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# Download e-book for kindle: Annales Henri Poincaré - Volume 4 by Vincent Rivasseau (Chief Editor)

By Vincent Rivasseau (Chief Editor)

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Example text

2. 1. Let P be as above, let the operators B, E, A be deﬁned as in section 3. 8). Therefore, in order to prove that A is an isomorphism, it is enough to prove that ker A = ker A∗ = {0}. It follows from the maximum principle that B is an isomorphism as long as trg k = 0. 2, P is an isomorphism. In view of the fact that A is lower triangular, the isomorphism property now follows from the isomorphism property for B and P . 2 apply. It remains to prove the continuation principle. 1, it is enough to estimate CN X in terms of Λ[¯ g] and (trg k)−1 .

Paris 272, 386–388 (1968). [4] Y. Choquet-Bruhat, Solutions globales des ´equations de Maxwell-Dirac-KleinGordon (masses nulles). (French) C. R. Acad. Sci. Paris S`eries. , 292, no. 2, 153–158 (1981). [5] Y. Choquet-Bruhat and D. Christodoulou, Existence of global solutions of the Yang-Mills, Higgs and spinor ﬁeld equations in 3+1 dimensions, Ann. Scient. ´ Norm. , 4e s´erie, t. 14, 481–505 (1981). Ec. [6] D. Christodoulou, The problem of a self-graviting scalar ﬁeld, Comm. Math. , 105, 337–361 (1986).

1) 8π √ where φ = φ1 + iφ2 , i = −1 is a complex scalar ﬁeld, and Dµ φ = ∂µ φ + iAµ is the (gauge) covariant derivative, Fµν = ∂µ Aν − ∂ν Aµ . V (·) is a continuously diﬀerentiable function, further conditions on which will be imposed in the later sections. The corresponding energy-momentum-stress tensor Tµν is LMH = − Tµν = 1 1 Fµα Fνβ g αβ − gµν Fρα Fσβ g ρσ g αβ 4π 4 1 +Re{Dµ φ(Dν φ)∗ } − gµν g αβ Dα φ(Dβ φ)∗ − gµν V (|φ|). 2) The coupled Einstein and Maxwell-Higgs equation is 1 Rµν − gµν R = 8πTµν .