A symmetry property for polyharmonic functions vanishing on equidistant hyperplanes

TL;DR

This paper proves a symmetry property for polyharmonic functions that vanish on multiple equidistant hyperplanes, showing they are odd at a central hyperplane, and establishes conditions under which such functions must be identically zero.

Contribution

It introduces a novel symmetry result for polyharmonic functions vanishing on equidistant hyperplanes and provides conditions for their triviality.

Findings

Polyharmonic functions are odd at a central hyperplane if they vanish on 2N-1 equidistant hyperplanes.

Such functions are identically zero if they vanish on 2N equidistant hyperplanes with a specific growth condition.

The results extend symmetry and uniqueness properties for polyharmonic functions in unbounded domains.

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 $$ \sup_{t\in K}\left\vert u\left( t,y\right) \right\vert \leq o\left( \left\vert y\right\vert ^{\left( 1-d\right) /2}e^{\frac{\pi}{c}\left\vert y\right\vert }\right) $$ for $\left\vert y\right\vert \rightarrow\infty$ 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 $c.$