Domain with a smooth boundary

We start with notations and preliminary remarks. Let C i be a bounded domain with a smooth boundary L having an exterior unit normal vector n = (ni,n2,n3). 

In the sequel we consider different functional spaces. To simplify the notation we write H Q) instead of [77 (12)] and so on. 

In the space (12) we shall consider different equivalent norms, in particular 

This means that in one can consider the equivalent norm 

Introduce some additional notations which are useful in the sequel. Consider the space 

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Let Q C he a bounded domain with a smooth boundary j. An external normal to 7 is denoted by n = (ni,ri2). Introduce the following operators defined at 7 by... 

In this section the existence of a solution to the three-dimensional elastoplastic problem with the Prandtl-Reuss constitutive law and the Neumann boundary conditions is obtained. The proof is based on a suitable combination of the parabolic regularization of equations and the penalty method for the elastoplastic yield condition. The method is applied in the case of the domain with a smooth boundary as well as in the case of an interior two-dimensional crack. It is shown that the weak solutions to the elastoplastic problem satisfying the variational inequality meet all boundary conditions. The results of this section can be found in (Khludnev, Sokolowski, 1998a). 

Again, let c i be a bounded domain with a smooth boundary T and Tc C H be a smooth orientable two-dimensional surface with a regular boundary. We assume that Tc can be extended in such a way that the domain fl is divided into two parts with Lipschitz boundaries. The surface Tc can be described parametrically... 

Let c be a bounded domain with a smooth boundary F, and Fc C be a smooth curve without selfintersections. Assume that Fc contains... 

Remark 2.1. The estimate above for the rate of decay at infinity holds for an arbitrary system of N bodies (w, du>i, 1 < i < N) confined to a bounded convex domain with a smooth boundary dQ. Indeed let... 

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