Hardy inequality and Pohozaev identity for operators with boundary singularities: Some applications

Abstract

We consider the Schrödinger operator A λ := −∆ − λ/|x| 2 , λ ∈ R, when the singularity is located on the boundary of a smooth domain Ω ⊂ R N , N ≥ 1The aim of this Note is two folded. Firstly, we justify the extension of the classical Pohozaev identity for the Laplacian to this case. The problem we address is very much related to Hardy-Poincaré inequalities with boundary singularities. Secondly, the new Pohozaev identity allows to develop the multiplier method for the wave and the Schrödinger equations. In this way we extend to the case of boundary singularities well known observability and control properties for the classical wave and Schrödinger equations when the singularity is placed in the interior of the domain (Vanconstenoble and Zuazua [17]). RésuméNous allons considérer l'operateur de Schrödinger A λ := −∆ − λ/|x| 2 , λ ∈ R, lorsque l'origine est située sur la frontiére d'un domaine borné et régulière Ω ⊂ R N , N ≥ 1.Cette Note a deux objectifs. Premièrement, nous montrons l'extension de l'identité classique de Pohozaev pour le Laplacien dans ce cas. Le problème que nous abordons est très lié aux inégalités de Hardy-Poincaré avec des singularités sur la frontiére. En second lieu, la nouvelle identité de Pohozaev permet de dériver le methode de multiplicateurs pour leséquations des ondes et de Schrödinger. De cette façon, nousétendons au cas de la singularité frontalière propriétés d'observabilité et contrôle pour leséquations des ondes classique et de Schrödinger bien connues dans le cas d'une singularitéà l'interieur (Vancostenoble et Zuazua [17])

Sharp Hardy and Hardy–Sobolev inequalities with point singularities on the boundary

“…We note however that whereas in the interior singularity case the geometry of Ω is irrelevant, in this work the curvature of the boundary introduces several technical difficulties even in the case of the plain Hardy inequality (5) as already noted in several recent works see e.g. [8,9,10,11,14,15,20,21]. To overcome these difficulties we produce new improved inequalities in the flat case, see Lemmas 1, 2, 3 and then we use suitable conformal transformations thus obtaining sharp inequalities under the exterior ball assumption.…”

“…We note however that whereas in the interior singularity case the geometry of Ω is irrelevant, in this work the curvature of the boundary introduces several technical difficulties even in the case of the plain Hardy inequality (5) as already noted in several recent works see e.g. [8,9,10,11,14,15,20,21].…”

“…In particular if the mean curvature at zero is negative then µ s (Ω) < µ s (R n + ) and there exists a minimizer for (3). In the limit case s = 2 the infimum µ 2 (Ω) is the best Hardy constant and under certain geometric assumptions on Ω has been studied in [8,9,10,11,15].…”

“…To prove estimate (9) we test inequality (8) with a function of the form u(x) = v(r)φ 1 (ω). Then an easy calculation gives…”

Schrödinger operators with Leray-Hardy potential singular on the boundary

“…Moreover, under the assumption (C-1) the imbedding of H µ (Ω) in L 2 (Ω) is compact (see e.g. [6]). We denote by γ Ω µ the positive eigenfunction, its satisfies…”