Sobolev inequalities for the Hardy–Schrödinger operator: extremals and critical dimensions

Nassif Ghoussoub (1) , Frédéric Robert (2)

1 UBC - University of British Columbia
2 IECL - Institut Élie Cartan de Lorraine

In this expository paper, we consider the Hardy-Schr\"odinger operator $-\Delta -\gamma/|x|^2$ on a smooth domain \Omega of R^n with 0\in\bar{\Omega}, and describe how the location of the singularity 0, be it in the interior of \Omega or on its boundary, affects its analytical properties. We compare the two settings by considering the optimal Hardy, Sobolev, and the Caffarelli-Kohn-Nirenberg inequalities. The latter rewrites: $C(\int_{\Omega}\frac{u^{p}}{|x|^s}dx)^{\frac{2}{p}}\leq \int_{\Omega} |\nabla u|^2dx-\gamma \int_{\Omega}\frac{u^2}{|x|^2}dx$ for all $u\in H^1_0(\Omega)$, where \gamma

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]; Géométrie différentielle [math.DG]

Nassif Ghoussoub, Frédéric Robert. Sobolev inequalities for the Hardy–Schrödinger operator: extremals and critical dimensions. Bulletin of Mathematical Sciences, 2016, 6 (1), pp.89-144. ⟨10.1007/s13373-015-0075-9⟩. ⟨hal-01165334⟩