A symmetry property for polyharmonic functions vanishing on equidistant hyperplanes

Abstract

Let $u\left(t,y\right)$ be a polyharmonic function of order $N$ defined on the strip $\left(a,b\right)\times {\mathbb{R}}^d$ satisfying the growth condition $$\underset{t\in K}{\sup }\left|u\left(t,y\right)\right|\le o\left({\left|y\right|}^{\left(1-d\right)/2}{e}^{\frac{\pi }{c}|y|}\right)$$ for $\left|y\right|\to \infty$ and any compact subinterval $K$ of $\left(a,b\right)$, and suppose that $u\left(t,y\right)$ vanishes on $2N-1$ equidistant hyperplanes of the form $\left\{t_j\right\}\times {\mathbb{R}}^d$ for $t_j=t_0+jc\in \left(a,b\right)$ and $j=-\left(N-1\right),...,N-1.$ Then it is shown that $u\left(t,y\right)$ is odd at $t_0,$ i.e. that $u\left(t_0+t,y\right)=-u\left(t_0-t,y\right)$ for $y\in {\mathbb{R}}^d$. The second main result states that $u$ is identically zero provided that $u$ satisfies the growth condition and vanishes on $2N$ equidistant hyperplanes with distance $c.$