8 problems

Let $B$ be a ball, let $p>1$, and let $u\in W^{1,r}(B)$ with $r>p-1$. Suppose that $u$ solves … in the distributional sense. Iwaniec-Sbordone conjecture. Then…

Let $p>1$, $n\geq 2$, and $r>\max\{p-1,1\}$. Let $u\in W^{1,r}_0(\mathbb{B}^n)\cap W^{1,p}_0(\mathbb{B}^n)$ and $f\in L^{\frac{r}{p-1}}\cap L^{p'}(\mathbb{B}^n,\mathbb{R}^n)$, wher…

The p-harmonic tug-of-war value-function conjecture. $p$-harmonic functions are related to the value functions of tug-of-war with noise and to solutions $u_{\varepsilon}$ of the dy…

Let $d,n\in\mathbb{N}$ and $1<p<\infty$. Let $h\colon\Omega\to\mathbb{R}^n$ be $p$-harmonic on an open set $\Omega\subset\mathbb{R}^d$, and let $V$ denote the standard nonlinear tr…

Let $u$ be a solution of the $p$-Laplace equation with an isolated singularity at the origin, and suppose that for every sufficiently small $\epsilon>0$ the sets … are connected; t…

Let $u$ solve the one-dimensional $p$-parabolic boundary-value problem … on $(-1,1)\times(0,T)$ with boundary values $u(-1,t)=0$, $u(1,t)=1$, and initial value…

Existence conjecture. Based on numerical experiments, for each $N\in\mathbb{N}$ there exists a solution $v=v_N$ to the linearized $p$-Laplace equation such that

Non-vanishing conjecture. If $\Delta_pu=0$ for $p>1$ and $p\ne2$, then $u\equiv 0$.