Square function estimates for the evolutionary p-Laplace equation

Kaj Nyström

doi:10.3934/dcds.2023044

Article Contents

2023, Volume 43, Issue 9: 3174-3212. Doi: 10.3934/dcds.2023044

This issue Previous Article Next Article Nonexpansive directions in the Jeandel-Rao Wang shift

Square function estimates for the evolutionary p-Laplace equation

Kaj Nyström

Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden

Received: September 2022

Revised: April 2023

Early access: May 10, 2023

Published online: May 10, 2023

The author was supported by grant 2022-03106 from the Swedish research council (VR).

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Abstract

We prove novel (local) square function/Carleson measure estimates for non-negative solutions to the evolutionary $ p $-Laplace equation in the complement of parabolic Ahlfors-David regular sets. In the case of the heat equation, the Laplace equation as well as the $ p $-Laplace equation, the corresponding square function estimates have proven fundamental in symmetry and inverse/free boundary type problems, and in particular in the study of (parabolic) uniform rectifiability. Though the implications of the square function estimates are less clear for the evolutionary $ p $-Laplace equation, mainly due its lack of homogeneity, we give some initial applications to parabolic uniform rectifiability, boundary behaviour and Fatou type theorems for $ \nabla_Xu $.

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Mathematics Subject Classification: Primary: 35K10, 35K92, 35R35; Secondary: 35N25.

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