Global regularity for a physically nonlinear version of the relaxed micromorphic model on Lipschitz domains

In this paper, we investigate the global higher regularity properties of weak solutions for a linear elliptic system coupled with a nonlinear Maxwell-type system defined on Lipschitz domains. The regularity result is established using a modified finite difference approach. These adjusted finite differences involve inner variations in conjunction with a Piola-type transformation to preserve the curl-structure within the matrix Maxwell system. The proposed method is further applied to the linear relaxed micromorphic model. As a result, for a physically nonlinear version of the relaxed micromorphic model, we demonstrate that for arbitrary ε > 0, the displacement vector u belongs to H3/2-ε(Ω), and the microdistortion tensor P belongs to H1/2-ε(Ω) while P belongs to H1/2-ε(Ω).