Seminar Analysis: Frédéric Robert (University of Lorraine)

On the Hardy-Schrödinger operator with a boundary singularity

We investigate the Hardy-Schrödinger operator L_γ=-Δ-γ/|x|2 on domains Ω⊂Rⁿ, whose boundary contain the singularity 0. The situation is quite different from the well-studied case when 0 is in the interior of Ω. For one, if 0∈Ω, then L is positive if and only if γ<(n-2)²/4, while if 0∈∂Ω the operator L could be positive for larger value of γ, potentially reaching the maximal constant n²/4 on convex domains.

We prove optimal regularity and a Hopf-type Lemma for variational solutions of corresponding linear Dirichlet boundary value problems of the form Lγ=a(x)u, but also for non-linear equations including Lγ=(|u|^β-2u)/(|x|^s), where γ < n²/4, s∈[0,2) and β:=2(n-s)/(n-2) is the critical Hardy-Sobolev exponent. We also provide a Harnack inequality and a complete description of the profile of all positive solutions–variational or not– of the corresponding linear equation on the punctured domain. The value γ=(n-1)²/4 turned out to be another critical threshold for the operator Lγ, and our analysis yields a corresponding notion of “Hardy singular boundary-mass” mγ(Ω) of a domain Ω having 0∈Ω, which could be defined whenever (n²-1)/4 < γ < n²/4.

As a byproduct, we give a complete answer to problems of existence of extremals for Hardy-Sobolev inequalities of the form

C( ∫_Ω (u^β)/(|x|^s) dx )^2/β ≤∫_Ω |∇u|² dx - γ∫_Ω (u²)/(|x|^s)dx

whenever γ<n<sup>2/4, and in particular, for those of Caffarelli-Kohn-Nirenberg. These resultsextend previous contributions by the authors in the case γ=0, and by Chern-Lin for the case γ<(n-2)²/4. Namely, if 0≤γ≤(n²-1)/4, then the negativity of the mean curvature of ∂Ω at 0 is sucient for the existence of extremals. This is however not sufficient for (n²-1)/4≤γ≤(n²)/4, which then requires the positivity of the Hardy singular boundary-massof the domain under consideration.

Joint work with Nassif Ghoussoub.

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