# Smooth dependence of convex functions on Monge-Ampère measure

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 $$t$$?