Analysis & PDE

Boundary behavior of solutions to the parabolic $p$-Laplace equation

Benny Avelin, Tuomo Kuusi, and Kaj Nyström

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We establish boundary estimates for nonnegative solutions to the p-parabolic equation in the degenerate range p>2. Our main results include new parabolic intrinsic Harnack chains in cylindrical NTA domains together with sharp boundary decay estimates. If the underlying domain is C1,1-regular, we establish a relatively complete theory of the boundary behavior, including boundary Harnack principles and Hölder continuity of the ratios of two solutions, as well as fine properties of associated boundary measures. There is an intrinsic waiting-time phenomenon present which plays a fundamental role throughout the paper. In particular, conditions on these waiting times rule out well-known examples of explicit solutions violating the boundary Harnack principle.

Article information

Anal. PDE, Volume 12, Number 1 (2019), 1-42.

Received: 28 October 2015
Accepted: 10 April 2018
First available in Project Euclid: 16 August 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K20: Initial-boundary value problems for second-order parabolic equations
Secondary: 35K65: Degenerate parabolic equations 35B65: Smoothness and regularity of solutions

$p$-parabolic equation degenerate intrinsic geometry waiting time phenomenon intrinsic Harnack chains boundary Harnack principle $p$-stability


Avelin, Benny; Kuusi, Tuomo; Nyström, Kaj. Boundary behavior of solutions to the parabolic $p$-Laplace equation. Anal. PDE 12 (2019), no. 1, 1--42. doi:10.2140/apde.2019.12.1.

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