Nous prouvons que la meilleure constante dans I'inCgalitC de Sobolev (HT C L,. avec 1 I-;;; = 1' r et 1 < p < n) est atteinte sur les variCtCs riemanniennes compactes, ou seulemen"t compktes sous certaines hypothbses. Nous Ctablissons aussi certaines irkgalitks plus fortes oh les normes sont BlevCes i? une puissance qui semble optimale. Pour la preuve nous montrons un r&hat g&&al de convergence dominCe en un point de concentration simple. 0 AcadCmie des SciencesMsevier, Paris On the best constant in the Sobolev inequality We prove that the best constant in the Sobolev inequality (Hy c L,* with 5 = 1 -i and 1 < p < n) is achieved on compact Riemannian manifolds, or only complete under some h.ypotheses. We also establish stronger inequalities where the norms are to some exponent which seems optimal. For the proof we show a general result of dominated convergence at a simple point qf concentration. 0 AcadCmie des Sciences/Elsevier,
Paris
Abridged
English Version On a complete Riemannian manifold (A4,,g) of dimension 7~ with injectivity radius greater than d > 0 and the Ricci curvature bounded from below, the Sobolev imbedding theorem holds: the inclusion Hy c L,* (with 5 = f -i and 1 5 y < n) is continuous. We can write the inequality in the form: there are constants C and A such that any cp E HT satisfies As the inclusion is not compact K = inf C such that A(C) exists is positive. It is proved [4] that K = K(n,p) depends only on 72 and p. A natural question arises: is the best constant achieved? Note pr&entCe par Thierry AUBIN.