Heat flow for \(p\)-harmonic maps with values in the circle
type: Item
instance of: scholarly article
title: Heat flow for \(p\)-harmonic maps with values in the circle
zbMATH Open document ID: 0979.35082
DOI: 10.1016/S0362-546X(98)00304-6
author: Patrick Courilleau
author: F. Demengel
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-02-21T00:00:00Z
review text: The authors investigate the initial-boundary value problem for the \(p\)-harmonic heat flow from the \(N\)-dimensional unit ball in \(\mathbb{R}^N\) into the unit circle \(S^1\subset \mathbb{R}^2\), \(p>1\). In explicit formulas, this problem asks for solutions \(u:[0,T]\times B^N \to S^1\) of the parabolic p. d. e. \[ \frac{\partial u}{\partial t}-\text{div}(|\nabla_x u|^{p-2}\nabla_x u)= |\nabla_x u|^p u \] with initial condition \(u(0, x) = u_0(x)\), \(x\in B^N\), and boundary condition \(u(t,x) =\Phi(x)\), \(t\geq 0\), \(x\in\partial B^N\). The authors prove the following result: given \(u_0\), \(\Phi\in W^{1,p}(B^N,S^1)\) with \(u_0 =\Phi\) on \(\partial B^N\), the above problem possesses a weak solution defined for all \(t > 0\) and such that \(\frac 1p\int|\nabla u(T,x)|^p dx+ \int^T_0 \int |\frac{\partial u}{\partial t}|^2dtdx\leq \frac 1p \int|\nabla u_0(x)|^p dx\). The authors show furthermore that, provided the initial datum \(u_0\) is weakly \(p\)-harmonic but not stationary with respect to variations of the independent variables, then the corresponding heat flow they construct is not constant in time. This shows that in this case the initial-boundary value problem has infinitely many solutions. The existence theorem is proved by a time-discretization scheme.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 1512502
zbMATH Keywords: infinitely many solutions
zbMATH Keywords: time-discretization scheme
reviewed by: F. Tomi
MaRDI profile type: Publication
full work available at URL: s0362-546x(98)00304-6
OpenAlex ID: W2090751519
description: scientific article; zbMATH DE number 1512502