The σ2$\sigma _{2}$‐curvature equation on a compact manifold with boundary

Abstract We first establish local estimates of solutions to the ‐curvature equation with nonlinear Neumann boundary condition. Then, under assumption that the mean curvature of a background metric is nonnegative on totally nonumbilic boundary, for dimensions 3 and 4 there exists a conformal metric having a prescribed positive ‐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.