CHAPTER 2 Notation and Preliminaries 2.1. Notation We assume that Ω R3 is a bounded domain with a connected C2-boundary. We fix R0 0 so that Ω ⊂⊂ BR 0 . We put E := R3\Ω. By the assumptions on Ω, E is an exterior domain. We will use BR to denote balls BR := {x Rn | |x| R} in Rn. We put ER := E BR for R R0. Moreover, we use the notation BR,r := BR \ Br. We let ei, i = 1, 2, 3, denote the standard basis vectors in R3. For x, y R3 we write x y to denote the vector product in R3. In expressions with multiple vector products, evaluation is always to be carried out from right to left. For example, we have x y z = x (y z). The tensor product x y R3×3 denotes the second order tensor (x y)ij = xiyj. The product of a second order tensor A R3×3 and first order tensor a R3 is defined as (Aa)i := 3 j=1 Aijaj, i = 1, 2, 3. The scalar product A : B of two second order tensors A, B R3×3 is defined as A : B := i,j=1,2,3 AijBij. In connection with products of tensors, we will typically make use of the Einstein summation convention and implicitly sum over all repeated indices. For example, we may write the previous definition as A : B := AijBij. By cof A we denote the co-factor matrix of A R3×3. Recall that cof A = det(A) A−T whenever A is invertible. We denote by Tr A := ∑3 i=1 Aii the trace of A. We use I to denote the identity tensor. For a differentiable vector field Φ : R3 R3 we define ∇Φ as the second order tensor field ∇Φ : R3 R3×3 given by (∇Φ)ij := ∂jΦi, i, j = 1, 2, 3. For a differentiable second order tensor field A : R3 R3×3 we define the divergence div A : R3 R3 as the vector field (div A)i := 3 j=1 ∂jAij, i = 1, 2, 3. We recall the Piola identity (2.1) div(cof ∇Φ) = 0 5
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