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Partial regularity of the gradient for subsolutions

Aram Hakobyan, Michael Poghosyan, Henrik Shahgholian

Abstract

We prove that the gradient of any bounded subharmonic function is upper semi-continuous, provided that its super-level sets can be touched from the exterior by uniform $C^{1,\text{Dini}}$ domains at every point. This idea extends to a class of general operators, as well as to the boundary behaviour of the gradient of solutions of the Dirichlet problem in a domain whose boundary satisfy this geometric condition.

Partial regularity of the gradient for subsolutions

Abstract

We prove that the gradient of any bounded subharmonic function is upper semi-continuous, provided that its super-level sets can be touched from the exterior by uniform domains at every point. This idea extends to a class of general operators, as well as to the boundary behaviour of the gradient of solutions of the Dirichlet problem in a domain whose boundary satisfy this geometric condition.
Paper Structure (3 sections, 6 theorems, 69 equations)

This paper contains 3 sections, 6 theorems, 69 equations.

Table of Contents

  1. Rationale
  2. Main Results
  3. Application to the Dirichlet problem

Key Result

Theorem 2.1

Let $u$ be a bounded function satisfying $\Delta u \geq -1$ in $B_1$, and suppose the exterior touching property eq:norm holds for all points in $B_{3/4}$. Then $\nabla u$ is upper semi-continuous in $B_{1/2}$.

Theorems & Definitions (12)

  • Theorem 2.1
  • Corollary 2.2
  • proof
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof
  • ...and 2 more