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Weak Harnack inequality and Cartan property for nonlocal $W^{s,1}$-minimizers

Panu Lahti, Yuxin Li, Khanh Nguyen

Abstract

We establish a weak Harnack inequality for nonlocal $W^{s,1}$-subminimizers in a complete, connected, doubling metric measure space where $0<s<1$. As a corollary, we prove that $W^{s,1}$-subminimizers are semicontinuous, up to a suitable choice of pointwise representative. We then prove \emph{Cartan-type properties} for $W^{s,1}$-superminimizers. The theory turns out to be mostly analogous with the local case of BV super- and subminimizers. Our results seem to be new even in the classical Euclidean setting.

Weak Harnack inequality and Cartan property for nonlocal $W^{s,1}$-minimizers

Abstract

We establish a weak Harnack inequality for nonlocal -subminimizers in a complete, connected, doubling metric measure space where . As a corollary, we prove that -subminimizers are semicontinuous, up to a suitable choice of pointwise representative. We then prove \emph{Cartan-type properties} for -superminimizers. The theory turns out to be mostly analogous with the local case of BV super- and subminimizers. Our results seem to be new even in the classical Euclidean setting.
Paper Structure (11 sections, 27 theorems, 200 equations)

This paper contains 11 sections, 27 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonlocal minimizers
  4. Doubling measure and Poincaré inequality
  5. Obstacle problem
  6. Capacity, thinness, and Hausdorff measure
  7. Proofs of main results
  8. Proof of Theorem \ref{['thm1']}
  9. Proof of Corollary \ref{['cor3.3-16oct']}
  10. Proof of Theorem \ref{['weakCartan-main']}
  11. Proof of Theorem \ref{['thm3.17-26nov']}

Key Result

Theorem 1.1

Let $(X,d,\mu)$ be a complete, connected metric measure space equipped with a doubling measure. Let $0<s<1$ and let $Q$ be a doubling dimension such that $Q>s$. Let $\Omega\subset X$ be a nonempty open set and let $u$ be an $s$-subminimizer in $\Omega$. Then there exists a constant $C>0$ depending o

Theorems & Definitions (50)

  • Theorem 1.1: Weak Harnack inequality
  • Corollary 1.2
  • Theorem 1.3: Weak Cartan property
  • Theorem 1.4: Strong Cartan property
  • Theorem 2.1: DLV23
  • Theorem 2.2: DLV23
  • Theorem 2.3: KLLZ25
  • Corollary 2.4
  • proof
  • Proposition 2.5
  • ...and 40 more