On Optimal Hardy Inequalities in Cones

Baptiste Devyver, Yehuda Pinchover, Georgios Psaradakis


For a given subcritical Schrödinger operator in a cone in ℝn with a given Hardy potential corresponding to the distance to the boundary of the cone, we present an explicit optimal Hardy-type improvement. In particular, we present an explicit expression for the associate best Hardy constant, and for the corresponding ground state.


Ground state; Hardy inequality; minimal growth; positive solutions

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S. Agmon, "Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-body Schrödinger Operators", Mathematical Notes, 29, Princeton University Press, Princeton, 1982

S. Agmon, On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds, in "Methods of Functional Analysis and Theory of Elliptic Equations", Liguori, Naples, 1983, pp. 19-52

A. Ancona, Negatively curved manifolds, elliptic operators and the Martin boundary, Annals of Mathematics, Second Series, 125 (1987), 495-536

A. Ancona, On positive harmonic functions in cones and cylinders, Rev. Mat. Iberoam. 28 (2012), 201-230

C. Cazacu, New estimates for the Hardy constant of multipolar Schrödinger operators with boundary singularities, arXiv:1402.5933

B. Devyver, M. Fraas, and Y. Pinchover, Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon, J. Functional Analysis 266 (2014), 4422-4489

B. Devyver, Y. Pinchover, and G. Psaradakis, Optimal Hardy inequalities in cones, submitted for publication, 2015. arXiv: 1502.05205

M. M. Fall, and R. Musina, Hardy-Poincaré inequalities with boundary singularities, Proc. Roy. Soc. Edinburgh 142 (2012), 769-786

S. Filippas, A. Tertikas and J. Tidblom, On the structure of Hardy-Sobolev-Maz'ya inequalities, J. Eur. Math. Soc. 11/6 (2009), 1165-1185

K.T Gkikas, Hardy-Sobolev inequalities in unbounded domains and heat kernel estimates, J. Funct. Anal. 264 (2013), 837-893

N. Ghoussoub and F. Robert, On the Hardy-Schrödinger operator with a boundary singularity, arXiv: 1410.1913v2

T. Hoffmann-Ostenhof and A. Laptev, Hardy inequalities with homogenuous weights, arXiv: 1408.5561

V.Ya. Lin and Y. Pinchover, Manifolds with group actions and elliptic operators, Mem. Amer. Math. Soc. 112 (1994)

V. Liskevich, S. Lyahkova and V. Moroz, Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains, Adv. Differential Equations 11/4 (2006), 361-398

V.M Miklyukov, and M.K. Vuorinen, Hardy's inequality for W01,p -functions on Riemannian manifolds, Proc. Amer. Math. Soc. 127 (1999), 2745-2754

M. Marcus, V. J. Mizel and Y. Pinchover, On the best constant for Hardy's inequality in Rn, Trans. Amer. Math. Soc. 350 (1998), 3237-3255

Y. Pinchover, On positive Liouville theorems and asymptotic behavior of solutions of Fuchsian type elliptic operators, Ann. Inst. Henri Poincaré. Analyse Non Linéaire 11 (1994), 313-341

Y. Pinchover and K. Tintarev, On positive solutions of p-Laplacian-type equations, in "Analysis, Partial Differential Equations and Applications - The Vladimir Maz'ya Anniversary Volume", eds. A. Cialdea et al., Operator Theory: Advances and Applications, Vol. 193, Birkäuser Verlag, Basel, 2009, 245-268

N. Rautenberg, A Hardy inequality on Riemannian manifolds and a classification of discrete Dirichlet spectra, ArXiv: 1401.5010

DOI: 10.6092/issn.2240-2829/4741


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Copyright (c) 2014 Baptiste Devyver, Yehuda Pinchover, Georgios Psaradakis

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