2015/06/18 by Nassif Ghoussoub, Ghoussoub, Nassif, Frédéric Robert +1 · 2 citations
Computer Science · Mathematics · #35B44 #35J35 #35J60 #58J05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP #msc:35B44 #msc:35J35 #msc:35J60 #msc:58J05
paper · pdf · doi:10.48550/arxiv.1506.05787
Expository paper. 48 pages
arxiv created 2015/06/18 · openalex publication_date 2015/06/18 · arxiv updated 2015/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this expository paper, we consider the Hardy-Schrödinger operator -Δ-γ/|x|2 on a smooth domain Ωof Rn with 0∈Ω, and describe how the location of the singularity 0, be it in the interior of Ωor 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(∫Ω\fracup|x|sdx)(2)/(p)≤ ∫Ω |∇ u|2dx-γ∫Ω(u2)/(|x|2)dx for all u∈ H10(Ω), where γ<n2/4, s∈ [0,2) and p:=2(n-s)/(n-2). We address questions regarding the explicit values of the optimal constant C, as well as the existence of non-trivial extremals attached to these inequalities. Scale invariance properties lead to situations where the best constants do not depend on the domain and are not attainable. We consider two different approaches to "break the homogeneity" of the problem: One approach was initiated by Brezis-Nirenberg and by Janelli. It is suitable for the case where 0 is in the interior of Ω, and consists of considering lower order perturbations of the critical nonlinearity. The other approach was initiated by Ghoussoub-Kang , C.S. Lin et al. and Ghoussoub-Robert. It consists of considering domains where the singularity is on the boundary. Both of these approaches are rich in structure and in challenging problems. If 0∈ Ω, a negative linear perturbation suffices for higher dimensions, while a positive "Hardy-singular interior mass" is required in lower dimensions. If the singularity is on the boundary, then the local geometry around 0 plays a crucial role in high dimensions, while a positive "Hardy-singular boundary mass" is needed for the lower dimensions.