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Porosity of the Free Boundary in a Class of Higher-Dimensional Elliptic Problems

Abdeslem Lyaghfouri

Abstract

We investigate a class of n-dimensional free boundary elliptic problems which includes the dam problem, the aluminum problem, and the lubrication problem. We establish that the free boundary in this class is a porous set, which implies its Hausdorff dimension being less than $n$, which in turn leads to its Lebesgue measure being zero. Our proof relies on the comparison of the solution with an appropriately constructed barrier function.

Porosity of the Free Boundary in a Class of Higher-Dimensional Elliptic Problems

Abstract

We investigate a class of n-dimensional free boundary elliptic problems which includes the dam problem, the aluminum problem, and the lubrication problem. We establish that the free boundary in this class is a porous set, which implies its Hausdorff dimension being less than , which in turn leads to its Lebesgue measure being zero. Our proof relies on the comparison of the solution with an appropriately constructed barrier function.
Paper Structure (3 sections, 11 theorems, 70 equations)

This paper contains 3 sections, 11 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. A Barrier Function
  3. Proof of the Main Results

Key Result

Theorem 1.1

The free boundary $(FB)$ is locally a porous set.

Theorems & Definitions (18)

  • Remark 1.1
  • Remark 1.2
  • Definition 1.1
  • Remark 1.3
  • Theorem 1.1
  • Corollary 1.1
  • Proposition 1.1
  • Remark 1.4
  • Proposition 1.2
  • Proposition 1.3
  • ...and 8 more