Let $\Omega\subset\mathbb{R}^2$ be a bounded convex domain, $f$ be a positive smooth function on $\Omega$ and $\phi:\partial\Omega\rightarrow\mathbb{R}$ be a continuous function. It is known that as long as $f$ does not tend to $+\infty$ too fast at the boundary, for every $t>0$ there is a unique convex function $u_t\in C^0(\overline{\Omega})\cap C^\infty(\Omega)$ satisfying $$ \det D^2 u_t=t\,f,\quad u_t|_{\partial\Omega}=\phi. $$

Question. Does $u_t$ depend smoothly on $t$?


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