Non-classical solutions to the p-Laplace equation
数学专题报告
报告题目(Title)：Non-classical solutions to the p-Laplace equation
报告人(Speaker)：Riccardo Tione (Max Planck Institute)
地点(Place)：Zoom ID：958 8198 9311, PWD: 485387
时间(Time)：2024 年 9月 26日 16：00—17：00
邀请人(Inviter)：熊金钢
报告摘要
In this talk we will consider the \(p\)-Laplace equation, \(\mbox{div}(|Du|^{p-2}Du) = 0\). In particular, we will focus on the very weak solutions, i.e. solutions \(u\) to the \(p\)-Laplace equation with \(u \in W^{1,q}\), where \(\max\{1,p-1\} < q < p\). In 1994, T. Iwaniec and C. Sbordone showed that if \(q\) is sufficiently close to \(p\), then very weak solutions belong to \(W^{1,p}\), and thus are classical solutions. They conjectured the same to happen for any \(\max\{1,p-1\} < q\). In this talk, I will present a positive result which shows that Iwaniec-Sbordone's conjecture is true if the gradient of \(u\) belongs to suitable cones, and next I will sketch the construction of a counterexample for this conjecture if this additional condition is not fulfilled. This is based on a joint work with Maria Colombo.
* This PDE seminar is co-organized with Tianling Jin at HKUST and Juncheng Wei at CUHK. See the seminar webpage: https://www.math.hkust.edu.hk/~tianlingjin/PDEseminar.html
