)=f&\mbox{ in }\Omega\\ u=0 &\mbox{ on }\partial\Omega \end{cases} $$ for any $\Omega'\subset\subset \Omega,$ we have that $u_k\to u$ in $C^{r,\beta}(\Omega')$ as $k\to\infty,$ where $r>n.$

In Second order stability for the Monge-Ampere equation and strong Sobolev convergence of optima transport maps it is proved that: if $f_k\chi_{\Omega_k}\to f$ in $L^1_{loc}(\Omega),$ then $\|u_k-u\|_{W^{2,1}(\Omega')}\to 0$ as $k\to\infty.$

Is there any hope to prove this conjecture (or a similar one)?

I am far from being an expert and I apologise if my question is too easy to answer.

Conjecture Let $\Omega_k$ be a family of convex (smooth) domains, and let $u_k$ be convex Alexandrov solution of $$ \begin{cases} det(D^2u_k)=f_k&\mbox{ in }\Omega_k\\ u_k=0 &\mbox{ on }\partial\Omega_k \end{cases} $$ with $0<\lambda\leq f_k\leq\Lambda,$ $f_k\in C^{n,\beta}.$ Assume that $\Omega_k$ converges to some domain $\Omega$ in ``some appropriate distance'' (the Hausdorff distance?) and $f_k\chi_{\Omega_k}\to f$ in $C_{loc}^{n,\beta},$ $f\in C^{n,\beta}.$ Then, if $u$ denotes the unique Alexandrov solution of $$ \begin{cases} det(D^2u)=f&\mbox{ in }\Omega\\ u=0 &\mbox{ on }\partial\Omega \end{cases} $$ for any $\Omega'\subset\subset \Omega,$ we have that $u_k\to u$ in $C^{r,\beta}(\Omega')$ as $k\to\infty,$ where $r>n.$

In Second order stability for the Monge-Ampere equation and strong Sobolev convergence of optima transport maps it is proved that: if $f_k\chi_{\Omega_k}\to f$ in $L^1_{loc}(\Omega),$ then $\|u_k-u\|_{W^{2,1}(\Omega')}\to 0$ as $k\to\infty.$

Is there any hope to prove this conjecture (or a similar one)?