Solutions to the Thin Obstacle Problem with non-2D frequency

Abstract

For all odd positive integers $m$, we construct $\hat{I}¼$-homogeneous solutions to the thin obstacle problem in ${\mathbb{R}}^3,$ with $\hat{I}¼\in (m,m+1)$. For $m$ large, $\hat{I}¼-m$ converges to $1$, so $\hat{I}¼\ne m+\frac{1}{2}$. The restriction to odd values of $m$ is necessary: we show that, for all $n\ge 2$, there are no $\hat{I}¼$-homogeneous solutions to the thin obstacle problem in ${\mathbb{R}}^n$ with $\hat{I}¼\in {\bigcup }_{k\ge 0}(2k,2k+1)$. These examples also apply to $2$-valued $C^{1,1/2}$ stationary harmonic functions or $\mathbb{Z}/2\mathbb{Z}$-eigenfunctions of the laplacian on the sphere.