equilibrium e.q.

Hi, guys,

 Do some of you know how Periodical Boundary Condition is implemented in equilibrium equation while using FEM:

div{ C:sym[grad(u)] } = 0 ,where the displacement u is periodical.

Is there any (detailed) documentation about this stuff?

Thank you very much!

continuity e.q.,equilibrium e.q. ,N-S , constitutive e.q. for elastic solid and compressible fluid in the space-frequency domain, as shown in the following Fig.

i don't understand how can this author deduce? why v=iwu? is this assumption?  why is this necessary?    thanks for your help and guidance!