Abstract: We study the $\sigma_k$-Loewner-Nirenberg problem on a bounded domain with smooth boundary in $n$-dimensional Euclidean space, $n\ge 3$ and $2\le k\le n$. We prove the existence and uniqueness of a locally Lipschitz solution on any such domain. When the domain is a ball, the solution, which corresponds to the Poincare metric, is smooth. We prove that when the boundary of the domain is disconnected, then the locally Lipschitz solution is not a differentiable function. These are joint works with Mar Gonzalez, Luc Nguyen and Jingang Xiong.

Bio. Prof. Yanyan Li is a Distinguished Professor at Rutgers University, and the Director of The Center for Nonlinear Analysis. He received his B.S. degree from University of Science and Technology of China in 1982, the M.S. from Institute of System Science, Academic Sinica in 1983, and the Ph.D. degree from Courant Institute of Mathematical Sciences, New York University in 1988.

Prof. Li’s main research interest is in nonlinear partial differential equations and applications. He was an invited speaker at the International Congress of Mathematicians in 2002, and has been a member of the Inaugural Class of Fellows of the American Mathematical Society since 2012. He was an Alfred P. Sloan Research Fellow in 1993-1995, a Simons Fellow in Mathematics and Theoretical Physics in 2020, and received the Rutgers Board of Trustees Award for excellence in Research in 2008. He is a member of the editorial board of Advances in Mathematics, among several other academic journals.