On the Dirichlet boundary value problem for the normalized $p$-laplacian evolution

Agnid Banerjee; Nicola Garofalo

doi:10.3934/cpaa.2015.14.1

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2015, Volume 14, Issue 1: 1-21. Doi: 10.3934/cpaa.2015.14.1

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On the Dirichlet boundary value problem for the normalized $p$-laplacian evolution

Agnid Banerjee^1, and
Nicola Garofalo^2,

1.
Department of Mathematics, Purdue University, West Lafayette, IN 47907
2.
Dipartimento di Ingegneria Civile, Edile e Ambientale (DICEA), Università di Padova, 35131 Padova

Received: April 2014

Revised: June 2014

Published: January 2015

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Abstract

In this paper, we study the potential theoretic aspects of the normalized $p$-Laplacian evolution, see (1.1) below. A systematic study of such equation was recently started in [1], [4] and [25]. Via the classical Perron approach, we address the question of solvability of the Cauchy-Dirichlet problem with "very weak" assumptions on the boundary of the domain. The regular boundary points for the Dirichlet problem are characterized in terms of barriers. For $p \geq 2 $, in the case of space - time cylinder $G \times (0,T)$, we show that $(x,t) \in \partial G \times (0, T]$ is a regular boundary point if and only if $x \in \partial G$ is a a regular boundary point for the p-Laplacian. This latter operator is the steady state corresponding to the evolution (1.1) below. Consequently, when $p\geq 2$ the Cauchy- Dirichlet problem for (1.1) can be solved in cylinders whose section is regular for the $p$-Laplacian. This can be thought of as an analogue of the results obtained in [17] for the standard parabolic $p$-Laplacian div$(|Du|^{p-2}Du) - u_t = 0 $.

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Mathematics Subject Classification: Primary: 35J25; Secondary: 35J70.

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