Advertisement

Optimal Normed Sobolev Type Inequalities on Manifolds I: The Nash Prototype

  • Jurandir Ceccon
  • Carlos DuránEmail author
  • Marcos Montenegro
Article
  • 20 Downloads

Abstract

Let M be a smooth compact manifold of dimension \(n \ge 1\) without boundary endowed with a volume form \(\omega \) and a fibrewise norm \(\mathcal {N}:T^*M \rightarrow \mathbb {R}\). For any \(p > q \ge 1\) and corresponding interpolation parameter \(\theta \), we prove that the optimal normed Nash inequality holds for any smooth function u on M,
$$\begin{aligned} \left( \int _M |u|^p\; \omega \right) ^{1/p \theta }\le & {} \left( N_{\mathrm{{opt}}} \left( \int _M \mathcal {N}^p(\mathrm{{d}}u)\; \omega \right) ^{1/p} \right. \\&+ \left. B\left( \int _M |u|^p\; \omega \right) ^{1/p} \right) \left( \int _M |u|^q\; \omega \right) ^{(1 - \theta )/\theta q} \end{aligned}$$
for some constant B, where \(N_{\mathrm{{opt}}}\) is the best possible constant. Its importance can be viewed from two perspectives. Firstly, this inequality is a powerful tool in the study of normed entropy and isoperimetrical inequalities on manifolds which have been established in the flat context by Gentil (J Funct Anal 202:591–599, 2003) and Cordero-Erausquin, Nazaret and Villani (Adv Math 182:307–332, 2004), , respectively. Secondly, this work introduces an appropriate framework to study Sobolev type inequalities on manifolds endowed with a very general way of measuring the involved quantities, instead of using the restricted Riemannian context.

Keywords

Optimal Sobolev inequalities Blow-up analysis Pointwise estimates Finsler geometry 

Mathematics Subject Classification

58C35 47J05 46E35 53C15 

Notes

Acknowledgements

The first author was supported by CNPq [PQ 308244/2017-6] and the third author was supported by CNPq (PQ 306855/2016-0) and Fapemig (APQ 02574-16). The authors are indebted to the anonymous referee, whose careful reading and invaluable suggestions, comments and remarks helped us improve substantially the presentation of this work.

References

  1. 1.
    Abraham, R., Marsden, J.: Foundations of Mechanics, 2nd edn. Addison-Wesley, Boston (1987)Google Scholar
  2. 2.
    Álvarez-Paiva, J.C., Thompson, A.C.: Volumes on normed and Finsler spaces. Riemann-Finsler Geom. MSRI Publ. 49, 1–46 (2009)zbMATHGoogle Scholar
  3. 3.
    Alves, M.T., Ceccon, J.: Sharp \(L^p\)-Moser inequality on Riemannian manifolds. J. Differ. Equ. 260(2), 1558–1584 (2016)zbMATHCrossRefGoogle Scholar
  4. 4.
    Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C., Scheffer, G.: Sur les inégalités de Sobolev logarithmiques, Panoramas et Synths̀es, vol. 10. Société Mathématique de France, Paris (2000)zbMATHGoogle Scholar
  5. 5.
    Aubin, T., Li, Y.Y.: On the best Sobolev inequality. J. Math. Pures Appl. 78, 353–387 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bakry, D.: L’hypercontractivitét son utilisation en thórie des semigroupes. In: Lectures on probability theory (Saint-Flour, 1992), Lecture Notes in Math. 1581, 1–114, Springer, Berlin, (1994)Google Scholar
  7. 7.
    Bakry, D., Coulhon, T., Ledoux, M., Sallof-Coste, L.: Sobolev inequalities in disguise. Indiana J. Math. 44(4), 1033–1074 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bao, D., CHern, S.S., Shen, Z.: An Introduction to Riemann-Finsler Geometry, Springer Graduate Text in Mathematics, vol. 200. Springer-Verlag, New York (2000)Google Scholar
  9. 9.
    Beckner, W.: Geometric proof of Nash’s inequality. Int. Math. Res. Not. 2, 67–71 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Beckner, W.: Geometric asymptotics and the logarithmic Sobolev inequality. Forum Math. 11(1), 105–137 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Bendikov, A., Coulhon, T., Saloff-Coste, L.: Ultracontractivity and embedding into \(L^\infty \). Math. Ann. 337, 817–853 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bobkov, S.G., Ledoux, M.: From Brunn-Minkowski to sharp Sobolev inequalities. Ann. di Matematica 187, 369–384 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Brouttelande, C.: The best-constant problem for a family of Gagliardo-Nirenberg inequalities on a compact Riemannian manifold. Proc. R. Soc. Edinb. 46, 147–157 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Carlen, E.A., Loss, M.: Sharp constant in Nash’s inequality. Int. Math. Res. Not. 7, 213–215 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Case, J.: A Yamabe-type problem on smooth metric measure spaces. J. Differ. Geom. 101(3), 467–505 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Ceccon, J.: General optimal Lp-Nash inequalities on Riemannian manifolds. Ann. Sci. Norm. Super. Pisa Cl. Sci. XV, 435–457 (2016)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ceccon, J., Durán, C.E.: Sharp constants in Riemannian Lp-Gagliardo-Nirenberg inequalities. J. Math. Anal. Appl. 433, 260–281 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Ceccon, J., Montenegro, M.: Homogeneous sharp Sobolev inequalities on product manifolds. Proc. R. Soc. Edinb. Sect. A 136(2), 277–300 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Ceccon, J., Montenegro, M.: Optimal Riemannian \(L^p\)-Gagliardo-Nirenberg inequalities revisited. J. Differ. Equ. 254, 2532–2555 (2013)zbMATHCrossRefGoogle Scholar
  20. 20.
    Ceccon, J., Montenegro, M.: Sharp \(L^p\)-entropy inequalities on manifolds. J. Funct. Anal. 269, 1591–1619 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Collion, S., Hebey, E., Vaugon, M.: Sharp Sobolev inequalities in the presence of a twist. Trans. Am. Math. Soc. 359, 2531–2537 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Cordero-Erausquin, D., Nazaret, B., Villani, C.: A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182, 307–332 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Cotsiolis, A., Labropoulos, N.: Sharp Nash inequalities on the unit sphere: the influence of symmetries. Nonlinear Anal. TMA 75, 612–624 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Coulhon, T.: Ultracontractivity and Nash type inequalities. J. Funct. Anal. 141(2), 510–539 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Davies, E.B.: Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press, Cambridge (1990)Google Scholar
  26. 26.
    Del Pino, M., Dolbeault, J.: The optimal Euclidean \(L^p\)-Sobolev logarithmic inequality. J. Funct. Anal. 1, 151–161 (2003)zbMATHCrossRefGoogle Scholar
  27. 27.
    Djadli, Z., Druet, O.: Extremal functions for optimal Sobolev inequalities on compact manifolds. Calc. Var. PDEs 12, 59–84 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Druet, O.: The best constants problem in Sobolev inequalities. Math. Ann. 314, 327–346 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Druet, O., Hebey, E.: The \(AB\) program in geometric analysis: sharp Sobolev inequalities and related problems. Mem. Am. Math. Soc. 160(761) (2002)Google Scholar
  30. 30.
    Druet, O., Hebey, E., Vaugon, M.: Optimal Nash’s inequalities on Riemannian manifolds: the influence of geometry. Int. Math. Res. Not. 14, 735–779 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Gentil, I.: The general optimal \(L^p\)-Euclidean logarithmic Sobolev inequality by Hamilton-Jacobi equations. J. Funct. Anal. 202, 591–599 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math. 97, 1061–1083 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Hebey, E.: Fonctions extrémales pour une inégalité de Sobolev optimale dans la classe conforme de la sphére. J. Math. Pures Appl. 77, 721–733 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Hebey, E.: Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Institute of Mathematical Sciences. Lecture Notes in Mathematics 5, (1999)Google Scholar
  35. 35.
    Hebey, E., Vaugon, M.: Meilleures constantes dans le théorème d’inclusion de Sobolev. Ann. Inst. H. Poincaré. 13, 57–93 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Hebey, E., Vaugon, M.: From best constants to critical functions. Math. Z. 237, 737–767 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Humbert, E.: Best constants in the \(L^2\)-Nash inequality. Proc. R. Soc. Edinb. 131, 621–646 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Humbert, E.: Extremal functions for the sharp \(L^2\)-Nash inequality. Calc. Var. Partial Differ. Equ. 22, 21–44 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Kato, T.: The Navier-Stokes equation for an incompressible fluid in \(\mathbb{R}^2\) with a measure as the initial vorticity. Differ. Integral Equ. 7, 949–966 (1994)zbMATHGoogle Scholar
  40. 40.
    Ledoux, M.: The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse Math. 6, 305–366 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Moser, J.: On Harnack’s theorem for elliptic differential equations. Commun. Pure Appl. Math. 14, 577–591 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Nazaret, B.: Best constant in Sobolev trace inequalities on the half-space. Nonlinear Anal. TMA 65, 1977–1985 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13, 115–162 (1959)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Ohta, S.: Finsler interpolation inequalities. Calc. Var. Partial Differ. Equ. 36, 211–249 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Ohta, S.: Heat flow on Finsler manifolds. Commun. Pure Appl. Math. 62, 1386–1433 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Palmer, B.: Anisotropic wavefronts and Laguerre geometry. J. Math. Phys. 56, 023503 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Serrin, J.: Local behavior of solutions of quasilinear equations. Acta Math. 111, 247–302 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51(1), 126–150 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Varopoulos, NTh, Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups, Cambridge Tracts in Mathematics, vol. 100. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  51. 51.
    Weissler, F.B.: Logarithmic Sobolev inequalities for the heat-diffusion semigroup. Trans. Am. Math. Soc. 237, 255–269 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Yajimaa, T., Yamasakib, K., Nagahamac, H.: Finsler metric and elastic constants for weak anisotropic media. Nonlinear Anal.: Real World Appl. 12(6), 3177–3184 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal do ParanáCuritibaBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal de Minas GeraisBelo HorizonteBrazil

Personalised recommendations