The ${\sigma }_2$-curvature equation on a compact manifold with boundary

We first establish local $C^2$ estimates of solutions to the ${\sigma }_2$-curvature equation with nonlinear Neumann boundary condition. Then, under assumption that the mean curvature of a background metric is nonnegative on totally non-umbilic boundary, for dimensions three and four there exists a conformal metric having a prescribed positive ${\sigma }_2$-curvature and a prescribed nonnegative boundary mean curvature. The local estimates play an important role in the blow up analysis for the latter existence result.