课程题目：REGULARITY OF STABLE SOLUTIONS TO SEMILINEAR ELLIPTIC EQUATIONS UP TO DIMENSION 9 (系列1-8)

报 告 人：Xavier Cabré 教授

所在单位：ICREA (Institució Catalana de Recerca i Estudis Avançats) & Universitat Politècnica de Catalunya

报告时间：2022年11月1日 16：00-18：00（lesson 1-2）

                2022年11月3日 15：00-17：00（lesson 3-4）

                2022年11月8日 16：00-18：00（lesson 5-6）

                2022年11月10日 16：00-18：00（lesson 7-8）

报告地点：ZOOM ID：581 880 5979

校内联系人：魏元鸿 weiyuanhong@jlu.edu.cn

课程摘要：In this mini-course, we prove the following long-standing conjecture: stable solutions to semilinear elliptic equations are bounded (and thus smooth) in dimension n≤9. This result, that was only known to be true for n≤4, is optimal: log(1/|x|2) is a W1,2 singular stable solution for n≥10. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension n≤9, stable solutions are bounded in terms only of their L1 norm, independently of the nonlinearity. In addition, in every dimension we establish a higher integrability result for the gradient and optimal integrability results for the solution in Morrey spaces.

As one can see by a series of classical examples, all our results are sharp. Furthermore, as a corollary we obtain that extremal solutions of Gelfand problems are W1,2 in every dimension and they are smooth in dimension n≤9. This answers to two famous open problems posed by Brezis and Brezis-Vázquez.

报告人简介：Xavier教授主要从事偏微分方程、变分法、非线性分析及动力系统的研究。在包括Acta Math.在内的数学重要期刊上公开发表论文70余篇。