Extremals for Hardy-Sobolev type inequalities: the influence of the curvature

Frédéric Robert (1)

1 LJAD - Laboratoire Jean Alexandre Dieudonné

We consider the optimal Hardy-Sobolev inequality on a smooth bounded domain of the Euclidean space. Roughly speaking, this inequality lies between the Hardy inequality and the Sobolev inequality. We address the questions of the value of the optimal constant and the existence of non-trivial extremals attached to this inequality. When the singularity of the Hardy part is located on the boundary of the domain, the geometry of the domain plays a crucial role: in particular, the convexity and the mean curvature are involved in these questions. The main difficulty to encounter is the possible bubbling phenomenon. We describe precisely this bubbling through refined concentration estimates. An offshot of these techniques allows us to provide general compactness properties for nonlinear equations, still under curvature conditions for the boundary of the domain.

Domaines: Equations aux dérivées partielles [math.AP]

Frédéric Robert. Extremals for Hardy-Sobolev type inequalities: the influence of the curvature . Analytic aspects of problems in Riemannian geometry: elliptic PDEs, solitons and computer imaging, Paul Baird, Rachid Begbaoui, Ali Fardoun, May 2005, Landéda, France. pp.1-15. ⟨hal-01279346⟩