Abstract: 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 and a prescribed nonnegative boundary mean curvature. The local estimates play an important role in the blow up analysis in the latter existence result. This is a joint work with Xuezhang Chen. 

Brief biography: Wei Wei is currently an assistant professor at Nanjing University. She obtained her Ph.D. at University of Science and Technology of China in 2019 under the supervision of Professor Xinan Ma. Her research focuses on fully nonlinear elliptic equations and geometric analysis.

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