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Riesz potential estimates for mixed local-nonlocal problems with measure data

Iwona Chlebicka, Kyeong Song, Yeonghun Youn, Anna Zatorska-Goldstein

Abstract

We study gradient regularity for mixed local-nonlocal problems modelled upon \[ -Δ_p u +(-Δ_p)^su=μ\qquad\text{for} \quad 2-\tfrac{1}{n}<p<\infty\quad \text{and}\quad s\in(0,1)\,,\] where $μ$ is a bounded Borel measure. We prove pointwise bounds for the gradient $Du$ in terms of the truncated 1-Riesz potential of $μ$.

Riesz potential estimates for mixed local-nonlocal problems with measure data

Abstract

We study gradient regularity for mixed local-nonlocal problems modelled upon where is a bounded Borel measure. We prove pointwise bounds for the gradient in terms of the truncated 1-Riesz potential of .
Paper Structure (24 sections, 209 equations)

This paper contains 24 sections, 209 equations.

Table of Contents

  1. Introduction
  2. Mixed local-nonlocal operators
  3. Nonlinear potential theory
  4. Assumptions and results
  5. Gradient regularity as corollary
  6. Strategy and novelty
  7. Organization
  8. Preliminaries
  9. Notation
  10. Weak solutions and SOLA
  11. Regularity for auxiliary homogeneous equations
  12. Quantitative gradient Hölder regularity for homogeneous mixed equations
  13. Comparison between homogeneous mixed problem and homogeneous local problem
  14. Excess decay estimates for homogeneous mixed equations
  15. Excess decay estimates for measure data mixed problems
  16. ...and 9 more sections

Theorems & Definitions (13)

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