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By Ruzhansky M., Wirth J. (eds.)

ISBN-10: 9814313165

ISBN-13: 9789814313162

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Here, we mention Dal Maso and Murat [6], Kozlov, Maz’ya and Movchan [8], Maz’ya, Nazarov and Plamenewskii [10], Ozawa [11], Ward and Keller [12]. We also mention the seminal paper of Ball [1] on nonlinear elastic cavitation. For more comments, see also [3]. Our main results in this sense are Theorems 1–3 and answer questions (j), (jj) in the spirit of [9]. We now consider case γm ∈ R by the following result of [2]. Theorem 1. Let γm ∈ R. Let a satisfy (4), (5). Let the superposition operator FGi which takes v ∈ C 0,α (∂Ωi , Rn ) to the function FGi [v] defined by FGi [v](x) ≡ Gi (x, v(x)) ∀x ∈ ∂Ωi , (8) map C 0,α (∂Ωi , Rn ) to itself and be real analytic.

2) G i dσ is invertible. If n = 2, we assume that the matrix I − 4π(ω+1) ∂Ωi If n ≥ 3, we assume that −G i satisfies assumptions (4), (5) on ∂Ωi . Then there exist ∈]0, 0 [ and a family {u( , ·)} ∈]0, [ such that u( , ·) belongs to C 1,α (clΩ( ), Rn ) and solves (7) for all ∈]0, [, and such that the family {u( , ·)} ∈]0, [ converges in clΩo \ {0} to u˜, and such that lim →0+ γ( ) u( , x) (log )δ2,n (17) v i (x) + = (1 − δ2,n )˜ δ2,n ω + 2 4π ω + 1 T (ω, D˜ v i )ν i dσ ∀x ∈ Rn \ Ωi . ∂Ωi Moreover, the following statements hold.

1 (see also §2 of Ref. ) Here ‘b’ stands for ‘body’ and ‘s’ stands for ‘small impurity’. We note that condition (1) in particular implies that Ωb and Ωs have no holes and that there exists a real number 0 such that 0 ∈]0, 1[ and clΩb ∩ ( clΩs ) = ∅ for all ∈]0, 0[ . (2) Then we denote by Ωe ( ) the exterior domain defined by Ωe ( ) ≡ Rn \ {clΩb ∪ ( clΩs )} ∀ ∈]0, 0 [. Next we introduce a function γ such that γ is defined from ]0, 0[ to [0, +∞[ and γ0 ≡ lim γ( ) ∈ [0, +∞[ . →0 (3) Now let f ∈ C 1,α (∂Ωs , Rn ).

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Progress in Analysis and Its Applications: Proceedings of the 7th International Isaac Congress by Ruzhansky M., Wirth J. (eds.)


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