On The Weak Harnack Estimate For Nonlocal Equations

2023/06/05 by Prasad, Harsh
#35B65 #35K65 #35K67 #35K92 #35R11 #47G20 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2306.02933

Abstract

We prove a weak Harnack estimate for a class of doubly nonlinear nonlocal equations modelled on the nonlocal Trudinger equation ∂_t(|u|^p-2u) + (-Δ_p)^s u = 0 for p∈ (1,∞) and s ∈ (0,1). Our proof relies on expansion of positivity arguments developed by DiBenedetto, Gianazza and Vespri adapted to the nonlocal setup. Even in the linear case of the nonlocal heat equation and in the time-independent case of fractional p-Laplace equation, our approach provides an alternate route to Harnack estimates without using Moser iteration, log estimates or Krylov-Safanov covering arguments.

Related

• Weak Harnack estimates for a doubly nonlinear nonlocal p-Laplace equation
• Harnack estimates for the nonlocal Trudinger equation
• Harnack's estimate for a mixed local-nonlocal doubly nonlinear parabolic equation
• Nonlocal Harnack inequalities for nonlocal heat equations
• Weak Harnack inequality for a mixed local and nonlocal parabolic equation