A note on local higher regularity in the dynamic linear relaxed micromorphic model

Abstract

We consider the regularity question of solutions for the dynamic initial-boundary value problem for the linear relaxed micromorphic model. This generalized continuum model couples a wave-type equation for the displacement with a generalized Maxwell-type wave equation for the microdistortion. Naturally, solutions are found in H 1 for the displacement u and H(Curl) for the microdistortion P. Using energy estimates for difference quotients, we improve this regularity. We show H 1 loc -regularity for the displacement field, H 1 loc -regularity for the microdistortion tensor P and that Curl P is H 1 -regular if the data are sufficiently smooth.

A local regularity result for the relaxed micromorphic model based on inner variations

“…The relaxed micromorphic model is a novel generalised continuum model allowing to describe size effects and band-gap behaviour of microstructured solids with effective equations ignoring the detailed microstructure [2,6,7,18,22,25]. As a micromorphic model it couples the classical displacement u : Ω ⊂ R 3 → R 3 with a non-symmetric tensor field P : Ω ⊂ R 3 → R 3×3 called the microdistortion through the variational problem Ω C e sym(∇u − P ), sym(∇u − P ) + C micro sym P, sym P + L c Curl P, Curl P − f, u − M, P dx −→ min w.r.t.…”

Homogenization of Complex Lattices for Metamaterials: Open Problems and Conjectures

“…The subsequent literature concerning prominent special cases is abundant (see, for example, recent results in references [7][8][9][10][11][12][13][14][15][16][17][18]). An open question is, however, whether we can have a unified view on material complexity.…”