Vol. 12, No. 1, 2019

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Boundary behavior of solutions to the parabolic $p$-Laplace equation

Benny Avelin, Tuomo Kuusi and Kaj Nyström

Vol. 12 (2019), No. 1, 1–42
Abstract

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.

Keywords
$p$-parabolic equation, degenerate, intrinsic geometry, waiting time phenomenon, intrinsic Harnack chains, boundary Harnack principle, $p$-stability
Mathematical Subject Classification 2010
Primary: 35K20
Secondary: 35K65, 35B65
Milestones
Received: 28 October 2015
Accepted: 10 April 2018
Published: 2 August 2018
Authors
Benny Avelin
Department of Mathematics
Uppsala University
Uppsala
Sweden
Tuomo Kuusi
Department of Mathematical Sciences
University of Oulu
Oulu
Finland
Kaj Nyström
Department of Mathematics
Uppsala University
Uppsala
Sweden