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On convex comparison for exterior Bernoulli problems with discontinuous anisotropy

William M Feldman, Norbert Pozar

Abstract

We give a new proof of a convex comparison principle for exterior Bernoulli free boundary problems with discontinuous anisotropy.

On convex comparison for exterior Bernoulli problems with discontinuous anisotropy

Abstract

We give a new proof of a convex comparison principle for exterior Bernoulli free boundary problems with discontinuous anisotropy.
Paper Structure (9 sections, 11 theorems, 23 equations)

This paper contains 9 sections, 11 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Convexity properties
  3. Extreme and exposed points of convex sets
  4. Convexity properties of a gradient of a harmonic function
  5. Comparison principle
  6. Outer regular points have a classical normal derivative
  7. Comparison principle: convex supersolution
  8. Comparison principle: convex subsolution
  9. Existence and uniqueness of a quasi-convex solution

Key Result

Theorem 1.4

A supersolution of (e.minsup) is minimal if and only if it satisfies the weak subsolution property Definition d.weaksub. In other words, a supersolution that is also a weak subsolution is the unique minimal supersolution. Furthermore the minimal supersolution has convex superlevel sets.

Theorems & Definitions (23)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Lemma 2.1: Rockafellar
  • Theorem 2.2: Straszewicz's TheoremRockafellar
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 3.1
  • ...and 13 more