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

Cazacu, Cristian

Comptes Rendus. Mathématique, Tome 349 (2011) no. 21-22, pp. 1167-1172.

On considère lʼopérateur de Schrödinger Aλ:=−Δ−λ/|x|2, λ∈R, lorsque lʼorigine est située sur la frontière dʼun domaine borné et régulier Ω⊂RN, N⩾1.

Cette Note a deux objectifs. Premièrement, on montre, dans ce cas, lʼextension de lʼidentité classique de Pohozaev pour le laplacien. Le problème abordé est très lié aux inégalités de Hardy–Poincaré avec des singularités sur la frontière. En second lieu, la nouvelle identité de Pohozaev permet de dʼobtenir la méthode des multiplicateurs pour lʼéquation des ondes et pour lʼéquation de Schrödinger. De cette façon on étend au cas de la singularité frontière les propriétés dʼobservabilité et contrôle pour lʼéquation des ondes classique et pour lʼéquation de Schrödinger bien connues dans le cas dʼune singularité intérieure (Vancostenoble et Zuazua (2009) [16]).

We consider the Schrödinger operator Aλ:=−Δ−λ/|x|2, λ∈R, when the singularity is located on the boundary of a smooth domain Ω⊂RN, N⩾1.

The 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 us 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 (Vancostenoble and Zuazua (2009) [16]).

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