Comptes Rendus
Partial Differential Equations/Functional Analysis
On Hardy inequalities with singularities on the boundary
Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 273-277.

In this Note we present some Hardy–Poincaré inequalities with one singularity localized on the boundary of a smooth domain. Then, we consider conical domains in dimension N3 whose vertex is on the singularity and we show upper and lower bounds for the corresponding optimal constants in the Hardy inequality. In particular, we prove the asymptotic behavior of the optimal constant when the amplitude of the cone tends to zero.

Dans ce travail nous présentons quelques inégalités de Hardy–Poincaré avec une singularité localisée sur la frontière dʼun domaine régulier. Ensuite, nous considérons des domaines coniques en dimension N3 dont le sommet est sur la singularité et nous établissons des bornes supérieure et inférieure pour les constantes optimales correspondantes dans lʼinégalité de Hardy. En particulier, nous montrons le comportement asymptotique de la constante optimale lorsque lʼamplitude du cône tend vers zéro.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2011.02.005

Cristian Cazacu 1, 2

1 BCAM – Basque Center for Applied Mathematics, Bizkaia Technology Park 500, 48160 Derio, Basque Country, Spain
2 Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain
@article{CRMATH_2011__349_5-6_273_0,
     author = {Cristian Cazacu},
     title = {On {Hardy} inequalities with singularities on the boundary},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {273--277},
     publisher = {Elsevier},
     volume = {349},
     number = {5-6},
     year = {2011},
     doi = {10.1016/j.crma.2011.02.005},
     language = {en},
}
TY  - JOUR
AU  - Cristian Cazacu
TI  - On Hardy inequalities with singularities on the boundary
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 273
EP  - 277
VL  - 349
IS  - 5-6
PB  - Elsevier
DO  - 10.1016/j.crma.2011.02.005
LA  - en
ID  - CRMATH_2011__349_5-6_273_0
ER  - 
%0 Journal Article
%A Cristian Cazacu
%T On Hardy inequalities with singularities on the boundary
%J Comptes Rendus. Mathématique
%D 2011
%P 273-277
%V 349
%N 5-6
%I Elsevier
%R 10.1016/j.crma.2011.02.005
%G en
%F CRMATH_2011__349_5-6_273_0
Cristian Cazacu. On Hardy inequalities with singularities on the boundary. Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 273-277. doi : 10.1016/j.crma.2011.02.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.02.005/

[1] N.C. Adimurthi; M. Ramaswamy An improved Hardy–Sobolev inequality and its application, Proc. Amer. Math. Soc., Volume 130 (2002) no. 2, pp. 489-505 (electronic)

[2] F. Bowman Introduction to Bessel Functions, Dover Publications Inc., New York, 1958

[3] H. Brezis; J.L. Vázquez Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, Volume 10 (1997) no. 2, pp. 443-469

[4] P. Caldiroli; R. Musina Stationary states for a two-dimensional singular Schrödinger equation, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), Volume 4 (2001) no. 3, pp. 609-633

[5] G.A. Cámera Some inequalities for the first eigenvalue of the Laplace–Beltrami operator, Mathematical Notes, vol. 100, Univ. de Los Andes, Mérida, 1989, pp. 67-82 (in Spanish)

[6] C. Cazacu Hardy inequalities with boundary singularities | arXiv

[7] C. Cazacu, E. Zuazua, Hardy inequalities and controllability of the wave equation with boundary singular quadratic potential, in: Proceedings Picof10, 2010, pp. 149–155.

[8] H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, Berlin, 1987.

[9] S. Filippas; A. Tertikas; J. Tidblom On the structure of Hardy–Sobolev–Mazʼya inequalities, J. Eur. Math. Soc. (JEMS), Volume 11 (2009) no. 6, pp. 1165-1185

[10] G.H. Hardy An inequality between integrals, Messenger of Math., Volume 54 (1925), pp. 150-156

[11] G.H. Hardy; J.E. Littlewood; G. Pólya Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988 (reprint of the 1952 edition)

[12] M.F. Mouhamed On the Hardy–Poincaré inequality with boundary singularities | arXiv

[13] M.F. Mouhamed; R. Musina Hardy–Poincaré inequalities with boundary singularities | arXiv

[14] M.F. Mouhamed; R. Musina Sharp nonexistence results for a linear elliptic inequality involving Hardy and Leray potentials | arXiv

[15] E.M. Stein; R. Shakarchi Fourier Analysis, Princeton Lectures in Analysis, vol. 1, Princeton University Press, 2003

[16] 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