Steklov eigenvalues for the Lamé operator in linear elasticity (Q2029431)

The paper deals with the Steklov eigenvalues for linearly elastic and isotropic materials described by the Lamé constitutive relation between the Cauchy stress and the infinitesimal strain tensor: \[ \operatorname{div}\boldsymbol{\sigma}(\mathbf{u})=0\,,\quad\mbox{on }\Omega\,,\quad\sigma(\mathbf{u})\mathbf{n}=pw\mathbf{u}\,,\quad\mbox{in }\partial\Omega\,, \] where \(\mathbf{n}\) is the outward unit normal to \(\partial\Omega\), \(p\in L^{\infty}(\Omega)\) a strictly positive parameter and \[ \boldsymbol{\sigma}(\mathbf{u})=\mu(\nabla\mathbf{u}+\nabla^{T}\mathbf{u})+\lambda(\operatorname{div}\mathbf{u})\mathbf{I}\,, \] with \(\lambda\in\mathbb{R}\) and \(\mu>0\) the Lamé constants. The originality of the paper is that a Robin condition is prescribed on the whole boundary whereas, e.g., in [\textit{D. Gómez} et al., Z. Angew. Math. Phys. 69, No. 2, Paper No. 35, 23 p. (2018; Zbl 1402.35033)] the displacement is zero on a subset of the boundary. In order to establish the existence of a countable spectrum of this Steklov-Lamé eigenproblem, the author proves first of all a Korn's type inequality which naturally extends that given in [\textit{A. Damlamian}, Chin. Ann. Math., Ser. B 39, No. 2, 335--344 (2018; Zbl 1393.35237)], then shows that the obtained inequality still holds over the very large class of the Jones domains [\textit{P. W. Jones}, Acta Math. 147, 71--88 (1981; Zbl 0489.30017)]. Furthermore, by following [\textit{I. Babuška} and \textit{J. Osborn}, Handb. Numer. Anal. 2, 641--787 (1991; Zbl 0875.65087)], the author obtains an interesting and useful result, namely that any conforming Galerkin scheme provides a stable approximation to the true eigencouples \((w\,,\mathbf{u})\), provided that for any finite-dimensional subspace \(H_{h}\subset H^{1}(\Omega)\) the following assumption holds: \[ \lim_{h\rightarrow 0}\inf_{\mathbf{v}_{h}\in H_{h}}\|\mathbf{u}-\mathbf{v}_{h}\|_{1\,,\Omega}=0\,. \] The paper finishes with some well-presented numerical examples in two dimensions (unit square, unit circle and L-shaped domains) with tabulated value of the first seven eigenvalues and graphical representation of the first seven eigenfunctions, and in three dimensions with the tabulated value of the first seven eigenvalues on the unit cube.