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Interacting free boundaries in obstacle problems

Damião J. Araújo, Rafayel Teymurazyan

TL;DR

The paper analyzes obstacle problems with two interacting diffusion operators, notably the Laplacian and the infinity Laplacian, introducing the intrinsic free boundary via $|(u,v)|=u^{1/2}+v^{1/3}$. Using a built-in obstacle framework and viscosity solutions, it establishes existence (via Schaefer's fixed point) and derives optimal regularity, non-degeneracy, and Hausdorff-measure estimates for the free boundary. A central finding is a strong coupling between the two coordinates: near regular points of the coordinate function, the free boundary is analytic, singular points lie on a smooth manifold, and all uncoupled free boundary points are singular; moreover, the free boundary of the coordinate function $u$ coincides with that of an intrinsic blend of $(u,v)$. This yields a Caffarelli-type dichotomy and a robust regularity theory, complemented by explicit examples that demonstrate sharpness of the assumptions.

Abstract

We study obstacle problems governed by two distinct types of diffusion operators involving interacting free boundaries. We obtain a somewhat surprising coupling property, leading to a comprehensive analysis of the free boundary. More precisely, we show that near regular points of a coordinate function, the free boundary is analytic, whereas singular points lie on a smooth manifold. Additionally, we prove that uncoupled free boundary points are singular, indicating that regular points lie exclusively on the coupled free boundary. Furthermore, optimal regularity, non-degeneracy, and lower dimensional Hausdorff measure estimates are obtained. Explicit examples illustrate the sharpness of assumptions.

Interacting free boundaries in obstacle problems

TL;DR

The paper analyzes obstacle problems with two interacting diffusion operators, notably the Laplacian and the infinity Laplacian, introducing the intrinsic free boundary via . Using a built-in obstacle framework and viscosity solutions, it establishes existence (via Schaefer's fixed point) and derives optimal regularity, non-degeneracy, and Hausdorff-measure estimates for the free boundary. A central finding is a strong coupling between the two coordinates: near regular points of the coordinate function, the free boundary is analytic, singular points lie on a smooth manifold, and all uncoupled free boundary points are singular; moreover, the free boundary of the coordinate function coincides with that of an intrinsic blend of . This yields a Caffarelli-type dichotomy and a robust regularity theory, complemented by explicit examples that demonstrate sharpness of the assumptions.

Abstract

We study obstacle problems governed by two distinct types of diffusion operators involving interacting free boundaries. We obtain a somewhat surprising coupling property, leading to a comprehensive analysis of the free boundary. More precisely, we show that near regular points of a coordinate function, the free boundary is analytic, whereas singular points lie on a smooth manifold. Additionally, we prove that uncoupled free boundary points are singular, indicating that regular points lie exclusively on the coupled free boundary. Furthermore, optimal regularity, non-degeneracy, and lower dimensional Hausdorff measure estimates are obtained. Explicit examples illustrate the sharpness of assumptions.
Paper Structure (7 sections, 15 theorems, 104 equations, 2 figures)

This paper contains 7 sections, 15 theorems, 104 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Existence of solutions
  3. A built-in obstacle problem
  4. Coupling properties and regularity
  5. Non-degeneracy and consequences
  6. Free boundary regularity
  7. Examples and beyond

Key Result

Theorem 2.1

If $X$ is a Banach space, $T:X\to X$ is continuous and compact, and the set is bounded, then $T$ has a fixed point.

Figures (2)

  • Figure 1: An illustration of uncoupled free boundary points shaped in a singular fashion. All regular free boundary points are in the coupled free boundary.
  • Figure 2: An illustration of a non-empty uncoupled free boundary $\partial\{v>0\}\setminus \partial\{u>0\}$ (see Example \ref{['sinclusion']}). The red line above is $\partial\{v>0\}$ and the blue point is $\partial\{u>0\}$. By Theorem \ref{['singularset']}, all points on $\partial\{v>0\}\setminus \partial\{u>0\}$ are singular. In this example, there are no regular points.

Theorems & Definitions (37)

  • Theorem 2.1: Schaefer
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • ...and 27 more