Weighted anisotropic Korn's inequality for a junction of a plate and a rod

Korn's inequality is proved for an elastic body obtained by attaching to a plate several rods with clamped farther ends. The thickness of the plate and the diameters of the rods are characterized by a single small parameter $h$, which also gauges the distinctions in the elastic properties of the elements of the junction. The selection of the weighted anisotropic norms distinguishing the longitudinal and transverse directions in the plate and in a rod ensures the asymptotic accuracy of the inequality, which is substantiated by examples of particular constructions. New results on single plates and rods are obtained in the course of the proof.