Solving the Dirichlet problem with prescribed density

Abstract

In this paper we prove the following result. Let $\Omega\subset\mathbb R^n, n\geq 3,$ be a bounded, strictly convex, smooth domain and $\varphi: \partial\Omega\rightarrow\mathbb R$ be a smooth function. Then for any $z\in\Omega,$ there exists $c_1=c_1(\Omega, \{z\}, n, \varphi)>0,$ such that if $c\geq c_1$ then the problem: $\sigma_{n-1}(D^2 u)=0$ in $\bar\Omega\setminus\{z\},$ $u|_{\partial\Omega}=\varphi,$ $\lim\limits_{r\rightarrow 0}\sup\limits_{B_r(z)}\frac{u(x)-u(z)}{|x-z|^{(n-2)/(n-1)}}=c,$ admits a smooth solution in $\bar\Omega\setminus\{z\}.$ Moreover, we obtain the optimal a priori estimates for this solution. In particular, we show for any integer $m\geq 0$ there exists $C_m=C_m(m, \Omega, \{z\}, n, \varphi, c)>0$ such that $|D^m u(x)|<C_m|x-z|^{\frac{n-2}{n-1}-m}.$ This work provides the first result demonstrating the existence of smooth solutions to the Dirichlet problem for fully nonlinear elliptic equations with prescribed density in Euclidean space. Previously, only continuous solutions were obtained.