An endpoint Littlewood-Paley inequality for BVP associated with the Laplacian on Lipschitz domains
type: Item
instance of: scholarly article
title: An endpoint Littlewood-Paley inequality for BVP associated with the Laplacian on Lipschitz domains
zbMATH Open document ID: 0959.35027
DOI: 10.5565/PUBLMAT_43299_09
author: Pascal Auscher
author: Philippe Tchamitchian
published in: Publicacions Matemàtiques
publication date: 2000-09-25T00:00:00Z
full work available at URL: https://eudml.org/doc/41381
review text: It was considered the heat equation \[ \frac {\partial u}{\partial t} - \Delta u=0\text{ in } \Omega \times \left(0,+\infty \right), \] \[ u\left(\cdot ,0\right) =0 \text{ in } \Omega , \] where \(\Delta \) denotes the Laplacian in \(\Omega \) with Dirichlet or Neumann boundary condition, \(u\) vanishes as \(t\) goes to \(\infty \) and \(\Omega \) is Lipschitz domain in \({\mathbb{R}}^n\). It was proved a commutator inequality of Littlewood-Paley type between partial derivatives and functions of the Laplacian on the Lipschitz domain, which gives interior energy estimates for some BVP: \[ \int\limits_{0}^{\infty }\int\limits_{\Omega }\left|c_{t}f\right|^2 \frac {dxdt}{t}\leq c\int\limits_{\Omega }\left|D_{x}f\right|^2dx, f\in V. \] Here on \(V=H^1_0\left(\Omega \right)\) for a Dirichlet boundary condition, \(V=H^1\left(\Omega \right)\) for a Neumann boundary condition, \(c_{t}=\left[D_{x},R_{t}\right], \left[A,B\right]=AB-BA, R_{t}=\left(1-t^2\Delta \right)^{-1}\). It can be seem as an endpoint inequality for a family of energy estimates. Besides, it was explained the result in the case of upper half-space. Then it was proved for a special Lipschitz domain and for bounded Lipschitz domains.
reviewed by: A. G. Shchelkov
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 1418173
zbMATH Keywords: commutator inequality of Littlewood-Paley type
zbMATH Keywords: energy estimates
zbMATH Keywords: Stefan problem
MaRDI profile type: Publication
OpenAlex ID: W2155189183
description: scientific article; zbMATH DE number 1418173