On the number of nodal domains of homogeneous caloric polynomials
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917030828965888 |
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| author | Badger, Matthew Jeznach, Cole |
| author_facet | Badger, Matthew Jeznach, Cole |
| contents | 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 $$p(λx, λ^2 t) = λ^d p(x,t)\quad\text{for all $x \in \mathbb{R}^n$, $t \in \mathbb{R}$, and $λ> 0$.}$$ 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 $d\not \equiv 0\pmod 4$ and is 3 if $d\equiv 0\pmod 4$. When $n\geq 3$, 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 $Θ(d^n)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_07268 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the number of nodal domains of homogeneous caloric polynomials Badger, Matthew Jeznach, Cole Analysis of PDEs Primary: 35K05. Secondary: 26C05, 26C10, 35R35 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 $$p(λx, λ^2 t) = λ^d p(x,t)\quad\text{for all $x \in \mathbb{R}^n$, $t \in \mathbb{R}$, and $λ> 0$.}$$ 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 $d\not \equiv 0\pmod 4$ and is 3 if $d\equiv 0\pmod 4$. When $n\geq 3$, 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 $Θ(d^n)$ 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. |
| title | On the number of nodal domains of homogeneous caloric polynomials |
| topic | Analysis of PDEs Primary: 35K05. Secondary: 26C05, 26C10, 35R35 |
| url | https://arxiv.org/abs/2401.07268 |