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The boundary Harnack principle on optimal domains

Francesco Paolo Maiale, Giorgio Tortone, Bozhidar Velichkov

Abstract

We give a short and self-contained proof of the Boundary Harnack inequality for a class of domains satisfying some geometric conditions given in terms of a state function that behaves as the distance function to the boundary, is subharmonic inside the domain and satisfies some suitable estimates on the measure of its level sets. We also discuss the applications of this result to some shape optimization and free boundary problems.

The boundary Harnack principle on optimal domains

Abstract

We give a short and self-contained proof of the Boundary Harnack inequality for a class of domains satisfying some geometric conditions given in terms of a state function that behaves as the distance function to the boundary, is subharmonic inside the domain and satisfies some suitable estimates on the measure of its level sets. We also discuss the applications of this result to some shape optimization and free boundary problems.
Paper Structure (12 sections, 14 theorems, 116 equations)

This paper contains 12 sections, 14 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Harnack chains and interior Harnack inequalities
  3. Harnack chains and Harnack inequality close to the boundary
  4. Harnack chains and Harnack inequality away from the boundary
  5. Boundary Harnack Inequality: proof of \ref{['t:main2']}
  6. Step 1
  7. Step 2
  8. Step 3
  9. Hölder continuity up to the boundary. Proof of \ref{['t:main']}
  10. Applications
  11. The vectorial free boundary problem
  12. Subsolutions and supersolutions

Key Result

Theorem 1.2

Let $\Omega\subset B_1$ be an open set with $0\in\partial\Omega$ and $\phi:B_1\to\mathbb{R}$ a continuous function such that: Then the Boundary Harnack Principle holds in $\Omega$ in the sense of d:boundary-harnack.

Theorems & Definitions (30)

  • Definition 1.1: Boundary Harnack Principle
  • Theorem 1.2
  • Theorem 1.3: Boundary Harnack Inequality
  • Remark 1.4: Scale invariance
  • Remark 1.5: On the assumption \ref{['item:7-nondegeneracy']}
  • Remark 1.6
  • Lemma 2.1: Short Harnack chains close to the boundary
  • proof
  • Lemma 2.2: Interior Harnack inequality close to the boundary
  • proof
  • ...and 20 more