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On the weak Harnack estimate for nonlocal equations

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Abstract

We prove a weak Harnack estimate for a class of doubly nonlinear nonlocal equations modelled on the nonlocal Trudinger equation

$$\begin{aligned} \partial _t(|u|^{p-2}u) + (-\Delta _p)^s u = 0 \end{aligned}$$

for \(p\in (1,\infty )\) and \(s \in (0,1)\). 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.

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Acknowledgements

The author would like to thank Karthik Adimurthi, Agnid Banerjee and Vivek Tewary for helpful comments and suggestions. The author was supported by the Department of Atomic Energy, Government of India, under project no. 12-R &D-TFR-5.01-0520.

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  1. Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Bangalore, Karnataka, 560065, India

    Harsh Prasad

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Prasad, H. On the weak Harnack estimate for nonlocal equations. Calc. Var. 63, 67 (2024). https://doi.org/10.1007/s00526-024-02670-3

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