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An estimate for the distance of a complex valued Sobolev function defined on the unit disc to the class of holomorphic functions

Fuchs M.

Journal of Applied Analysis

2011

Vol. 17, nr 1

131-135

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.

Weak solutions to anti-plane boundary value problems in a linear theory of elasticity with microstructure

Shmoylova E., Potapenko S., Dorfmann A.

Archives of Mechanics

2007

Vol. 59, nr 6

519-539

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.