stability of the Monge-Ampère equation

Question

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 $k\to\infty.$

Answer 1

Yes, this follows from Schauder theory for the Monge-Ampere equation and for linear equations. Subtracting the equations $\det D^2u_k = f_k$ and $\det D^2u = f$ gives, for $v_k = u_k - u$, the equation $$a_{ij}(x) (v_k)_{ij} = f_k - f$$ where $a_{ij}$ are coefficients depending on $D^2u_k$ and $D^2u$ (to see this observe that $f_k - f = \int_{0}^1 \frac{d}{dt} \det(tD^2u_k + (1-t)D^2u)\,dt$). By Caffarelli's Schauder estimates, the $a_{ij}$ are $C^{n,\beta}$ and uniformly elliptic when we step away from the boundary. Say $B_2 \subset \Omega_k,\,\Omega$ after an affine transformation. Linear Schauder theory gives (take $n = 0$ for simplicity) $$\|v_k\|_{C^{2,\beta}(B_{1/2})} < C(\|f_k-f\|_{C^{\beta}(B_1)} + \|v_k\|_{L^{\infty}(B_1)}).$$ By hypothesis, the first term on the right side goes to zero, and it is easy to see that the second term goes to zero using the maximum principle (e.g. apply the ABP maximum principle to $v_k$, which is small on the boundary of the common domain of definition for $u_k$ and $u$ by the Alexandrov maximum principle).