By Sommerfeld A.

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Assume a : V x V + JR is a continuous, V-elliptic bilinear form and e : V -+ R is a linear continuous functional. 25 Let V be a Hilbert space. c. on V and e : V -+ IR is a linear continuousfunctional. 28). Therefore, elliptic variational inequalities of the first kind can be viewed as special cases of elliptic variational inequalities of the second kind. 29) where j is a real valued functional. In this connection, one can also study the following mixed type variational inequality where the set K, the bilinear form a ( .

0 Each K is a triangle or quadrilateral with a nonempty interior K . For distinct K 1 ,K z E Ph, K 1 f l K 2 is either empty, or a common vertex, a common side or a common face of K l and K 2 . For any K E Ph, its diameter diam ( K ) 5 h. The assumption that the domain R is a polygon ensures that it can be partitioned into straight-sided triangles and quadrilaterals. 7 for a finite element mesh. For convenience in practical implementation as well as in theoretical analysis, it is assumed that there exist a finite number of fixed polyhedra, called reference elements or master elements, ambiguously represented by one symbol K ,such that for each element K , there is an invertible affine mapping function FK with K = FK ( K ).

For a vector v , we will use its normal component v, = v v and tangential component v, = v - v,v at a point on the boundary. Similarly for a tensor a E Sd, we define its normal component a, = a v . v and tangential component a, = a v - a,v. For a detailed treatment of traces for vector and tensor fields in contact problems and related spaces see [94] or [81]. ~ t = ~ 33 Preliminaries The material we consider here is linearly elastic. We denote by C : R x Sd -+ S d the elasticity tensor of the material.

### About the Production of the Continuous X-Ray Spectrum by Sommerfeld A.

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