The Dirichlet problem for the prescribed curvature quotient equations
The authors consider the classical Dirichlet problem for equations of the form \(f(\kappa) = \psi(x, u)\) in domains \(\Omega\) in the Euclidean \(n\)- space, where \(\kappa = (\kappa_ 1, \dots,\kappa_ n)\) denotes the principal curvatures of the graph of \(u\) over \(\Omega\), \(\Psi\) is a prescribed positive function on \(\Omega \times \mathbb{R}\) and \(f\) is a symmetric function of the form \(f(\kappa) = {S_ k\over S_ l}\) where \(0 \leq l < k \leq n\) and \(S_ k = \sum_{i_ 1 < \dots < i_ k} \kappa_{i_ 1} \kappa_{i_ 2} \cdots \kappa_{i_ k}\). Then they prove that under conditions on \(\partial \Omega\) and \(\Psi\) and conditions guaranteeing a priori solution bounds, the above problem has a unique solution \(u\) satisfying \(u = 0\) on \(\partial \Omega\).
scientific article; zbMATH DE number 683315