An extremal case of the equation of prescribed Weingarten curvature (Q376587)

The author studies the existence and regularity of solutions to the Dirichlet problem for a Hessian quotient equation on the sphere \(S_k(k(\mathrm{graph}(u)))=\psi(x,v)\) in \(\Omega\), subject to the boundary condition \(|Du(x)|\to\infty\) as \(x\to\partial\Omega\). The equation in question arises as the determining equation for the support function of a convex surface which is required to meet a given enclosing cylinder tangentially and whose \(k\)-th Weingarten curvature is a given function. Under certain regularity assumptions on \(\psi\) and \(\Omega\) it is shown the existence of a solution whose graph is \(C^{3,\alpha}\) provided that \(\psi^{-\frac{1}{k}}=\psi^{-\frac{1}{k}}(x,v)\) is convex in \(x\) and a certain compatibility condition holds.