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Regularity near the fixed boundary for transmission systems

Alessio Figalli, Somayeh Khademloo, Sunghan Kim, Henrik Shahgholian

Abstract

Given $Ω\subset \mathbb{R}^n$ with $n\geq 2$, $D\subset Ω$ open, and $u:Ω\to \mathbb{R}^m$, we study elliptic systems of the type $$ {\rm div} \big( ( A + (B- A)χ_D)\nabla u\big) = 0 \quad \text{in $Ω\cap B_1$,} $$ for some uniformly elliptic tensors $A$ and $B$ with Hölder continuous entries. We show that, given appropriate boundary data, the Lipschitz regularity of $u$ inside $B_1 \cap D$ is transmitted to $B_{1/2}\cap Ω$ up to the boundary of $Ω$. This corresponds to the boundary counterpart of the interior regularity results in Figalli-Kim-Shahgholian, Nonlinear Anal. 2022.

Regularity near the fixed boundary for transmission systems

Abstract

Given with , open, and , we study elliptic systems of the type for some uniformly elliptic tensors and with Hölder continuous entries. We show that, given appropriate boundary data, the Lipschitz regularity of inside is transmitted to up to the boundary of . This corresponds to the boundary counterpart of the interior regularity results in Figalli-Kim-Shahgholian, Nonlinear Anal. 2022.
Paper Structure (7 sections, 6 theorems, 51 equations)

This paper contains 7 sections, 6 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Standing assumptions and main results
  4. Reduction to the flat boundary case
  5. $BMO$-type estimate
  6. Decay of the density
  7. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $D\subset\Omega$ be an open set. Assume that $B_1\cap\partial\Omega$ is of class $C^{1,\sigma}$, and $g\in C^{1,\sigma}(B_1 \cap\partial\Omega;{\mathbb R}^m)$. Let ${\mathcal{A}},{\mathcal{B}}: B_1 \cap\Omega\to{\mathbb R}^{n^2m^2}$ satisfy eq:ellip and eq:A-Ca, and $u\in W^{1,2}(B_1 \cap\Omega; where $C$ depends only on $n$, $m$, $\lambda$, $K$, the $C^{1,\sigma}$-character of $B_1 \cap\parti

Theorems & Definitions (11)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 4.1
  • proof
  • proof : Proof of Proposition \ref{['prop:main']}
  • ...and 1 more