12-17【Qiqi Zhang】管楼1418 偏微分方程讨论班系列报告


Title:Dirichlet boundary of some Fully nonlinear PDEs

Speaker:Qiqi Zhang  (Fujian Normal University)

Time:2019年12月17日          下午    15:00-15:50

Room:东区管理科研楼  1418教室

Abstract:In this talk, we study the Dirichlet problem for the following fully nonlinear second order elliptic PDEs. 
$$\left\{\begin{array}{ll}\sigma_k(D^2u)+\alpha(z)\sigma_{k-1}(D^2u)=\Sigma_{\ell=0}^{k-2}\alpha_{\ell}(z)\sigma_{\ell}(D^2u) & \quad \hbox{if}\ z\in \Omega\\ u(z)=\phi(z) & \quad \hbox{if}\ z\in \partial\Omega\end{array}\right.$$
where $\sigma_k$ is the $k$th symmetric function of complex Hessian $D^2u$ of $u$. We prove that there exists an admissible solution (i.e. $\lambda(D^2u)\in\Gamma_{k-1}$) $u\in C^{\infty}(\bar\Omega)$ in a bounded domain with smooth boundary for without any sign  requirement for $\alpha(z)$ and $\alpha_{\ell}>0$ for all $1\le \ell\le k-2$. It is based on the concave property of
$${\sigma_k\over \sigma_{k-1}}(D^2u)-\sum\limits_{\ell=0}^{k-2}\alpha_{\ell}(z){\sigma_\ell\over \sigma_{k-1}}(D^2u)=-\alpha(z) \quad \hbox{if} \Omega$$

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