arXiv subjects
Publications and source records attributed to Jiguang Bao.
We establish upper bounds on the blow up rate of the gradients of solutions of the Lamé system with partially infinite coefficients in dimension two as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero.
In this paper, we show the global well-posedness for periodic gKdV equations in the space $H^s(\mathbb{T})$, $s\ge \frac12$ for quartic case, and $s> \frac59$ for quintic case. These improve the previous results of I-team in 2004. In particular, the result is sharp for quintic case. The main approaches are the I-method combining with the resonance decomposition developed by Miao et al in 2010, and a bilinear Strichartz estimate in periodic setting.
We consider the Monge-Ampère equation $\det(D^2u)=f$ where $f$ is a positive function in $\mathbb R^n$ and $f=1+O(|x|^{-β})$ for some $β>2$ at infinity. If the equation is globally defined on $\mathbb R^n$ we classify the asymptotic behavior of solutions at infinity. If the equation is defined outside a convex bounded set we solve the corresponding exterior Dirichlet problem. Finally we prove for $n\ge 3$ the existence of global solutions with prescribed asymptotic behavior at infinity. The assumption $β>2$ is sharp for all the results in this article.
In this paper, we establish a theorem on the existence of the solutions of the exterior Dirichlet problem for Hessian equations with prescribed asymptotic behavior at infinity. This extends a result of Caffarelli and Li for the Monge-Ampère equation to Hessian equations.