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Boundary behavior of
solutions to the parabolic $p$-Laplace equation
Benny Avelin, Tuomo Kuusi and Kaj Nyström |
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Vol. 12 (2019), No. 1, 1–42
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Abstract |
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We establish boundary estimates for nonnegative solutions to the -parabolic equation in the degenerate range . Our main results include new parabolic intrinsic Harnack chains in cylindrical NTA domains together with sharp boundary decay estimates. If the underlying domain is -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
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Mathematical Subject Classification 2010
Primary: 35K20
Secondary: 35K65, 35B65
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Milestones
Received: 28 October 2015
Accepted: 10 April 2018
Published: 2 August 2018
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Authors |
| Benny Avelin | |
| Department of Mathematics Uppsala University Uppsala Sweden |
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| Tuomo Kuusi | |
| Department of Mathematical
Sciences University of Oulu Oulu Finland |
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| Kaj Nyström | |
| Department of Mathematics Uppsala University Uppsala Sweden |
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