Abstract
We prove Hardy-type inequality with weight singular at \({0 \in \Omega}\) in the class of functions which are not zero on the boundary \({\partial \Omega}\). The Hardy constant is optimal and the inequality is sharp due to the additional boundary term.
Similar content being viewed by others
References
Adams R.A.: Sobolev Spaces. Academic Press, Boston (1975)
Adimurthi: Hardy–Sobolev inequality in \({H^1 (\Omega)}\) and its applications. Commun. Contemp. Math. 4, 409–434 (2002)
Adimurthi, Esteban, M.: An improved Hardy–Sobolev inequality in W 1,p and its applications to Srödinger operators. Nonlinear Differ. Equ. Appl. 12, 243–236 (2005)
Adimurthi, Filippas, S., Tertikas, A.: On the best constant of Hardy–Sobolev inequalities. Nonlinear Anal. T.M.A. 70, 2826–2833 (2009)
Alvino A., Volpicelli R., Volzone B.: On Hardy inequalities with a remainder term. Ricerche Math. 59, 265–280 (2010)
Barbitas G., Filippas S., Tertikas A.: Series expantion for Lp Hardy inequalities. Indiana Univ. Math. J. 52, 171–189 (2003)
Barbitas G., Filippas S., Tertikas A.: A unified approach to improved Lp Hardy inequalities with best constants. Trans. Am. Math. Soc 356, 2169–2196 (2003)
Barbitas G., Filippas S., Tertikas A.: Critical heat kernel estimates for Schrödinger operator via Hardy–Sobolev inequalities. J. Funct. Anal. 208, 1–30 (2004)
Berchio E., Gazzola F., Pierotti D.: Gelfand type elliptic problems under Steclov boundary condiions. Ann. Inst. Henri Poincaré Anal. Non Linéaire 27, 315–335 (2010)
Brezis H., Vazquez J.: Blow–up solutions of some nonlinear elliptic problem. Rev. Mat. Univ. Complut. Madrid 10, 443–469 (1997)
Evans, L.: Partial Differential Equations. Graduate Studies of Mathematics, vol. 19. AMS, Providence (1998)
Hardy G.: Note on a theorem of Hilbert. Math. Zeit 6, 314–317 (1920)
Hardy G.: Notes on some points in the integral calculus, LX. An inequality between integrals. Messenger Math. 54, 150–156 (1925)
Leray J.: Etude de diverses équationes integralès, nonlinéaires et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933)
Maz’ja V.G.: Sobolev Spaces. Springer, Berlin (1985)
Opic, B., Kufner, A.: Hardy-type inequalities. In: Longman, Pitman research notes in Math, vol. 219 (1990)
Peral I., Vazquez J.: The semilinear heat equation with exponential reaction term. Arch. Ration. Mech. Anal. 129, 201–224 (1995)
Pinchover Y., Tintarev K.: Existence of minimizers for Schrödinger operators under domain perturbations with application to Hardy’s inequality. Indiana Univ. Math. J. 129, 201–224 (1995)
Wang Z.-Q., Zhu M.: Hardy inequalities with boundary terms, electron. J. Differ. Equ. 43, 1–8 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fabricant, A., Kutev, N. & Rangelov, T. Note on Sharp Hardy-Type Inequality. Mediterr. J. Math. 11, 31–44 (2014). https://doi.org/10.1007/s00009-013-0367-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-013-0367-9