A note on the frequency gaps between integers in the thin obstacle problem

TL;DR

This paper proves that in all dimensions, the thin obstacle problem has no homogeneous solutions with frequencies in certain intervals, specifically excluding frequencies between 2 and 3, simplifying previous understanding of solution structures.

Contribution

It provides a straightforward proof that excludes the existence of homogeneous solutions with frequencies in specific intervals for the thin obstacle problem across all dimensions.

Findings

No homogeneous solutions with frequencies in (2k, 2k+1) for all k in natural numbers.

In particular, no solutions with frequencies in (2, 3).

Simplifies the understanding of frequency gaps in the thin obstacle problem.

Abstract

We give a simple proof of the fact that - in all dimensions - there are no homogeneous solutions to the thin obstacle problem with frequency $\lambda$ belonging to intervals of the form $(2k,2k+1)$, $k \in \mathbb{N}$. In particular, there are no frequencies in the interval $(2,3)$.