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Proof. 22) hence v2 is well-defined. Similarly, v1 is well-defined for φ ∈ [γ1 , π/2 − ε1/3]. 17) we have 1 cos γ1 (ε1/2 ) ε ε Eε (uε , Aγ2 ,γ1 ) ≥ 2 (dW (p1 , p2 ) − 2ε1/2 ) 2π ε 2 2 1 (ε1/2 ) ≥ cos φ0 + αj (mj,ε − mj ) + O(ε ln ) (dW (pε1 , pε2 ) − 2ε1/2 ) ε ε j=1 (ε1/2 ) ≥ 2dW 2 (pε1 , pε2 ) 4 cos φ0 + O(| ln ε|) . 9) we deduce that Eε (uε , S 2 \ Aγ1 ,γ2 ) ≤ C ε1/2 . 24) γ2 γ2 Ψ2 (uε ) cos φ ≤ C 0 0 W (uε ) cos φ ≤ Cε2 Eε (uε , S 2 \ Aγ1 ,γ2 ) ≤ Cε2 ε−1/2 = Cε3/2 , so there exists θ2 ∈ (0, γ2 ) such that Ψ2 (uε (θ2 )) ≤ Cε3/2 .

12) is that for each η > 0 we have π meas {φ ∈ [φ0 + δ, − δ]} : |v1,ε (φ)| − m1 > η ≤ C(η)ε1/2 . e. on (φ0 , π2 ). Consider then a point φ˜ ∈ [φ0 + δ, π2 − δ] such that (possibly for a further subsequence) ˜ = limε →0 v1,ε (φ) ˜ = x1 ∈ M1 . 1). 19) between two distinct points of Γ1 is positive. The December 17, 2007 16:30 Proceedings Trim Size: 9in x 6in Itai1 42 above argument implies that if limεn →0 uεn (φn ) = limεn →0 v1,εn (φn ) = x, with {φn } ⊂ [φ0 +δ, π2 −δ], then x = x(1) . The claimed uniform convergence on [φ0 + δ, π2 − δ] follows.

2) does not admit nontrivial bounded solutions. Finally in the last section we give an almost sharp criterion for the existence of nontrivial solutions. 2. A preliminary estimate Let us denote by Ω a bounded open subset of Rk with Lipschitz boundary and starshaped with respect to the origin. For any r ∈ R we set Ωr = rΩ. 3) where c denotes some positive constant. 1. 1). 4) 1 2 , Ωr \Ωr/2 where | · | denotes the usual euclidean norm in Rk . Proof. 2) we have for every v ∈ H01 (Ωr ) A∇u · ∇v + auv dx = 0.

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Analysis of the conservation law for matter and gravitational field introduced by Einstein

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