×

On families of optimal Hardy-weights for linear second-order elliptic operators. (English) Zbl 1447.35022

Given a nonnegative linear second order elliptic operator \(\mathcal{P}\), the authors contruct families of optimal (in the sense of criticality theory) Hardy-weights. The results generalize those of [B. Devyver et al., J. Funct. Anal. 266, No. 7, 4422–4489 (2014; Zbl 1298.47057)]. In fact, a characterization of the set of all optimal Hardy weights in the one-dimensional case is provided first, and then a one-dimensional reduction argument is used to tackle the case for \(\mathcal{P}\).

MSC:

35B09 Positive solutions to PDEs
35J08 Green’s functions for elliptic equations
35J20 Variational methods for second-order elliptic equations
47J20 Variational and other types of inequalities involving nonlinear operators (general)
49J40 Variational inequalities

Citations:

Zbl 1298.47057
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Agmon, S., Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-Body Schrödinger Operators, Math. Notes, vol. 29 (1982), Princeton University Press: Princeton University Press Princeton · Zbl 0503.35001
[2] Agmon, S., On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds, (Methods of Functional Analysis and Theory of Elliptic Equations. Methods of Functional Analysis and Theory of Elliptic Equations, Naples, 1982 (1983), Liguori: Liguori Naples), 19-52 · Zbl 0595.58044
[3] Ancona, A., Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. Math., 125, 495-536 (1987) · Zbl 0652.31008
[4] Ancona, A., Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators, Nagoya Math. J., 165, 123-158 (2002) · Zbl 1028.31003
[5] Balinsky, A. A.; Evans, W. D.; Lewis, R. T., The Analysis and Geometry of Hardy’s Inequality, Universitext (2015), Springer: Springer Cham · Zbl 1332.26005
[6] Barbatis, G.; Filippas, S.; Tertikas, A., Series expansion for \(L^p\) Hardy inequalities, Indiana Univ. Math. J., 52, 171-190 (2003) · Zbl 1035.26014
[7] Barbatis, G.; Filippas, S.; Tertikas, A., Sharp Hardy and Hardy-Sobolev inequalities with point singularities on the boundary, J. Math. Pures Appl., 117, 146-184 (2018) · Zbl 1395.35004
[8] Birkhoff, G.; Rota, G., Ordinary Differential Equations (1989), John Wiley and Sons, Inc.: John Wiley and Sons, Inc. New York · Zbl 0183.35601
[9] Brezis, H.; Marcus, M., Hardy’s inequalities revisited, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), 25, 217-237 (1997) · Zbl 1011.46027
[10] Cycon, H. L.; Froese, R. G.; Kirsch, W.; Simon, B., Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics, Springer Study Edition (1987), Springer-Verlag: Springer-Verlag Berlin · Zbl 0619.47005
[11] Davies, E. B., A review of Hardy inequalities, (The Maz’ya Anniversary Collection, vol. 2. The Maz’ya Anniversary Collection, vol. 2, Oper. Theory Adv. Appl., vol. 110 (1999), Birkhäuser: Birkhäuser Basel), 55-67 · Zbl 0936.35121
[12] Devyver, B.; Fraas, M.; Pinchover, Y., Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon, J. Funct. Anal., 266, 4422-4489 (2014) · Zbl 1298.47057
[13] Devyver, B.; Pinchover, Y., Optimal \(L^p\) Hardy-type inequalities, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 33, 93-118 (2016) · Zbl 1331.35013
[14] Ermakov, V. P., Second order differential equations: conditions of complete integrability, Appl. Anal. Discrete Math., 2, 123-145 (2008), Universita Izvestia Kiev, Series III 9, pp. 1-25 (in Russian); Translated from the 1880 Russian original by A.O. Harin and edited by P.G.L. Leach · Zbl 1199.34004
[15] Filippas, S.; Tertikas, A., Optimizing improved Hardy inequalities, J. Funct. Anal., 192, 186-233 (2002) · Zbl 1030.26018
[16] Haas, F., The damped Pinney equation and its applications to dissipative quantum mechanics, Phys. Scr., 81, Article 025004 pp. (2010) · Zbl 1190.81074
[17] Keller, M.; Pinchover, Y.; Pogorzelski, F., Optimal Hardy inequalities for Schrödinger operators on graphs, Commun. Math. Phys., 358, 767-790 (2018) · Zbl 1390.26042
[18] Keller, M.; Pinchover, Y.; Pogorzelski, F., Criticality theory for Schrödinger operators on graphs, J. Spectr. Theory (2019), 34 pp., in press
[19] Kovařík, H.; Pinchover, Y., On minimal decay at infinity of Hardy-weights, Commun. Contemp. Math. (2019), 16 pp., in press
[20] Kufner, A.; Maligranda, L.; Persson, L., The Hardy Inequality: About Its History and Some Related Results (2007), Vydavatelsý Servis: Vydavatelsý Servis Plzen · Zbl 1213.42001
[21] Kufner, A.; Opic, B., Hardy-Type Inequalities, Pitman Research Notes in Math., vol. 219 (1990), Longman: Longman Harlow · Zbl 0698.26007
[22] Lieb, E.; Loss, M., Analysis, Graduate Studies in Mathematics, vol. 14 (2001), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0966.26002
[23] Murata, M., Structure of positive solutions to \((- \operatorname{\Delta} + V) u = 0\) in \(\mathbb{R}^n\), Duke Math. J., 53, 869-943 (1986) · Zbl 0624.35023
[24] Pinchover, Y., Topics in the theory of positive solutions of second-order elliptic and parabolic partial differential equations, (Gesztesy, F.; etal., Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon’s 60th Birthday. Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon’s 60th Birthday, Proceedings of Symposia in Pure Mathematics, vol. 76 (2007), American Mathematical Society: American Mathematical Society Providence, RI), 329-356, (Part 1) · Zbl 1138.35008
[25] Pinchover, Y.; Psaradakis, G., On positive solutions of the \((p, A)\)-Laplacian with potential in Morrey space, Anal. PDE, 9, 1317-1358 (2016) · Zbl 1351.35065
[26] Pinchover, Y.; Tintarev, K., Ground state alternative for p-Laplacian with potential term, Calc. Var. Partial Differ. Equ., 28, 179-201 (2007) · Zbl 1208.35032
[27] Pinney, E., The nonlinear differential equation \(y'' + p(x) y + c y^{- 3} = 0\), Proc. Am. Math. Soc., 1, 681 (1950) · Zbl 0038.24303
[28] Zettl, A., Sturm-Liouville Theory, Mathematical Surveys and Monographs, vol. 121 (2005), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1074.34030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.