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 background metric has nonnegative mean curvature on totally non-umbilic boundary, for dimensions three and four we prove the existence of a conformal metric with a prescribed positive $\sigma_2$-curvature function and a prescribed nonnegative boundary mean curvature function. The local estimates play an important role in the blow up analysis in the latter existence result.