Solutions to the Thin Obstacle Problem with non-2D frequency
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917996063096832 |
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| author | Franceschini, Federico Savin, Ovidiu |
| author_facet | Franceschini, Federico Savin, Ovidiu |
| contents | For all odd positive integers $m$, we construct $μ$-homogeneous solutions to the thin obstacle problem in $\mathbb{R}^3,$ with $μ\in(m,m+1)$. For $m$ large, $μ-m$ converges to $1$, so $μ\neq m+\tfrac 1 2$. The restriction to odd values of $m$ is necessary: we show that, for all $n\ge 2$, there are no $μ$-homogeneous solutions to the thin obstacle problem in $\mathbb{R}^n$ with $μ\in \bigcup_{k\ge 0}(2k,2k+1)$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_16486 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solutions to the Thin Obstacle Problem with non-2D frequency Franceschini, Federico Savin, Ovidiu Analysis of PDEs 35R35 For all odd positive integers $m$, we construct $μ$-homogeneous solutions to the thin obstacle problem in $\mathbb{R}^3,$ with $μ\in(m,m+1)$. For $m$ large, $μ-m$ converges to $1$, so $μ\neq m+\tfrac 1 2$. The restriction to odd values of $m$ is necessary: we show that, for all $n\ge 2$, there are no $μ$-homogeneous solutions to the thin obstacle problem in $\mathbb{R}^n$ with $μ\in \bigcup_{k\ge 0}(2k,2k+1)$. 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. |
| title | Solutions to the Thin Obstacle Problem with non-2D frequency |
| topic | Analysis of PDEs 35R35 |
| url | https://arxiv.org/abs/2504.16486 |