Korn's inequality for periodic solids and convergence rate of homogenization

Abstract

In a three-dimensional solid with arbitrary periodic Lipschitz perforation the Korn inequality is proved with a constant independent of the perforation size. The convergence rate of homogenization as a function of the Sobolev–Slobodetskii smoothness of data is also estimated. We improve foregoing results in elasticity dropping customary restrictions on the shape of the periodicity cell and superfluous smoothness and smallness assumptions on the external forces and traction.

Keywords:

• Korn's inequality
• homogenization
• convergence rate

AMS Subject Classifications::

• 74Q05
• 74B20

Notes

Notes

1. The periods π _i can differ from 3, 14 …

2. In 14 traction is prescribed over the entire exterior boundary ∂Ω while perforation does not occur in the ch-neighbourhood of the boundary. Since we assume that the surface (7) is clamped, perforation covers the whole solid.

3. To much the present geometrical construction with the example of a skyscraper, in Section 4.1, two of the doors must be welded onto the corresponding frame and can be called a ceiling and a floor while there may be windows among the other four doors.