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Abstract. Let Ω be a domain in R n and p ∈ (1, ∞). We consider the (generalized) Hardy inequality Ω |∇u| p ≥ K Ω |u/δ| p , where δ(x) = dist (x, ∂Ω). The inequality is valid for a large family of domains, including all bounded domains with Lipschitz boundary. We here explore the connection between the value of the Hardy constantΩ |∇u| p / Ω |u/δ| p and the existence of a minimizer for this Rayleigh quotient. It is shown that for all smooth n-dimensional domains, µp(Ω) ≤ cp, where cp = (1 − 1 p ) p is the one-dimensional Hardy constant. Moreover it is shown that µp(Ω) = cp for all those domains not possessing a minimizer for the above Rayleigh quotient. Finally, for p = 2, it is proved that µ 2 (Ω) < c 2 = 1/4 if and only if the Rayleigh quotient possesses a minimizer. Examples show that strict inequality may occur even for bounded smooth domains, but µp = cp for convex domains.

We study the generalized boundary value problem for nonnegative solutions of of −∆u + g(u) = 0 in a bounded Lipschitz domain Ω, when g is continuous and nondecreasing. Using the harmonic measure of Ω, we define a trace in the class of outer regular Borel measures. We amphasize the case where g(u) = |u| q−1 u, q > 1. When Ω is (locally) a cone with vertex y, we prove sharp results of removability and characterization of singular behavior. In the general case, assuming that Ω possesses a tangent cone at every boundary point and q is subcritical, we prove an existence and uniqueness result for positive solutions with arbitrary boundary trace.1991 Mathematics Subject Classification. 35K60; 31A20; 31C15; 44A25; 46E35.

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L'accès aux archives de la revue « Annales de l'I. H. P., section C » (http://www.elsevier.com/locate/anihpc) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/