Area integral estimates for higher order elliptic equations and systems
type: Item
instance of: scholarly article
title: Area integral estimates for higher order elliptic equations and systems
zbMATH Open document ID: 0892.35053
author: Björn E. J. Dahlberg
author: C. E. Kenig
author: Jill Pipher
author: Gregory Verchota
publication date: 1997-12-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75269
review text: Let \(\Omega\) be a domain in \(\mathbb{R}^n\), and \(L\) an elliptic equation (or system). When one studies boundary value problems for \(L\) in \(\Omega\), and the given data are in some \(L^p\), the boundary is to be understood in the sense of nontangential \(L^p\) estimates. When the boundary \(\partial \Omega\) of \(\Omega\) is just Lipschitz, there are additional difficulties. In this situation, it is known that the nontangential maximal function gives a very precise control over the growth of solutions of the boundary value problem. But there is another approach, via the square function of solutions, which presents some advanges. The goal of this article is to show the \(L^p\) equivalence (for \(0<p <\infty)\) between the nontangential maximal function and the square function of a solution \(u\). Namely, let \(\Gamma(Q) = \Gamma_a (Q) =\{x\in \Omega,| X- Q| \leq(i+a) d(X,\partial \Omega)\}\) be the nontangential approach region when \(Q\in \partial \Omega\), \(a>0\) depends only on the Lipschitz constant of \(\Omega\). Then the maximal nontangential function \(N(u)\) is defined by \(N(u)(Q)= \sup_{x\in \Gamma(Q)} | u(X) |\). The square function for a function \(u\) in \(\Omega\) is defined by \[ S(u)(Q)= \left(\int_{\Gamma (Q)} {\bigl| \nabla u(X) \bigr|^2 \over | X- Q|^{n-2}} dX \right)^{1/2}. \] The elliptic systems considered by the authors, called elliptic symmetric \(K\)-systems are of the form \(Lu= \sum^K_{l=1} L^{ql}u^l\) for \(u= (u^1, \dots, u^K)\), where \(L^{kl}= \sum_{| \alpha| =|\beta |=m} D^\alpha a^{kl}_{\alpha \beta} D^\beta\) for \(m,k,l\) position integers, \(a^{kl}_{\alpha \beta}\) real constants, with \(L^{kl}= L^{lk}\), and the system is strongly elliptic, i.e. \(\text{Re} \sum^k_{k,l=1} L^{kl} (\xi) \zeta_k \overline \zeta_lz \geq| \xi |^{2w} |\zeta |^2\) for \(\zeta\in \mathbb{C}^n\), \(\xi\in \mathbb{R}^n) \). Because of the fact that the boundary of \(\Omega\) is not necessarily smooth, the authors have to use an adapted distance fun…
Mathematics Subject Classification ID: 35J40
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 35J55
zbMATH DE Number: 1092700
zbMATH Keywords: elliptic symmetric \(K\)-systems
zbMATH Keywords: nontangential maximal function
zbMATH Keywords: square function
zbMATH Keywords: tangential derivatives
reviewed by: Gheorghe Gussi
MaRDI profile type: Publication
OpenAlex ID: W1999424146
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DOI: 10.5802/AIF.1605
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1092700