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General sharp weighted Caffarelli–Kohn–Nirenberg inequalities

Part of: Inequalities Linear function spaces and their duals

Published online by Cambridge University Press:  27 December 2018

Nguyen Lam
Nguyen Lam*
Affiliation:
Department of Mathematics, University of British Columbia and The Pacific Institute for the Mathematical Sciences, Vancouver, BC V6T1Z4, Canada (nlam@math.ubc.ca)
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Abstract

In this paper, we will use optimal mass transport combining with suitable transforms to study the sharp constants and optimizers for a class of the Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities. Moreover, we will investigate these inequalities with and without the monomial weights $x_{1}^{A_{1}} \cdots x_{N}^{A_{N}}$ on ℝN.

Keywords

Gagliardo–Nirenberg inequalitiesCaffarelli–Kohn–Nirenberg inequalitiesbest constantsextremal functionsmass transportduality

MSC classification

Primary: 26D10: Inequalities involving derivatives and differential and integral operators
Secondary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 3 , June 2019 , pp. 691 - 718
Copyright
Copyright © Royal Society of Edinburgh 2018 

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References

1Adimurthi, Chaudhuri, N. and Ramaswamy, M.. An improved Hardy–Sobolev inequality and its application. Proc. Amer. Math. Soc. 130 (2002), 489–505 (electronic).Google Scholar
2Agueh, M.. Gagliardo–Nirenberg inequalities involving the gradient L 2-norm. C. R. Math. Acad. Sci. Paris 346 (2008a), 757–762.Google Scholar
3Agueh, M.. Sharp Gagliardo–Nirenberg inequalities via p-Laplacian type equations. NoDEA Nonlinear Differ. Equ. Appl. 15 (2008b), 457–472.Google Scholar
4Agueh, M., Ghoussoub, N. and Kang, X.. Geometric inequalities via a general comparison principle for interacting gases. Geom. Funct. Anal. 14 (2004), 215–244.Google Scholar
5Aubin, T.. Problèmes isopérimétriques et espaces de Sobolev. (French) J. Differ. Geom. 11 (1976), 573–598.Google Scholar
6Bakry, D., Gentil, I. and Ledoux, M.. Analysis and geometry of Markov diffusion operators. Grundlehren der Mathematischen Wissenschaften,vol. 348 (Berlin: Springer, 2013).Google Scholar
7Beckner, W.. Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality. Ann. of Math. (2) 138 (1993), 213–242.Google Scholar
8Beckner, W. and Pearson, M.. On sharp Sobolev embedding and the logarithmic Sobolev inequality. Bull. London Math. Soc. 30 (1998), 80–84.Google Scholar
9Bellazzini, J., Frank, R. L. and Visciglia, N.. Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems. Math. Ann. 360 (2014), 653–673.Google Scholar
10Brenier, Y.. Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44 (1991), 375–417.Google Scholar
11Brezis, H. and Marcus, M.. Hardy's inequalities revisited. Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997a), 217–237 (1998).Google Scholar
12Brezis, H. and Vázquez, J. L.. Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complut. Madrid 10 (1997b), 443–469.Google Scholar
13Cabré, X. and Ros-Oton, X.. Sobolev and isoperimetric inequalities with monomial weights. J. Differ. Equ. 255 (2013), 4312–4336.Google Scholar
14Caffarelli, L., Kohn, R. and Nirenberg, L.. First order interpolation inequalities with weights. Compositio Math. 53 (1984), 259–275.Google Scholar
15Caldiroli, P. and Musina, R.. Symmetry breaking of extremals for the Caffarelli–Kohn–Nirenberg inequalities in a non-Hilbertian setting. Milan J. Math. 81 (2013), 421–430.Google Scholar
16Catrina, F. and Wang, Z.-Q.. On the Caffarelli–Kohn–Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. Comm. Pure Appl. Math. 54 (2001), 229–258.Google Scholar
17Chou, K. S. and Chu, C. W.. On the best constant for a weighted Sobolev–Hardy inequality. J. London Math. Soc. (2) 48 (1993), 137–151.Google Scholar
18Cordero-Erausquin, D., Nazaret, B. and Villani, C.. A mass transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities. Adv. Math. 182 (2004), 307–332.Google Scholar
19Del-Pino, M. and Dolbeault, J.. Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. 81 (2002), 847–875.Google Scholar
20Del-Pino, M. and Dolbeault, J.. The optimal Euclidean L p-Sobolev logarithmic inequaity. J. Funct. Anal. 197 (2003), 151–161.Google Scholar
21Dolbeault, J., Felmer, P., Loss, M. and Paturel, E.. Lieb–Thirring type inequalities and Gagliardo–Nirenberg inequalities for systems. J. Funct. Anal. 238 (2006), 193–220.Google Scholar
22Dolbeault, J., Esteban, M. J., Laptev, A. and Loss, M.. One-dimensional Gagliardo–Nirenberg–Sobolev inequalities: remarks on duality and flows. J. Lond. Math. Soc. (2) 90 (2014), 525–550.Google Scholar
23Dolbeault, J., Esteban, M. J. and Loss, M.. Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces. Invent. Math. 206 (2016), 397–440.Google Scholar
24Dong, M. and Lu, G.. Best constants and existence of maximizers for weighted Moser-Trudinger inequalities. Calc. Var. Partial Differ. Equ. 55 (2016), 26, Paper No. 88.Google Scholar
25Filippas, S., Maz'ya, V. and Tertikas, A.. Critical Hardy–Sobolev inequalities. J. Math. Pures Appl. (9) 87 (2007), 37–56.Google Scholar
26Gazzola, F., Grunau, H.-C. and Mitidieri, E.. Hardy inequalities with optimal constants and remainder terms. Trans. Amer. Math. Soc. 356 (2004), 2149–2168.Google Scholar
27Gentil, I.. The general optimal L p-Euclidean logarithmic Sobolev inequality by Hamilton–Jacobi equations. J. Funct. Anal. 202 (2003), 591–599.Google Scholar
28Ghoussoub, N. and Moradifam, A.. Functional inequalities: new perspectives and new applications. Mathematical Surveys and Monographs,vol. 187 (Providence, RI: American Mathematical Society, 2013), xxiv+299.Google Scholar
29Ghoussoub, N. and Yuan, C.. Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans. Amer. Math. Soc. 352 (2000), 5703–5743.Google Scholar
30Ha, H. B. and Mai, T. T.. A Gagliardo–Nirenberg inequality for Orlicz and Lorentz spaces on ${{\open R}}_+^n$. Vietnam J. Math. 35 (2007), 415–427.Google Scholar
31Lam, N.. Hardy and Caffarelli–Kohn–Nirenberg inequalities with monomial weights. Preprint (2017a).Google Scholar
32Lam, N.. Sharp Trudinger–Moser inequalities with monomial weights. NoDEA Nonlinear Differential Equations Appl. 24 (2017), 24:39.Google Scholar
33Lam, N. and Lu, G.. Sharp constants and optimizers for a class of the Caffarelli–Kohn–Nirenberg inequalities. Adv. Nonlinear Stud. 17 (2017), 457–480.Google Scholar
34Lieb, E. H.. Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. of Math. (2) 118 (1983), 349–374.Google Scholar
35Lions, P.-L.. The concentration-compactness principle in the calculus of variations. The locally compact case I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984a), 109–145.Google Scholar
36Lions, P.-L.. The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984b), 223–283.Google Scholar
37Lions, P.-L.. The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoamericana 1 (1985a), 145–201.Google Scholar
38Lions, P.-L.. The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoamericana 1 (1985b), 45–121.Google Scholar
39McCann, R. J.. Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80 (1995), 309–323.Google Scholar
40McCann, R. J.. A convexity principle for interacting gases. Adv. Math. 128 (1997), 153–179.Google Scholar
41Moroz, V. and Van Schaftingen, J.. Nonlocal Hardy type inequalities with optimal constants and remainder terms. Ann. Univ. Buchar. Math. Ser. 3(LXI) (2012), 187–200.Google Scholar
42Nguyen, V. H.. Sharp weighted Sobolev and Gagliardo–Nirenberg inequalities on half-spaces via mass transport and consequences. Proc. Lond. Math. Soc. (3) 111 (2015), 127–148.Google Scholar
43Talenti, G.. Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976), 353–372.Google Scholar
44Wang, Z.-Q. and Willem, M.. Caffarelli–Kohn–Nirenberg inequalities with remainder terms. J. Funct. Anal. 203 (2003), 550–568.Google Scholar
45Zhong, X. and Xou, W.. Existence of extremal functions for a family of Caffarelli–Kohn–Nirenberg inequalities. arXiv:1504.00433.Google Scholar