On well-posedness of the first boundary-value problem within linear isotropic Toupin-Mindlin strain gradient elasticity and constraints for elastic moduli

Within the linear Toupin-Mindlin strain gradient elasticity we discuss the well-posedness of the first boundary-value problem, that is, a boundary-value problem with Dirichlet-type boundary conditions on the whole boundary. For an isotropic material we formulate the necessary and sufficient conditions which guarantee existence and uniqueness of a weak solution. These conditions include strong ellipticity written in terms of higher-order elastic moduli and two inequalities for the Lame moduli. The conditions are less restrictive than those followed from the positive definiteness of the deformation energy.