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The parabolic Harnack inequality for nonlocal equations

Moritz Kassmann, Marvin Weidner

Abstract

We complete the local regularity program for weak solutions to linear parabolic nonlocal equations with bounded measurable coefficients. Within the variational framework we prove the parabolic Harnack inequality and Hölder regularity estimates. We discuss in detail the shortcomings of previous results in this direction. The key element of our approach is a fine study of the nonlocal tail term.

The parabolic Harnack inequality for nonlocal equations

Abstract

We complete the local regularity program for weak solutions to linear parabolic nonlocal equations with bounded measurable coefficients. Within the variational framework we prove the parabolic Harnack inequality and Hölder regularity estimates. We discuss in detail the shortcomings of previous results in this direction. The key element of our approach is a fine study of the nonlocal tail term.
Paper Structure (12 sections, 15 theorems, 93 equations)

This paper contains 12 sections, 15 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of local boundedness with $L^1$-tail
  4. Proof via localization argument
  5. Proof via modified level set truncation in the De Giorgi iteration
  6. Proof of weak Harnack inequalities with $L^1$-tail
  7. Proof of the main results
  8. Full Harnack inequalities
  9. Hölder regularity estimates
  10. Extension of our techniques
  11. Full parabolic Harnack inequalities under weaker assumptions
  12. Harnack-type inequalities for singular nonlocal operators

Key Result

theorem 1

Assume that $K$ satisfies eq:Kcomp. Then, for every $R > 0$, $t_0 \in I$, $x_0 \in \Omega$ with $I_{4R}(t_0) \times B_{4R}(x_0) \subset I \times \Omega$, and any globally nonnegative solution $u$ to $\partial_t u - L_t u = 0$ in $I \times \Omega$ it holds Here, $c = c(d,\alpha_0,\lambda,\Lambda) > 0$ is a constant.

Theorems & Definitions (41)

  • theorem 1: full Harnack inequality
  • remark 2
  • remark 3
  • corollary 4: full Harnack inequality with tail
  • theorem 5: parabolic Hölder regularity
  • remark 6
  • remark 7
  • theorem 8: local boundedness with $L^1$-tail
  • theorem 9: weak Harnack inequality with $L^1$-tail
  • remark 10: Next steps
  • ...and 31 more