A nonlinear inequality arising in geometry and Calabi-Bernstein type problems
The authors study the following nonlinear differential inequality \[ H(u)^ 2 \leq \frac{f'(u)^ 2}{f(u)^ 2}, \] under the assumption \[ | Du| < \lambda f(u) \quad 0<\lambda <1, \] where \(f\) is a positive smooth function, \(u \in C^{\infty}(\Omega)\), \(| Du| \) is the length of the gradient \(Du\) of \(u\) and \[ H(u)=-\text{div}\Big(\frac{Du}{2f(u)\sqrt{f(u)^ 2 -| Du|^ 2}}\Big)-\frac{f'(u)}{2\sqrt{f(u)^ 2 -| Du|^ 2}}\Big(2+\frac{| Du|^ 2}{f(u)^ 2}\Big). \] A characterization of the entire solutions of the above nonlinear inequality is presented, which has a natural interpretation in terms of certain nonflat Robertson-Walker spacetimes. Furthermore, new Calabi-Bernstein-type problems are raised and solved.
scientific article; zbMATH DE number 5823208