Celebratio Mathematica

Y. Choquet-Bruhat, D. Christodoulou, and M. Francaviglia: “Cauchy data on a manifold,” Ann. Inst. H. Poincaré Sect. A (N.S.) 29 : 3 (1978), pp. 241–255. MR 519694 Zbl 0412.35018 article

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Y. Choquet-Bruhat, D. Christodoulou, and M. Francaviglia: “Problème de Cauchy sur une variété,” C. R. Acad. Sci. Paris Sér. A-B 287 : 5 (1978), pp. A373–A375. MR 524042 Zbl 0412.35017 article

Mauro Francaviglia	Related
Demetrios Christodoulou	Related

Y. Choquet-Bruhat, D. Christodoulou, and M. Francaviglia: “On the wave equation in curved spacetime,” Ann. Inst. H. Poincaré Sect. A (N.S.) 31 : 4 (1979), pp. 399–414. MR 574143 Zbl 0454.58016 article

Mauro Francaviglia	Related
Demetrios Christodoulou	Related

Y. Choquet-Bruhat and D. Christodoulou: “Systèmes elliptiques sur une variété euclidienne à l’infini,” C. R. Acad. Sci. Paris Sér. A-B 290 : 17 (1980), pp. A781–A785. MR 580565 Zbl 0453.58021 article

Y. Choquet-Bruhat and D. Christodoulou: “Elliptic systems in \( H_{s,\delta } \) spaces on manifolds which are Euclidean at infinity,” Acta Math. 146 : 1–2 (1981), pp. 129–150. MR 594629 article

In this paper we study elliptic differential systems of order \( m \) on non-compact manifolds which are euclidean at infinity, in weighted Sobolev spaces \( H_{s,\delta} \). Such a study has been done in weighted Hölder spaces \( C^{1,\alpha}_{\beta} \), for equations of order 2 in [Chaljub-Simon and Choquet-Bruhat 1978]. On the other hand, M. Cantor has proved [Cantor 1979] closed range and isomorphism theorems for elliptic operators of order \( m \) in \( \mathbf{R}^n \), in weighted Sobolev spaces \( W^{p}_{s,\delta} \), where \( p > n/(n-m) \). His paper is based on a work by L. Nirenberg and H. Walker [Nirenberg and Walker 1973] on the null spaces of such operators with continuous coefficients. In the present article we show that this restriction on \( p \) is unnecessary. Although we shall treat explicitly only the case \( p = 2 \) which is of special interest since \( W^2_{s,\delta}=H_{s,\delta} \) is a Hilbert space, the results extend trivially to any \( p > 1 \). The hypotheses on the coefficients which we make, permit the study of nonlinear systems in the same framework.

Y. Choquet-Bruhat and D. Christodoulou: “Existence de solutions globales des équations classiques des théories de jauge,” C. R. Acad. Sci. Paris Sér. I Math. 293 : 3 (1981), pp. 195–199. MR 635980 Zbl 0478.58027 article

Y. Choquet-Bruhat and D. Christodoulou: “Existence of global solutions of the Yang–Mills, Higgs and spinor field equations in \( 3+1 \) dimensions,” Ann. Sci. École Norm. Sup. (4) 14 : 4 (1981), pp. 481–506. MR 654209 Zbl 0499.35076 article

Y. Choquet-Bruhat and D. Christodoulou: “Cauchy problem at past infinity for nonlinear equations in curved spacetime,” pp. 73–91 in Studies in applied mathematics. Edited by V. Guillemin. Adv. Math. Suppl. Stud. 8. Academic Press (New York), 1983. MR 759906 Zbl 0517.53028 incollection

The authors consider the Cauchy problem at past infinity for both linear and non-linear wave equations on a globally hyperbolic curved spacetime whose metric tends to be stationary at past infinity. The first five sections consider the problem of a linear wave equation with a source term, and the final two sections consider weakly coupled systems of non-linear wave equations. The last section applies these results to a general Yang–Mills system in the Lorentz gauge with a source on a curved spacetime. Contents include sections on definitions and hypotheses, function spaces, energy estimates, weighted function spaces, existence theorems for linear equations, quasi-linear wave equations, existence theorems.

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Yvonne Choquet-Bruhat

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