Nonlinear oblique boundary value problems for two-dimensional curvature equations
The paper is devoted to the study of the following nonlinear oblique boundary value problem on a bounded uniformly convex plane domain \(\Omega\): \[ F[D^2 u]= g(x, u)\quad \text{in }\Omega,\quad b(x, u, Du)= 0\quad \text{on } \partial\Omega. \] The operator \(F\) is defined as \(F[D^2 u]= f(k_1, k_2)\), where \(k_1\), \(k_2\) denote the curvatures of the graph of \(u\) relative to the upward unit normal \(\nu\), and \(f\) is a suitable symmetric function defined on an open, symmetric (under interchange of \(k_1\) and \(k_2\)) region \(\Sigma\subset \mathbb{R}^2\). The solution \(u\in C^2(\Omega)\) is called admissible if the corresponding \((k_1, k_2)\in \Sigma\). The function \(f\) is assumed to satisfy \(\partial f/\partial k_i> 0\), \(i= 1,2\), in \(\Sigma\) and \(f= 0\) on \(\partial\Sigma\). Moreover, \(f\) must satisfy some structural conditions. The function \(g(x, z)\) is assumed to be smooth, positive and nondecreasing with respect to \(z\). The semilinear boundary condition \(D_\beta u+ \phi(x, u)= 0\) for suitable \(\phi\) and a vector field \(\beta\) with \(\beta\cdot \nu> 0\) is investigated as well as the fully nonlinear boundary condition \(D_\nu u+ \phi(x, u, \delta u)= 0\). Here \(\delta\) denotes the tangential gradient operator. To prove existence, the method of continuity is used by proving a priori bounds for \(u\), \(Du\) and \(D^2 u\). The techniques are often similar to those used by the same author in a previous paper on the Hessian equation [Ann. Inst. H. Poincaré, Anal. Non Linéaire 12, 507-575 (1995; Zbl 0841.35042)].
scientific article; zbMATH DE number 889012