On a vector version of a fundamental lemma of J. L. Lions
type: Item
instance of: scholarly article
title: On a vector version of a fundamental lemma of J. L. Lions
zbMATH Open document ID: 1398.46028
author: Philippe G. Ciarlet
author: Maria Mălin
author: Cristinel Mardare
published in: Chinese Annals of Mathematics. Series B
publication date: 2018-02-14T00:00:00Z
review text: The scalar version of the fundamental lemma of J. L. Lions says that, if \(\Omega\) is a bounded domain in \(\mathbb R^n\), \(m \geq 1\), and \(f \in \mathcal{D}^{\prime}(\Omega)\), \(\mathbf{grad} \;f \in H^{-m}(\Omega)\), then \(f \in H^{-m +1}(\Omega)\), where \(H^{-m}(\Omega)\) is the Sobolev space of (possibly) negative order. It has been generalized by \textit{C. Amrouche} et al. [J. Math. Pures Appl. (9) 86, No. 2, 116--132 (2006; Zbl 1115.35131)] to vectors by replacing \(\mathbf {grad}\) by the symmetrized gradient, \((\mathbf {grad}_s \mathbf{v})_{ij} = \frac12(\partial_jv_i + \partial_i v_j)\), \(\mathbf{v} = (v_1, v_2, \dots, v_n)\). The authors consider the vector version for \(m = 1:\mathbf{v} \in \mathcal{D}^{\prime}(\Omega)\), \(\mathbf{grad}_s \mathbf{v} \in \mathbf{H}^{-1}(\Omega)\), implies \(\mathbf{v} \in \mathbf{L}^2(\Omega)\). They prove that this is equivalent to 1. the weak Lions lemma, where it is assumed that \(\mathbf{v} \in \mathbf{H}^{-1}(\Omega)\), 2. a vector-valued Nečas inequality: there exists a constant \(C_0(\Omega)\) such that \[ || \mathbf{v} ||_{L^2(\Omega)} \leq C_0 \left( || \mathbf{v}||_{\mathbf{H}^{-1}(\Omega)} + ||\mathbf{\nabla}_s \mathbf{v}||_{\mathbf{H}^{-1}(\Omega)} \right), \] 3. the operator \(\mathbf{\nabla}_s \) has closed range, 4. the operator \( \mathbf {div}: \mathbf{H}^1_0(\Omega) \mapsto \mathbf{L}^2_0(\Omega)\) is onto, as well as two other conditions requiring a longer explanation. I refer to the paper for the weak approximation property and the weak Donati compatibility condition, which asserts the solvability of certain equations of the form \(\mathbf{\nabla}_s \mathbf{v} = \mathbf{e}\) under compatibility conditions provided that \(\mathbf{v} \in \mathcal{D}^{\prime}(\Omega)\). The authors conclude by showing the equivalence with a weak version of the weak Donati compatibility condition where one assumes \(\mathbf{v} \in \mathbf{H}^{-1}(\Omega)\), and with a weak version of the Saint-Venant compatibilit…
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 6838911
zbMATH Keywords: J. L. Lions lemma
zbMATH Keywords: Nečas inequality
zbMATH Keywords: Donati compatibility conditions
zbMATH Keywords: Saint-Venant compatibility conditions
reviewed by: Raymond Johnson
MaRDI profile type: Publication
full work available at URL: s11401-018-1049-5
OpenAlex ID: W2781679978
Wikidata QID: Q125030379
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DOI: 10.1007/S11401-018-1049-5
description: scientific article; zbMATH DE number 6838911