On the number of nodal domains of homogeneous caloric polynomials

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 ${\partial }_{t}p\not\equiv 0$ and $$p(\hat{I}»x,\hat{I}{»}^{2}t)=\hat{I}{»}^{d}p(x,t)\text{for all x∈Rn, t∈R, and I^»>0.}$$ 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 $d\not\equiv 0(\mathrm{mod}4)$ and is 3 if $d\equiv 0(\mathrm{mod}4)$. 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 $\hat{I}˜(d^n)$ as $d\to \infty$ and lies between $\lfloor \frac{d}{n}{\rfloor }^n$ and $\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ 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.

31 pages, 12 figures. Comments welcome