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

Federico Franceschini, Ovidiu Savin

TL;DR

This work identifies new non-2D frequencies for the thin obstacle problem by constructing $\mu$-homogeneous solutions in $\mathbb{R}^3$ with $\mu\in(m,m+1)$ for odd $m$, and proves a sharp nonexistence result in other dimensions for frequencies in $(2k,2k+1)$. The core approach reduces the problem to eigenfunctions on carefully chosen spherical domains, using Legendre functions to control boundary data and a continuity argument to locate parameters where the global solution satisfies the obstacle conditions. The authors obtain precise asymptotics, showing $\mu-m\to1$ and the contact-set width $\bar{\sigma}_m\to0$ as $m\to\infty$, and provide a variant construction that yields additional families of solutions via higher eigenfunctions. These results establish significant gaps in the spectrum of admissible frequencies, with implications for the structure of singular sets and connections to 2-valued harmonic functions and $\mathbb{Z}/2\mathbb{Z}$-eigenfunctions. The work advances understanding of frequency sets in the thin obstacle problem and informs regularity theory for related free-boundary problems.

Abstract

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.

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

TL;DR

This work identifies new non-2D frequencies for the thin obstacle problem by constructing -homogeneous solutions in with for odd , and proves a sharp nonexistence result in other dimensions for frequencies in . The core approach reduces the problem to eigenfunctions on carefully chosen spherical domains, using Legendre functions to control boundary data and a continuity argument to locate parameters where the global solution satisfies the obstacle conditions. The authors obtain precise asymptotics, showing and the contact-set width as , and provide a variant construction that yields additional families of solutions via higher eigenfunctions. These results establish significant gaps in the spectrum of admissible frequencies, with implications for the structure of singular sets and connections to 2-valued harmonic functions and -eigenfunctions. The work advances understanding of frequency sets in the thin obstacle problem and informs regularity theory for related free-boundary problems.

Abstract

For all odd positive integers , we construct -homogeneous solutions to the thin obstacle problem in with . For large, converges to , so . The restriction to odd values of is necessary: we show that, for all , there are no -homogeneous solutions to the thin obstacle problem in with . These examples also apply to -valued stationary harmonic functions or -eigenfunctions of the laplacian on the sphere.

Paper Structure

This paper contains 15 sections, 10 theorems, 180 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The thin obstacle problem
  3. Main results
  4. Comments
  5. Connection with 2-valued harmonic functions and $\mathbb{Z}/2\mathbb{Z}$ eigenfunctions
  6. Structure of the paper
  7. Acknowledgments
  8. Proof of \ref{['thm:main']}
  9. Notation and Legendre functions
  10. Construction of $\bm {v,}$ $\bm{h,}$ and $\bm u$
  11. Existence of solutions
  12. Non-existence of solutions --- Proof of \ref{['thm:gaps']}
  13. Asymptotics for ${m\to+\infty}$
  14. A variant of the construction
  15. Legendre Functions

Key Result

Theorem 1

For each odd integer $m\ge1$ there is a homogeneous solution of the thin obstacle problem $u\colon \mathbb{R}^3\to \mathbb{R},$ with homogeneity $\bar{\mu}_m\in (m,m+1)$ and contact set for some $\bar{\sigma}_m \in (0,1)$. Furthermore, $m+1-\bar{\mu}_m\to 0^+$ and $\bar{\sigma}_m\to0^+$ as $m\to+\infty$ along odd numbers.

Figures (4)

  • Figure 1: The contact set $Z(u) = \{u =0 \}\cap\{ z=0\}$.
  • Figure 2: Plots of $p_\mu(\varphi)$ as $\mu$ ranges between two consecutive integers and $\varphi\in[0,\tfrac{\pi}{2}]$.
  • Figure 3: Numerical plots of $h$ for $m=3$ and different values of $\sigma$.
  • Figure 4: Plots of $v$ (left) and $u$ (right) for $m=3$ and $\sigma=.42$

Theorems & Definitions (21)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Proposition 3: Legendre functions
  • proof : Proof of \ref{['thm:gaps']}
  • Proposition 4: Blow-up
  • Lemma 5: Properties of $V$
  • proof
  • Lemma 6: $v$ and $V$
  • ...and 11 more