Bounding domain

Let a solid occupy a bounded domain fl c with the smooth boundary r (see Fig.1.1). Let fl contain a smooth unclosed surface Fc, probably intersecting F. We assume that Fc is an oriented surface such that there exists a mapping... 

Firstly, let us formulate an auxiliary statement concerning boundary values for the vector-functions having square integrable divergence (Baiocchi, Capelo, 1984 Temam, 1979). Consider a bounded domain H c i . Introduce the Hilbert space... 

Let C i be a bounded domain with a smooth boundary L, and n = (ni,n2,n3) be a unit outward normal vector to L. Introduce the stress and strain tensors of linear elasticity (see Section 1.1.1),... 

Now we consider a two-dimensional solid occupying a bounded domain fl C with a smooth boundary T. Let the bilinear form B be introduced by the formula... 

Here H C is a bounded domain with boundary F, and n is the unit exterior normal to F. The results of this section can be found in (Khludneva, 1990a). 

Let C be a bounded domain with smooth boundary T, <3 = x (0, T). Our object is to study a contact problem for a plate under creep conditions (see Khludneva, 1990b). The formulation of the problem is as follows. In the domain Q, it is required to find functions w, Mij, i,j = 1,2, satisfying the relations... 

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... 

Let a plate occupy a bounded domain fl c with smooth boundary F. Inside fl there is a graph Fc of a sufficiently smooth function. The graph Fc corresponds to the crack in the plate (see Section 1.1.7). A unit vector n = being normal to Fc defines the surfaces of the crack. 

Let (9 C be a bounded domain with smooth boundary 7 and outward normal n = (ni,ri2). We introduce the following notation for the bending moment and transverse forces on 7 ... 

A thin isotropic homogeneous plate is assumed to occupy a bounded domain C with the smooth boundary T. The crack Tc inside 0 is described by a sufficiently smooth function. The chosen direction of the normal n = to Tc defines positive T+ and negative T crack faces. 

To estimate the third-order derivatives of the function w with respect to y, we make use of the following fact (see Duvaut, Lions, 1972). Let O d E be a bounded domain with smooth boundary and let u be a distribution on O such that u, Du G Then u G L 0) and there is a constant c,... 

Let the mid-surface of the Kirchhoff-Love plate occupy a domain flc = fl Tc, where C is a bounded domain with the smooth boundary T, and Tc is the smooth curve without self-intersections recumbent in fl (see Fig.3.4). The mid-surface of the plate is in the plane z = 0. Coordinate system (xi,X2,z) is assumed to be Descartes and orthogonal, x = xi,X2)-... 

Let C be a bounded domain with the smooth boundary L, which has an inside smooth curve Lc without self-intersections. We denote flc = fl Tc. Let n = (ni,ri2) be a unit normal vector at L, and n = ( 1,1 2) be a unit normal vector at Lc, which defines a positive and a negative surface of the crack. We assume that there exists a closed continuation S of Lc dividing fl into two domains the domain fl with the outside normal n at S, and the domain 12+ with the outside normal —n at S (see Section 1.4). By doing so, for a smooth function w in flc, we define the traces of w at boundaries 912+ and, in particular, the traces w+ and the jump [w] = w+ — w at Lc. Let us consider the bilinear form... 

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... 

Assume that D C is a bounded domain with a Lipschitz boundary 7, and 7o is a curve being a part of 7 such that the length of 70 is positive. Denote by the subspace of the space IFf(D) consisting of all... 

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

Let CO be a finite set of nodes (a grid) in some bounded domain of the n-dimensional Euclidean space and let P G co be a point of the grid u>. Consider the equation... 

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