On a vector version of a fundamental lemma of J. L. Lions
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 compatibility condition.
scientific article; zbMATH DE number 6838911
scientific article