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Free boundary regularity in obstacle problems with a degenerate forcing term

Yong Liu

TL;DR

Addresses the obstacle problem with degenerate forcing term $f(x)=|x|$ in $B_1\subset\mathbb{R}^2$, i.e., $\Delta u=|x|\chi_{\{u>0\}}$, focusing on free boundary behavior at degenerate points. The authors extend Weiss's epiperimetric inequality to this setting, obtain a decay rate for the Weiss energy and deduce the uniqueness of blow-ups at the origin. Leveraging blow-up uniqueness, they prove a weak directional monotonicity and, at a regular point, establish the free boundary regularity by showing it is a $C^1$-graph in a small ball with a computable tangent. They also classify all degree-3 homogeneous global blow-ups and provide nondegeneracy and growth estimates that underpin the analysis, connecting the degenerate problem to the classical theory. Overall, the work broadens the applicability of energy-based free boundary techniques to obstacle problems with degenerate forcing and sets a framework for analogous problems.

Abstract

In this paper, we consider the properties of a special free boundary point in the following obstacle problem: The Laplacian of u equals f(x) multiplied by the characteristic function of the set where u is positive within the two-dimensional unit ball, where $f(x)=|x|$ is a degenerate forcing term. The key challenge stems from the degeneracy of $f(x)$, which leads to a more complex structure of the free boundary compared to the classical setting. To analyze it, we introduce the epiperimetric inequality developed by Weiss (Invent Math 138:23-50, 1999). Although this powerful tool was firstly introduced for the classical obstacle problem characterized by $f(x)>0$ in B_1, it also proves effective in our degenerate setting. This allows us to first obtain the decay rate of the Weiss energy for all blow-ups at the origin, which in turn implies the uniqueness of the blow-up profiles. With this uniqueness established, we then prove a very weak directional monotonicity properties satisfied by the solutions. This finally yields the regularity of the free boundary at the origin if the origin is a regular point.

Free boundary regularity in obstacle problems with a degenerate forcing term

TL;DR

Addresses the obstacle problem with degenerate forcing term in , i.e., , focusing on free boundary behavior at degenerate points. The authors extend Weiss's epiperimetric inequality to this setting, obtain a decay rate for the Weiss energy and deduce the uniqueness of blow-ups at the origin. Leveraging blow-up uniqueness, they prove a weak directional monotonicity and, at a regular point, establish the free boundary regularity by showing it is a -graph in a small ball with a computable tangent. They also classify all degree-3 homogeneous global blow-ups and provide nondegeneracy and growth estimates that underpin the analysis, connecting the degenerate problem to the classical theory. Overall, the work broadens the applicability of energy-based free boundary techniques to obstacle problems with degenerate forcing and sets a framework for analogous problems.

Abstract

In this paper, we consider the properties of a special free boundary point in the following obstacle problem: The Laplacian of u equals f(x) multiplied by the characteristic function of the set where u is positive within the two-dimensional unit ball, where is a degenerate forcing term. The key challenge stems from the degeneracy of , which leads to a more complex structure of the free boundary compared to the classical setting. To analyze it, we introduce the epiperimetric inequality developed by Weiss (Invent Math 138:23-50, 1999). Although this powerful tool was firstly introduced for the classical obstacle problem characterized by in B_1, it also proves effective in our degenerate setting. This allows us to first obtain the decay rate of the Weiss energy for all blow-ups at the origin, which in turn implies the uniqueness of the blow-up profiles. With this uniqueness established, we then prove a very weak directional monotonicity properties satisfied by the solutions. This finally yields the regularity of the free boundary at the origin if the origin is a regular point.
Paper Structure (1 section, 4 theorems, 21 equations)

This paper contains 1 section, 4 theorems, 21 equations.

Table of Contents

  1. Introduction

Key Result

Theorem 1.1

There exist $\kappa\in(0,1)$ and $\delta\in(0,1)$ such that the following holds for every non-negative function $c\in W^{1,2}(B_1)$ that is homogeneous of degree 3, if for some $h_{\theta_1}\in\mathbb{H}$ (defined in Definition Def3), then there exists a function $v\in W^{1,2}(B_1)$ satisfying $v=c$ on $\partial B_1$ and

Theorems & Definitions (6)

  • Theorem 1.1: Epiperimetric inequality
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5: Energy decay and uniqueness
  • Theorem 1.6