Estimate for the second derivatives of curvature-type equations (Q2714015)

Let \(A(x,z,p)\in\mathbb S^n\), \((x,z,p)\in\mathbb R^n\times\mathbb R\times\mathbb R^n\), be a positive definite matrix, where \({\mathbb S}^{n}\) is the space of real symmetric \(n\times n\) matrices, let \(\Omega\) be a bounded domain in \(\mathbb R^n\), and let \(F\in C^2\) be a given function on a domain \(D_0\subset\mathbb S^n\). Denote \(u_i = \partial u/\partial x^i\), \(u_x = (u_i)\), \(u_{xx} = (u_{ij})\), and \(u_{(xx)} = A^{-1/2}(x,u,u_x)\cdot u_{xx}\cdot A^{-1/2}(x,u,u_x)\).NEWLINENEWLINE The aim of the article is to study solutions to the curvature-type equation NEWLINE\[NEWLINE F(u_{(xx)}) = f(x,u,u_x),\quad x\in\Omega, NEWLINE\]NEWLINE where \(f\in C^2(\Gamma)\), \(\Gamma = \Omega\times\mathbb R\times\mathbb R^n\). The authors deal with the class of the so-called admissible solutions. As a results, an a priori estimate is established for the second derivatives of the admissible solutions \(u\in C^4\) to the above-mentioned equation.