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 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 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.
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Cristian Cazacu 1, 2
@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}, }
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] An improved Hardy–Sobolev inequality and its application, Proc. Amer. Math. Soc., Volume 130 (2002) no. 2, pp. 489-505 (electronic)
[2] Introduction to Bessel Functions, Dover Publications Inc., New York, 1958
[3] Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, Volume 10 (1997) no. 2, pp. 443-469
[4] 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] 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] 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] On the structure of Hardy–Sobolev–Mazʼya inequalities, J. Eur. Math. Soc. (JEMS), Volume 11 (2009) no. 6, pp. 1165-1185
[10] An inequality between integrals, Messenger of Math., Volume 54 (1925), pp. 150-156
[11] Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988 (reprint of the 1952 edition)
[12] On the Hardy–Poincaré inequality with boundary singularities | arXiv
[13] Hardy–Poincaré inequalities with boundary singularities | arXiv
[14] Sharp nonexistence results for a linear elliptic inequality involving Hardy and Leray potentials | arXiv
[15] Fourier Analysis, Princeton Lectures in Analysis, vol. 1, Princeton University Press, 2003
[16] 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
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