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This will be true for the majority of examples studied in this book. 8 is carried out. This proof is intended to introduce—at an easy example—two important techniques, namely the localization technique and the technique of local tangential translations in arguments of the employed functions, which we call translation method or shift technique. These techniques are also needed in most of the problems studied later. 4 for more general cases). This norm essentially represents a higher-order Sobolev norm (the À « -norm) by taking a lowerorder norm (the Ä2 -norm) of a diﬀerence of shifted function and non-shifted function to some power, multiplying the result by another power of the shift parameter and integrating with respect to .
30) be valid. Then the contact problem with Coulomb friction has a solution. 1. 0 Geometry of domains ¨ © We now give a more detailed description of the geometrical properties of the boundary of a domain ª and its relation to the deﬁnition of norms. Let Æ 1 ( ) := Ü ¾ ÊÆ 1 ; Ü denote the (Æ 1)-dimensional ball ÜÆ 1 ) . A bounded domain of radius and for Ü ¾ ÊÆ let Ü¼ (Ü1 ª is said to have a Lipschitz boundary or to be of the class 0 1 , if for any Ü0 ¾ there exists a transformation Ü0 composed of a rotation and a (Ü¼ ÜÆ Ü0 (Ü¼ )) translation, taking Ü0 to the origin, and a map ©Ü0 : Ü Ç 36 Chapter 1.
For ( 1) simplicity of the presentation we assume ÙØ = , as it is satisﬁed in the ﬁrst step of the time discretization for initial datum (Ù0 )Ø = . This restriction is not necessary for the mathematical analysis to be carried out, ( 1) but the consideration of a non-zero ÙØ = 0 leads to an enlargement of already large formulae without any substantial mathematical insight. All the proofs can be modiﬁed to include also the case of a non-zero, suﬃciently ( 1) . smooth ÙØ In order to derive the variational formulation of the problem it is necessary to ﬁnd a variational formulation for the non-linear contact and friction boundary conditions.
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