Metadata
Abstract
We investigate the minimum and maximum number of nodal domains across all time-dependent homogeneous caloric polynomials of degree $d$ in $\mathbb{R}^{n}\times \mathbb{R}$ (space $\times $ time), i.e., polynomial solutions of the heat equation satisfying $\partial _t p\lnot \equiv 0$ and
| \[ p\left(\lambda x, \lambda ^2 t\right) = \lambda ^d p~(x,t)\quad \text{for all }x \in \mathbb{R}^n, t \in \mathbb{R}, \quad \text{and}\quad \lambda > 0. \] |
When $n=1$, it is classically known that the number of nodal domains is precisely $2\lceil d/2\rceil $. When $n=2$, we prove that the minimum number of nodal domains is 2 if $d\lnot \equiv 0\pmod {4}$ and is 3 if $d\equiv 0\pmod {4}$. When $n\ge 3$, we prove that the minimum number of nodal domains is $2$ for all $d$. Finally, we show that the maximum number of nodal domains is $\Theta (d^n)$ as $d\rightarrow \infty $ and lies between $\lfloor \frac{d}{n}\rfloor ^n$ and $\binom{n+d}{n}$ for all $n$ and $d$. As an application and motivation for counting nodal domains, we confirm existence of the singular strata in Mourgoglou and Puliatti’s two-phase free boundary regularity theorem for caloric measure.
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