Comptes Rendus
Partial Differential Equations/Functional Analysis
Hardy inequality and Pohozaev identity for operators with boundary singularities: Some applications
Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1167-1172.

We consider the Schrödinger operator Aλ:=Δλ/|x|2, λR, when the singularity is located on the boundary of a smooth domain ΩRN, N1.

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]).

On considère lʼopérateur de Schrödinger Aλ:=Δλ/|x|2, λR, lorsque lʼorigine est située sur la frontière dʼun domaine borné et régulier ΩRN, N1.

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]).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2011.10.009

Cristian Cazacu 1, 2

1 BCAM – Basque Center for Applied Mathematics, Bizkaia Technology Park, Building 500, 48160 Derio, Spain
2 Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
@article{CRMATH_2011__349_21-22_1167_0,
     author = {Cristian Cazacu},
     title = {Hardy inequality and {Pohozaev} identity for operators with boundary singularities: {Some} applications},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1167--1172},
     publisher = {Elsevier},
     volume = {349},
     number = {21-22},
     year = {2011},
     doi = {10.1016/j.crma.2011.10.009},
     language = {en},
}
TY  - JOUR
AU  - Cristian Cazacu
TI  - Hardy inequality and Pohozaev identity for operators with boundary singularities: Some applications
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 1167
EP  - 1172
VL  - 349
IS  - 21-22
PB  - Elsevier
DO  - 10.1016/j.crma.2011.10.009
LA  - en
ID  - CRMATH_2011__349_21-22_1167_0
ER  - 
%0 Journal Article
%A Cristian Cazacu
%T Hardy inequality and Pohozaev identity for operators with boundary singularities: Some applications
%J Comptes Rendus. Mathématique
%D 2011
%P 1167-1172
%V 349
%N 21-22
%I Elsevier
%R 10.1016/j.crma.2011.10.009
%G en
%F CRMATH_2011__349_21-22_1167_0
Cristian Cazacu. Hardy inequality and Pohozaev identity for operators with boundary singularities: Some applications. Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1167-1172. doi : 10.1016/j.crma.2011.10.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.10.009/

[1] A. Adimurthi, C. Cazacu, E. Zuazua, Best constants and Pohozaev identity for Hardy–Sobolev type operators, in preparation.

[2] C. Bardos; G. Lebeau; J. Rauch Control and stabilization for hyperbolic equations, Strasbourg, 1991, SIAM, Philadelphia, PA (1991), pp. 252-266

[3] H. Brezis; M. Marcus; I. Shafrir Extremal functions for Hardyʼs inequality with weight, J. Funct. Anal., Volume 171 (2000) no. 1, pp. 177-191

[4] C. Cazacu On Hardy inequalities with singularities on the boundary, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011), pp. 273-277

[5] J. Dávila; I. Peral Nonlinear elliptic problems with a singular weight on the boundary, Calc. Var., Volume 41 (2011) no. 3–4, pp. 567-586 | DOI

[6] S. Ervedoza Control and stabilization properties for a singular heat equation with an inverse-square potential, Comm. Partial Differential Equations, Volume 33 (2008), pp. 10-12 (1996–2019)

[7] L.C. Evans Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010 (pp. xxii+749)

[8] M.M. Fall, On the Hardy Poincaré inequality with boundary singularities, Commun. Contemp. Math., in press.

[9] M.M. Fall, R. Musina, Hardy–Poincaré inequality with boundary singularities. Proc. Roy. Soc. Edinburgh, in press.

[10] N. Ghoussoub; C. Yuan Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., Volume 352 (2000) no. 12, pp. 5703-5743

[11] D. Jerison; C.E. Kenig Unique continuation and absence of positive eigenvalues for Schrödinger operators, with an appendix by E.M. Stein, Ann. of Math. (2), Volume 121 (1985) no. 3, pp. 463-494

[12] J.-L. Lions Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1. Contrôlabilité exacte (Exact controllability) with appendices by E. Zuazua, C. Bardos, G. Lebeau, J. Rauch, Recherches en Mathématiques Appliquées, Research in Applied Mathematics, vol. 8, Masson, Paris, 1988

[13] E. Machtyngier Exact controllability for the Schrödinger equation, SIAM J. Control Optim., Volume 32 (1994) no. 1, pp. 24-34

[14] M. Tucsnak; G. Weiss Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Advanced Texts: Basel Textbooks, Birkhäuser Verlag, Basel, 2009 (pp. xii+483)

[15] J. Vancostenoble; E. Zuazua Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal., Volume 254 (2008) no. 7, pp. 1864-1902

[16] J. Vancostenoble; E. Zuazua Hardy inequalities, observability, and control for the wave and Schrödinger equations with singular potentials, SIAM J. Math. Anal., Volume 41 (2009) no. 4, pp. 1508-1532

[17] J.L. Vázquez; N.B. Zographopoulos Functional aspects of the Hardy inequality. Appearance of a hidden energy | arXiv

[18] J.L. Vázquez; E. Zuazua The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., Volume 173 (2000) no. 1, pp. 103-153

Cited by Sources:

Comments - Policy