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Classification of global solutions to the obstacle problem in the plane

Anthony Salib, Georg Weiss

Abstract

Global solutions to the obstacle problem were first completely classified in two dimensions by Sakai using complex analysis techniques. Although the complex analysis approach produced a very succinct proof in two dimensions, it left the higher dimensional cases, and even closely related problems in two dimensions, unresolved. A complete classification in dimensions $n\geq 3$ was recently given by Eberle, Figalli and Weiss, forty years after Sakai published his proof. In this paper we give a proof of Sakai's classification result for unbounded coincidence sets in the spirit of the recent proof by Eberle, Figalli and Weiss. Our approach, in particular, avoids the need for complex analysis techniques and offers new perspectives on two-dimensional problems that complex analysis cannot address.

Classification of global solutions to the obstacle problem in the plane

Abstract

Global solutions to the obstacle problem were first completely classified in two dimensions by Sakai using complex analysis techniques. Although the complex analysis approach produced a very succinct proof in two dimensions, it left the higher dimensional cases, and even closely related problems in two dimensions, unresolved. A complete classification in dimensions was recently given by Eberle, Figalli and Weiss, forty years after Sakai published his proof. In this paper we give a proof of Sakai's classification result for unbounded coincidence sets in the spirit of the recent proof by Eberle, Figalli and Weiss. Our approach, in particular, avoids the need for complex analysis techniques and offers new perspectives on two-dimensional problems that complex analysis cannot address.
Paper Structure (15 sections, 39 theorems, 153 equations, 2 figures)

This paper contains 15 sections, 39 theorems, 153 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Notation and Preliminaries
  4. Notation
  5. Basic properties of global solutions
  6. Paraboloid solutions
  7. The thin obstacle problem
  8. Modified ACF functional
  9. Blow-down analysis
  10. Growth estimates for the coincidence set
  11. Sharp growth estimates of $u-p$
  12. Refined blow-down and matching
  13. Proof of Theorem \ref{['maintheorem']}
  14. Proof of Lemma \ref{['holderboundlemma']}
  15. Proof of Proposition \ref{['expansionthm']}

Key Result

Theorem 1.1

Let $n=2$ and let $u$ be a solution of obstacleproblem such that $\{u=0\}$ has non-empty interior. Then $\{u=0\}$ is either a half-plane, ellipse, parabola or a strip.

Figures (2)

  • Figure 1: Setting of the proof of Corollary \ref{['corollary:around:the:tip']}
  • Figure 2: The sliding of $\gamma P$ downwards produces at least three unbounded regions. The dashed lines represent their boundaries.

Theorems & Definitions (72)

  • Theorem 1.1: Sakai, 1981
  • Theorem 1.2: Eberle, Figalli and Weiss (2022)
  • Theorem 1.3
  • Lemma 2.1: Basic Properties
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4: Classification of blow-downs
  • Remark 2.5
  • Lemma 2.6: caffarelli2000regularity
  • Definition 2.7: $x_2$-monotone solutions
  • ...and 62 more