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Global solutions to the thin obstacle problem with superquadratic growth

Xavier Fernández-Real, Hui Yu

TL;DR

The paper analyzes global solutions to the thin obstacle problem in $\mathbb{R}^{n+1}$ and reveals two main advances: (i) a complete classification of global solutions with polynomial growth and bounded positivity sets through an intrinsic correspondence with odd polynomials in $\mathcal{P}^o_{n+1}$, yielding $|u(x)+|x_{n+1}|q(x)|\to0$ as $|x|\to\infty$; and (ii) a striking flexibility of compact contact sets, showing they can be nonconvex or arbitrarily closely approximate any compact subset of the thin space, including via level-set approximations of polynomials. The results resolve conjectures about non-rigidity of compact contact sets and establish a universal approximation mechanism for compact sets by global thin obstacle solutions, contrasting with the rigidity known for the classical obstacle problem. The work employs exterior Liouville theory, Perron-type constructions, comparison principles, and careful barrier arguments tailored to the thin obstacle framework, with implications for the structure of free boundaries and potential extensions to cubic and higher-order growth.

Abstract

We study rigidity/flexibility properties of global solutions to the thin obstacle problem. For solutions with bounded positive sets, we give a classification in terms of their expansions at infinity. For solutions with bounded contact sets, we show that the contact sets are highly flexible and can approximate arbitrary compact sets. These phenomena have no counterparts in the classical obstacle problem.

Global solutions to the thin obstacle problem with superquadratic growth

TL;DR

The paper analyzes global solutions to the thin obstacle problem in and reveals two main advances: (i) a complete classification of global solutions with polynomial growth and bounded positivity sets through an intrinsic correspondence with odd polynomials in , yielding as ; and (ii) a striking flexibility of compact contact sets, showing they can be nonconvex or arbitrarily closely approximate any compact subset of the thin space, including via level-set approximations of polynomials. The results resolve conjectures about non-rigidity of compact contact sets and establish a universal approximation mechanism for compact sets by global thin obstacle solutions, contrasting with the rigidity known for the classical obstacle problem. The work employs exterior Liouville theory, Perron-type constructions, comparison principles, and careful barrier arguments tailored to the thin obstacle framework, with implications for the structure of free boundaries and potential extensions to cubic and higher-order growth.

Abstract

We study rigidity/flexibility properties of global solutions to the thin obstacle problem. For solutions with bounded positive sets, we give a classification in terms of their expansions at infinity. For solutions with bounded contact sets, we show that the contact sets are highly flexible and can approximate arbitrary compact sets. These phenomena have no counterparts in the classical obstacle problem.
Paper Structure (12 sections, 15 theorems, 96 equations, 3 figures)

This paper contains 12 sections, 15 theorems, 96 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Global solutions with bounded positivity set
  3. Flexibility of compact contact sets
  4. Structure of the paper
  5. Preliminaries
  6. Notation
  7. Classification of global solutions with compact contact set
  8. Comparison principle
  9. Proof of Theorem \ref{['thm:main1']}
  10. A conjecture on cubic solutions
  11. A nonconvex compact contact set
  12. Approximating any compact set

Key Result

Theorem 1.1

For $n \ge 2$, let $u$ be a global solution to eq:main_TOP with polynomial growth. Then, the positivity set $\{u > 0, x_{n+1} = 0\}$ is bounded if and only if for some $q\in \mathcal{P}^o_{n+1}$. Moreover, given any $q\in \mathcal{P}^o_{n+1}$, there is a unique $u$ solution to eq:main_TOP such that eq:cond_inf_odd holds.

Figures (3)

  • Figure 1: Representation of the thin space in the counter-example constructed in Lemma \ref{['lem:twoballs']} for $n = 2$, where the greyed area corresponds to the contact set $\{u = 0\}$.
  • Figure 2: Representation of the thin space in the counter-example constructed in Lemma \ref{['lem:disk']} for $n = 2$, where the greyed area corresponds to the contact set $\{u = 0\}$.
  • Figure 3: Representation of the thin space in the counter-example constructed in the proof Lemma \ref{['lem:starshaped']} for $n = 2$, where the greyed area corresponds to the contact set $\{u = 0\}$, and the blacked lines correspond to the zero level set of $q(x, y)$ in a ball. (This is, in fact, a numerical representation in MatLab of the corresponding setting.)

Theorems & Definitions (31)

  • Theorem 1.1
  • Lemma 1.2
  • Proposition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 2.1: ERW
  • Lemma 2.2
  • Corollary 2.3
  • proof
  • ...and 21 more