Boundary behavior in the Loewner-Nirenberg problem (Q1779215)

Let \(\Omega\subset \mathbb R^n\) be a bounded open domain of class \(C^{2+\alpha}\), \(0<\alpha<1\). The author considers the maximal solution \(u_\Omega\) of the problem (in \(\Omega\)) \[ \Delta u = n(n-2) u^{\frac{n+2}{n-2}} \] He proves that \(v_\Omega = u_\Omega^{-\frac{2}{n-2}}\) is of class \(C^{2+\alpha}\) up to the boundary. The proof uses the reduction of the problem to a nonlinear Fuchsian elliptic PDE.