Let f : B →C denote a Sobolev function of class W1p defined on the unit disc. We show that the distance of f to the class of all holomorphic functions measured in the norm of the space W1p(B;C) is bounded by the Lp-norm of theWirtinger derivative ∂-zf. As a consequence we obtain a Korn type inequality for vector fields B →R2.
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In this paper we formulate the interior and exterior Dirichlet and Neumann boundary value problems of anti-plane rnicropolar elasticity in a weak (Sobolev) space setting, we show that these problems are well-posed and the corresponding weak solutions depend continuously on the data. We show that the problem of torsion of a rnicropolar beam of (non-smooth) arbitrary cross-section can be reduced to an interior Neumann boundary value problem in antiplane micropolar elasticity and consider an example which demonstrates the significance of material microstructure.
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