Dilation invariant estimates and the boundary Gårding inequality for higher order elliptic operators
type: Item
instance of: scholarly article
title: Dilation invariant estimates and the boundary Gårding inequality for higher order elliptic operators
zbMATH Open document ID: 0878.35035
DOI: 10.2307/2118610
author: Jill Pipher
author: Gregory Verchota
published in: Annals of Mathematics. Second Series
publication date: 1995-09-06T00:00:00Z
review text: The Dirichlet problem for \(Lu=0\) with data in \(L^2(\partial \Omega,d \sigma)\) or the Sobolev space \(L^2_1(\partial \Omega,d \sigma)\) is solved for all real homogeneous constant coefficient elliptic operators \(L=\sum_{|\alpha|=2m} a_\alpha D^\alpha\), \(m\in \mathbb{N}\), in bounded Lipschitz domains \(\Omega\subseteq \mathbb{R}^n\). Then, a perturbation technique due to Dahlberg and Kenig immediately implies solvability in \(L^p\) and \(L^p_1\) for \(2-\varepsilon <p<2+\varepsilon\) where \(\varepsilon>0\) depends on the Lipschitz character of \(\Omega\), \(m,n\), and the ellipticity constant associated with \(L\). The central question we are addressing here is what exactly does ellipticity (in the homogeneous case) imply about dilation invariant estimates near the boundary for solutions to the Dirichlet problem. Our Theorems provide a complete answer in the following sense. For any \(p\neq 2\) it is possible to find an \(m\), an \(n\), and a bounded Lipschitz domain in \(\mathbb{R}^n\) so that the Dirichlet problem is not solvable for data in \(L^p\) (and \(L^p_1)\) for \(L\) of order \(2m\). However, for a given \(m\) or \(n\) the situation is more complicated. All of the previous methods for solving boundary value problems for higher order operators on Lipschitz (or \({\mathcal C}^1)\) boundaries relied on special algebraic properties of the operators in question, namely that they were powers of second order operators. In this paper our main contribution is to overcome the difficulties in relaxing this assumption. Our approach is to obtain a priori dilation invariant estimates for known solutions in smooth domains and then to obtain the same estimates together with point-wise limits at the boundary in Lipschitz domains by an approximation argument. The two main tools used to do this are some new Rellich inequalities and the representation of solutions by means of layer potentials.
Mathematics Subject Classification ID: 35J40
Mathematics Subject Classification ID: 35C15
zbMATH DE Number: 784170
zbMATH Keywords: \(L^ p\)-Dirichlet problem
zbMATH Keywords: a priori dilation invariant estimates
zbMATH Keywords: Rellich inequalities
zbMATH Keywords: layer potentials
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2118610
OpenAlex ID: W2071790509
description: scientific article; zbMATH DE number 784170