Mathematical Study of the Boundary Integral Formulations of the Signorini-Fichera B.V.P.
Heinz Antes, Panagiotis D. Panagiotopoulos · Birkhäuser Basel eBooks · 1992
Let Ω ⊂ ℝ3 be an open bounded set with a Lipschitz boundary Γ, decomposed into three mutually disjoint parts Γ1 Γ2 and Γ3, open in Γ and let mes Γ1 > 0, mes Γ3 > 0. Let H k (Ω), k ≥ 0 integer, be the classical Sobolev space and let V 0 be the subspace of [H 1(Ω)]3 defined by 8.1 $$ {V_0} = \{ v|v = \{ {v_i}\} ,{v_i} \in {H^1}(\Omega ),\gamma v = 0on{\Gamma _1}\} $$ where γυ denotes the trace of υ on Γ. The space of all traces of functions, belonging to V 0, is denoted by $$ \left[ {H^{1/2} (\tilde \Gamma )} \right]^3 $$ where $$ \tilde \Gamma = \Gamma _2 \cup \Gamma _3. $$ For the sake of simplicity the following notation will be used: We denote by $$ H^{1/2} (\tilde \Gamma ) $$ the space $$ \left[ {H^{1/2} (\tilde \Gamma )} \right]^3 $$ , by $$ H^{ - 1/2} (\tilde \Gamma ) $$ the dual space to $$ \left[ {H^{1/2} (\tilde \Gamma )} \right]^3 $$ and by V 0′ the dual space of V 0. The corresponding norms are defined in the usual way [Nee] and will be denoted by ‖·‖ k , and ‖·‖1/2 whereas the dual norms are denoted by $$\parallel \cdot \parallel v'$$ , and ‖·‖−1/2 respectively. If on Γ1, u i = Ū i , then this case will be transformed to the homogeneous one by the transformation ̄v = v − sv 0 where $${v_{0i}}|{\Gamma _1} = {{\bar U}_i}$$ and ̅v ∈ V 0. Therefore this case will not be treated separately here. We assume now that Ω is occupied by a linear elastic body in its undeformed state. Then, the Signorini-Fichera B.V.P. is defined as in the previous Chapter by the eqs.(7.1) (with ̅U i = 0), (7.2), (1.80), (7.3)÷(7.6).