7 problems

Let $D$ be a bounded domain in $\mathbb{R}^2$. Let $A_x(r_1,r_2)$ be an admissible annular domain centred at $x$, with $r_2>r_1>0$, and let $B_{r_2}(x)$ be admissible. Denote by…

Let $D$ be a bounded domain in $\mathbb{R}^m$, with $m\geqslant 2$. Let $A_x(r_1,r_2)$ be an admissible annular domain centred at $x$, with $r_2>r_1>0$, and let $B_{r_2}(x)$ be adm…

Let $\Omega\subset \operatorname{\mathbb{R}}^d$ be a domain, and let $u\geq 0$ be subharmonic in $\Omega$, with $u$ not identically zero in $\Omega$. For an admissible disc…

Let $D \subset \mathbb{R}^m$, with $m \geqslant 2$, be a bounded domain with sufficiently smooth boundary, and let $r>0$ satisfy $|B_r|=|D|$. A function $u$ is panharmonic if it be…

Veech's conjecture. Every non-negative, $r$-median function on $D$ is harmonic.

Littlewood's one circle problem. Is $u$ necessarily harmonic in $D$?

Let $(\mathbb R^n,F)$ be a Minkowski space, and let $\boldsymbol{\Delta}$ denote its Finsler Laplacian. The mean value property refers to the property that $\boldsymbol{\Delta}$-ha…