3 problems

Let $\varphi_p$ and $\varphi_q$ be the first eigenfunctions of the $p$- and $q$-Laplacians, respectively. Linear independence conjecture. The eigenfunctions $\varphi_p$ and…

Let $p\in(2,\infty)$, let $B_1$ be the unit ball, and let $\Delta_p u\coloneqq\operatorname{div}(|Du|^{p-2}Du)$. A function is locally of class $C^{1,\alpha}$ if it has locally Höl…

Let $p>2$, let $p^{\prime}=p/(p-1)$, and consider regular solutions in the range $\alpha_{c}<\alpha<-p^{\prime}$. For a regular solution, let $y$ denote the corresponding phase-pla…