Some remarks on the Sobolev inequality in Riemannian manifolds

Andreucci, Daniele; Tedeev, Anatoli

doi:10.1090/proc/15774

Some remarks on the Sobolev inequality in Riemannian manifolds
HTML articles powered by AMS MathViewer

by Daniele Andreucci and Anatoli F. Tedeev

Proc. Amer. Math. Soc. 150 (2022), 1657-1667

DOI: https://doi.org/10.1090/proc/15774

Published electronically: January 27, 2022

PDF | Request permission

Abstract:

We investigate Sobolev and Hardy inequalities, specifically weighted Minerbe’s type estimates, in noncompact complete connected Riemannian manifolds whose geometry is described by an isoperimetric profile. In particular, we assume that the manifold satisfies the $p$-hyperbolicity property, stated in terms of a necessary integral Dini condition on the isoperimetric profile. Our method seems to us to combine sharply the knowledge of the isoperimetric profile and the optimal Bliss type Hardy inequality depending on the geometry of the manifold. We recover the well known best Sobolev constant in the Euclidean case.

References

D. Andreucci and A. F. Tedeev, Extinction in a finite time for parabolic equations of fast diffusion type on manifolds, Operator theory and differential equations, Trends Math., Birkhäuser/Springer, Cham, [2021] ©2021, pp. 1–6. MR 4221137, DOI 10.1007/978-3-030-49763-7_{1}
Daniele Andreucci and Anatoli F. Tedeev, Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity, Adv. Differential Equations 5 (2000), no. 7-9, 833–860. MR 1776343
Daniele Andreucci and Anatoli F. Tedeev, Optimal decay rate for degenerate parabolic equations on noncompact manifolds, Methods Appl. Anal. 22 (2015), no. 4, 359–376. MR 3457534, DOI 10.4310/MAA.2015.v22.n4.a2
Thierry Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry 11 (1976), no. 4, 573–598 (French). MR 448404
Thierry Aubin, Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1636569, DOI 10.1007/978-3-662-13006-3
Elvise Berchio, Debdip Ganguly, and Gabriele Grillo, Sharp Poincaré-Hardy and Poincaré-Rellich inequalities on the hyperbolic space, J. Funct. Anal. 272 (2017), no. 4, 1661–1703. MR 3590248, DOI 10.1016/j.jfa.2016.11.018
G. A. Bliss, An Integral Inequality, J. London Math. Soc. 5 (1930), no. 1, 40–46. MR 1574997, DOI 10.1112/jlms/s1-5.1.40
Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR 936419, DOI 10.1007/978-3-662-07441-1
Isaac Chavel, Isoperimetric inequalities, Cambridge Tracts in Mathematics, vol. 145, Cambridge University Press, Cambridge, 2001. Differential geometric and analytic perspectives. MR 1849187
Mu-Fa Chen, The optimal constant in Hardy-type inequalities, Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 5, 731–754. MR 3333168, DOI 10.1007/s10114-015-4731-5
Andrea Cianchi, Symmetrization in anisotropic elliptic problems, Comm. Partial Differential Equations 32 (2007), no. 4-6, 693–717. MR 2334829, DOI 10.1080/03605300600634973
D. Cordero-Erausquin, B. Nazaret, and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math. 182 (2004), no. 2, 307–332. MR 2032031, DOI 10.1016/S0001-8708(03)00080-X
Lorenzo D’Ambrosio and Serena Dipierro, Hardy inequalities on Riemannian manifolds and applications, Ann. Inst. H. Poincaré C Anal. Non Linéaire 31 (2014), no. 3, 449–475. MR 3208450, DOI 10.1016/j.anihpc.2013.04.004
Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520. MR 123260, DOI 10.2307/1970227
Alexander Grigor′yan, Isoperimetric inequalities and capacities on Riemannian manifolds, The Maz′ya anniversary collection, Vol. 1 (Rostock, 1998) Oper. Theory Adv. Appl., vol. 109, Birkhäuser, Basel, 1999, pp. 139–153. MR 1747869
Emmanuel Hebey and Michel Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. J. 79 (1995), no. 1, 235–279. MR 1340298, DOI 10.1215/S0012-7094-95-07906-X
M. Ledoux, On manifolds with non-negative Ricci curvature and Sobolev inequalities, Comm. Anal. Geom. 7 (1999), no. 2, 347–353. MR 1685586, DOI 10.4310/CAG.1999.v7.n2.a7
Ying Li and Yong-Hua Mao, The optimal constant in generalized Hardy’s inequality, Math. Inequal. Appl. 23 (2020), no. 1, 257–266. MR 4061539, DOI 10.7153/mia-2020-23-20
Joaquim Martín and Mario Milman, Towards a unified theory of Sobolev inequalities, Special functions, partial differential equations, and harmonic analysis, Springer Proc. Math. Stat., vol. 108, Springer, Cham, 2014, pp. 163–201. MR 3297660, DOI 10.1007/978-3-319-10545-1_{1}3
Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
Vladimir M. Miklyukov and Matti K. Vuorinen, Hardy’s inequality for $W^{1,p}_0$-functions on Riemannian manifolds, Proc. Amer. Math. Soc. 127 (1999), no. 9, 2745–2754. MR 1600117, DOI 10.1090/S0002-9939-99-04849-2
Vincent Minerbe, Weighted Sobolev inequalities and Ricci flat manifolds, Geom. Funct. Anal. 18 (2009), no. 5, 1696–1749. MR 2481740, DOI 10.1007/s00039-009-0701-3
A. I. Nazarov, The Hardy-Sobolev inequalities in a cone, Algebra i Analiz 22 (2010), no. 6, 200–213 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 22 (2011), no. 6, 997–1006. MR 2760091, DOI 10.1090/S1061-0022-2011-01180-X
B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Scientific & Technical, Harlow, 1990. MR 1069756
Renato H. L. Pedrosa and Manuel Ritoré, Isoperimetric domains in the Riemannian product of a circle with a simply connected space form and applications to free boundary problems, Indiana Univ. Math. J. 48 (1999), no. 4, 1357–1394. MR 1757077, DOI 10.1512/iumj.1999.48.1614
Jimmy Petean and Juan Miguel Ruiz, Isoperimetric profile comparisons and Yamabe constants, Ann. Global Anal. Geom. 40 (2011), no. 2, 177–189. MR 2811624, DOI 10.1007/s10455-011-9252-6
Manuel Ritoré and Efstratios Vernadakis, Large isoperimetric regions in the product of a compact manifold with Euclidean space, Adv. Math. 306 (2017), 958–972. MR 3581323, DOI 10.1016/j.aim.2016.11.001
Antonio Ros, The isoperimetric problem, Global theory of minimal surfaces, Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 2005, pp. 175–209. MR 2167260
Juan Miguel Ruiz and Areli Vázquez Juárez, Isoperimetric estimates in low-dimensional Riemannian products, Ann. Global Anal. Geom. 59 (2021), no. 4, 417–434. MR 4258014, DOI 10.1007/s10455-021-09757-6
Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. MR 463908, DOI 10.1007/BF02418013
M. Troyanov, Parabolicity of manifolds, Siberian Adv. Math. 9 (1999), no. 4, 125–150. MR 1749853
Changyu Xia, Complete manifolds with nonnegative Ricci curvature and almost best Sobolev constant, Illinois J. Math. 45 (2001), no. 4, 1253–1259. MR 1894894