A multi-point maximum principle to prove the global Harnack inequality for the doubly nonlinear heat equation
Abstract
This work aims to present a new proof of the global pointwise Harnack inequality satisfied by positive viscosity solutions of the doubly nonlinear heat equation on ℝd. Our approach is based on a multi-point maximum principle argument introduced by the authors in a recent article [Calc. Var., 2026]. Required for the proof is a new variation of the maximum principle for semicontinuous functions, which is a standard tool in the analysis of viscosity solutions. In particular, our techniques provide a new proof of the Harnack inequalities satisfied by positive solutions of the porous medium equation and the p-heat equation, which were derived by Auchmuty and Bao [Proc. Amer. Math. Soc., 1994] using results of Aronson and Bénilan [C. R. Acad. Sci. Paris, 1979] and Esteban and Vázquez [Arch. Rational Mech. Anal., 1988]. The results contained in this article have also been discussed in the third author's doctoral thesis.
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BibTeX
@misc{andrews2026multipoint,
title = {{A multi-point maximum principle to prove the global Harnack inequality for the doubly nonlinear heat equation}},
author = {Ben Andrews and Daniel Hauer and Jessica Slegers},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21732},
eprint = {2610.21732},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21732},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - A multi-point maximum principle to prove the global Harnack inequality for the doubly nonlinear heat equation
AU - Ben Andrews
AU - Daniel Hauer
AU - Jessica Slegers
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21732
UR - https://arxiv.org/abs/2610.21732
AB - This work aims to present a new proof of the global pointwise Harnack inequality satisfied by positive viscosity solutions of the doubly nonlinear heat equation on ℝ^d. Our approach is based on a multi-point maximum principle argument introduced by the authors in a recent article [Calc. Var., 2026]. Required for the proof is a new variation of the maximum principle for semicontinuous functions, which is a standard tool in the analysis of viscosity solutions. In particular, our techniques provide a new proof of the Harnack inequalities satisfied by positive solutions of the porous medium equation and the p-heat equation, which were derived by Auchmuty and Bao [Proc. Amer. Math. Soc., 1994] using results of Aronson and Bénilan [C. R. Acad. Sci. Paris, 1979] and Esteban and Vázquez [Arch. Rational Mech. Anal., 1988]. The results contained in this article have also been discussed in the third author's doctoral thesis.
ER -