Hardy inequalities on Riemannian manifolds and applications

D'Ambrosio, Lorenzo; Dipierro, Serena

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 3, pp. 449-475.

We prove a simple sufficient criterion to obtain some Hardy inequalities on Riemannian manifolds related to quasilinear second order differential operator ${\Delta }_{p}u:=\mathrm{div}\left({\left|\nabla u\right|}^{p-2}\nabla u\right)$. Namely, if ρ is a nonnegative weight such that $-{\Delta }_p\rho ⩾0$, then the Hardy inequality

$$c\underset{M}{\int }\frac{{\left|u\right|}^p}{{\rho }^p}{\left|\nabla \rho \right|}^{p}d{v}_g⩽\underset{M}{\int }{\left|\nabla u\right|}^{p}d{v}_g,u\in {\mathrm{C}}_0^{\infty }\left(M\right),$$

holds. We show concrete examples specializing the function ρ.Our approach allows to obtain a characterization of p-hyperbolic manifolds as well as other inequalities related to Caccioppoli inequalities, weighted Gagliardo–Nirenberg inequalities, uncertain principle and first order Caffarelli–Kohn–Nirenberg interpolation inequality.

Classification : 58J05, 31C12, 26D10
Mots clés : Hardy inequality, Riemannian manifolds, Parabolic manifolds, Caccioppoli inequality, Weighted Gagliardo–Nirenberg inequality, Interpolation inequality

[1] Adimurthi, A. Sekar, Role of the fundamental solution in Hardy–Sobolev-type inequalities, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 1111 -1130 | MR | Zbl

[2] L. Adriano, C. Xia, Hardy type inequalities on complete Riemannian manifolds, Monatsh. Math. 163 (2011), 115 -129 | MR | Zbl

[3] T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer-Verlag, Berlin (1998) | MR | Zbl

[4] P. Baras, J. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc. 284 (1984), 121 -139 | MR | Zbl

[5] G. Barbatis, S. Filippas, A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants, Trans. Amer. Math. Soc. 356 (2004), 2169 -2196 | MR | Zbl

[6] M. Van Den Berg, P. Gilkey, A. Grigor'Yan, K. Kirsten, Hardy inequality and heat semigroup estimates for Riemannian manifolds with singular data, Comm. Partial Differential Equations 37 (2012), 885 -900 | MR | Zbl

[7] Y. Bozhkov, E. Mitidieri, Conformal Killing vector fields and Rellich type identities on Riemannian manifolds, I, Lect. Notes Semin. Interdiscip. Mat. vol. 7 (2008), 65 -80 | MR | Zbl

[8] Y. Bozhkov, E. Mitidieri, Conformal Killing vector fields and Rellich type identities on Riemannian manifolds, II, Mediterr. J. Math. 9 (2012), 1 -20 | MR | Zbl

[9] H. Brezis, X. Cabré, Some simple nonlinear PDE's without solutions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 8 no. 1 (1998), 223 -262 | EuDML | MR | Zbl

[10] H. Brezis, M. Marcus, Hardy's inequalities revisited, Ann. Sc. Norm. Super. Cl. Sci. (4) 25 (1997), 217 -237 | EuDML | Numdam | MR | Zbl

[11] H. Brezis, J.L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madr. 10 (1997), 443 -469 | EuDML | MR | Zbl

[12] V. Burenkov, E.B. Davies, Spectral stability of the Neumann Laplacian, J. Differential Equations 186 (2002), 485 -508 | MR | Zbl

[13] L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequalities with weights, Compos. Math. 53 no. 3 (1984), 259 -275 | EuDML | Numdam | MR | Zbl

[14] M.P. Do Carmo, C. Xia, Complete manifolds with non-negative Ricci curvature and the Caffarelli–Kohn–Nirenberg inequalities, Compos. Math. 140 (2004), 818 -826 | MR | Zbl

[15] G. Carron, Inégalités isopérimétriques de Faber–Krahn et conséquences, Actes de la Table Ronde de Géométrie Différentielle, Luminy, 1992, Semin. Congr. vol. 1 (1996), 205 -232 | MR | Zbl

[16] G. Carron, Inégalités de Hardy sur les variétés Riemanniennes non-compactes, J. Math. Pures Appl. (9) 76 (1997), 883 -891 | MR

[17] J. Cheeger, D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. 96 (1972), 413 -443 | MR | Zbl

[18] L. D'Ambrosio, Hardy inequalities related to Grushin type operators, Proc. Amer. Math. Soc. 132 (2004), 725 -734 | MR | Zbl

[19] L. D'Ambrosio, Some Hardy inequalities on the Heisenberg group, Differ. Equ. 40 (2004), 552 -564 | MR | Zbl

[20] L. D'Ambrosio, Hardy-type inequalities related to degenerate elliptic differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) IV (2005), 451 -586 | EuDML | Numdam | MR | Zbl

[21] E.B. Davies, A. Hinz, Explicit constants for Rellich inequalities in $L^p$ , Math. Z. 227 (1998), 511 -523 | MR | Zbl

[22] E.B. Davies, M. Lianantonakis, Heat kernel and Hardy estimates for locally Euclidean manifolds with fractal boundaries, Geom. Funct. Anal. 3 (1993), 527 -563 | EuDML | MR | Zbl

[23] B. Devyver, M. Fraas, Y. Pinchover, Optimal Hardy-type inequality for second-order elliptic operator and applications, arXiv:1208.2342 | MR

[24] B. Devyver, M. Fraas, Y. Pinchover, Optimal Hardy-type inequalities for elliptic operators, C. R. Acad. Sci. Paris 350 (2012), 475 -479 | MR | Zbl

[25] J. Duoandikoetxea, L. Vega, Some weighted Gagliardo–Nirenberg inequalities and applications, Proc. Amer. Math. Soc. 135 (2007), 2795 -2802 | MR | Zbl

[26] J. Escobar, A. Freire, The spectrum of the Laplacian of manifolds of positive curvature, Duke Math. J. 65 (1992), 1 -21 | MR | Zbl

[27] F. Gazzola, H. Grunau, E. Mitidieri, Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc. 356 (2004), 2149 -2168 | MR | Zbl

[28] A.A. Grigor'Yan, On the existence of a Green function on a manifold, Russian Math. Surveys 38 no. 1 (1983), 190 -191 | Zbl

[29] A.A. Grigor'Yan, On the existence of positive fundamental solutions of the Laplace equation on Riemannian manifold, Math. USSR Sb. 56 no. 2 (1987), 349 -357 | MR | Zbl

[30] A.A. Grigor'Yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.) 36 no. 2 (1999), 135 -249 | MR | Zbl

[31] A.A. Grigor'Yan, Isoperimetric inequalities and capacities on Riemannian manifolds, Oper. Theory Adv. Appl. 109 (1999), 139 -153 | MR | Zbl

[32] H. Hedenmalm, The Beurling operator for the hyperbolic plane, Ann. Acad. Sci. Fenn. Math. 37 (2012), 3 -18 | MR | Zbl

[33] J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford (1993) | MR | Zbl

[34] I. Holopainen, Asymptotic Dirichlet problem for the p-Laplacian on Cartan–Hadamard manifolds, Proc. Amer. Math. Soc. 130 no. 11 (2002), 3393 -3400 | MR | Zbl

[35] I. Holopainen, Non linear potential theory and quasiregular mappings on Riemannian manifolds, Ann. Acad. Sci. Fenn. Ser. A 74 (1990) | MR | Zbl

[36] I. Holopainen, P. Pankka, p-Laplace operator, quasiregular mappings, and Picard-type theorems, Quasiconformal Mappings and Their Applications, Narosa, New Delhi (2007), 117 -150 | MR | Zbl

[37] E. Jannelli, The role played by space dimension in elliptic critical problems, J. Differential Equations 156 (1999), 407 -426 | MR | Zbl

[38] I. Kombe, M. Özaydin, Improved Hardy and Rellich inequalities on Riemannian manifolds, Trans. Amer. Math. Soc. 361 no. 12 (2009), 6191 -6203 | MR | Zbl

[39] P. Li, L. Tam, Symmetric Green's functions on complete manifolds, Amer. J. Math. 109 (1987), 1129 -1154 | MR | Zbl

[40] P. Li, J. Wang, Weighted Poincaré inequality and rigidity of complete manifolds, Ann. Sci. Éc. Norm. Supér. (4) 39 no. 6 (2006), 921 -982 | EuDML | Numdam | MR | Zbl

[41] M. Marcus, V. Mizel, Y. Pinchover, On the best constant for Hardy's inequality in ${ℝ}^n$ , Trans. Amer. Math. Soc. 350 (1998), 3237 -3255 | MR | Zbl

[42] T. Matskewich, P. Sobolevskii, The best possible constant in generalized Hardy's inequality for convex domain in ${ℝ}^n$ , Nonlinear Anal. 28 (1997), 1601 -1610 | MR | Zbl

[43] T. Matskewich, P. Sobolevskii, The sharp constant in Hardy's inequality for complement of bounded domain, Nonlinear Anal. 33 (1998), 105 -120 | MR | Zbl

[44] V. Miklyukov, M. Vuorinen, Hardy's inequality for $W_0^{1,p}$-functions on Riemannian manifolds, Proc. Amer. Math. Soc. 127 no. 9 (1999), 2745 -2754 | MR | Zbl

[45] E. Mitidieri, A simple approach to Hardy inequalities, Mat. Zametki 67 (2000), 563 -572 | MR | Zbl

[46] E. Mitidieri, S. Pohozaev, A-priori estimates and blow-up of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math. 234 (2001), 1 -362 | Zbl

[47] S. Pigola, M. Rigoli, A. Setti, Vanishing and Finiteness Results in Geometric Analysis, Birkhäuser Verlag, Berlin (2008) | MR | Zbl

[48] M. Troyanov, Parabolicity of manifolds, Siberian Adv. Math. 9 (1999), 125 -150 | MR | Zbl

[49] M. Troyanov, Solving the p-Laplacian on manifolds, Proc. Amer. Math. Soc. 128 no. 2 (2000), 541 -545 | MR | Zbl

[50] C. Xia, The Gagliardo–Nirenberg inequalities and manifolds of non-negative Ricci curvature, J. Funct. Anal. 224 (2005), 230 -241 | MR | Zbl