Nontrivial-domain existence conjecture for finite-time overdetermination

Let $m\in\mathbb{N}$, let $N\ge 2$, and let $\Omega\subset\mathbb{R}^N$ be a bounded domain. Let $u$ solve the heat equation problem referred to as Problem finite times, and require that there exist times $t_1,\dots,t_m>0$ and constants $b_1,\dots,b_m$ such that

$$\partial_\nu u(t_n)\equiv b_n\quad\text{on }\partial\Omega\quad\text{for }n=1,\dots,m.$$

Finite-time nontriviality conjecture. For every $m\in\mathbb{N}$, this problem admits nontrivial solutions, meaning solutions whose domains are not Euclidean balls.

This asks whether finitely many overdetermined times permit domains beyond balls, in contrast with rigidity results requiring stronger or asymptotic information. The conjecture remains open in the supplied source.