Advertisement

A weighted anisotropic Sobolev type inequality and its applications to Hardy inequalities

  • 38 Accesses

Abstract

In this paper we focus our attention on an embedding result for a weighted Sobolev space that involves as weight the distance function from the boundary taken with respect to a general smooth gauge function F. Starting from this type of inequalities we prove some refined Hardy-type inequalities.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 199

This is the net price. Taxes to be calculated in checkout.

Notes

  1. 1.

    Since \(d_F\in \mathrm{BV}(\varOmega )\), its level sets \(\{d_F<r\}\) are of finite perimeter for a.e. \(r\in (0,\infty )\) (see [12, Theorem 1–§5.5])

References

  1. 1.

    Alvino, A., Ferone, A., Mercaldo, A., Takahashi, F., Volpicelli, R.: Finsler Hardy–Kato’s inequality. J. Math. Anal. Appl. 470, 360–374 (2019)

  2. 2.

    Alvino, A., Ferone, V., Trombetti, G., Lions, P.-L.: Convex symmetrization and applications. Ann. Inst. Henri Poincaré Anal. Non Linéaire 14, 275–293 (1997)

  3. 3.

    Bal, K.: Hardy inequalities for Finsler \(p\)-Laplacian in the exterior domain. Mediterr. J. Math. 14, Art. 165 (2017)

  4. 4.

    Ball, K., Carlen, E., Lieb, E.: Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math. 115, 463–482 (1994)

  5. 5.

    Barbatis, G., Filippas, S., Tertikas, A.: A unified approach to improved \(L^p\) Hardy inequalities with best constants. Trans. Am. Math. Soc. 356, 2169–2196 (2003)

  6. 6.

    Brasco, L., Franzina, G.: Convexity properties of Dirichlet integrals and Picone-type inequalities Kodai. Math. J. 37, 769–799 (2014)

  7. 7.

    Cabré, X., Ros-Oton, X.: Sobolev and isoperimetric inequalities with monomial weights. J. Differ. Equ. 255, 4312–4336 (2013)

  8. 8.

    Cianchi, A., Salani, P.: Overdetermined anisotropic elliptic problems. Math. Ann. 345, 859–881 (2009)

  9. 9.

    Crasta, G., Malusa, A.: The distance function from the boundary in a Minkowski space. Trans. Am. Math. Soc. 359, 5725–5759 (2007)

  10. 10.

    Della Pietra, F., di Blasio, G., Gavitone, N.: Anisotropic Hardy inequalities. Proc. R. Soc. Edinb. Sect. A 148, 483–498 (2018)

  11. 11.

    Dyda, B., Lehrbäck, J., Vähäkangas, A.V.: Fractional Hardy–Sobolev type inequalities for half spaces and John domains. Proc. Am. Math. Soc. 146, 3393–3402 (2018)

  12. 12.

    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Funcions. Stud. Adv. Math. CRC Press, Boca Raton (1991)

  13. 13.

    Filippas, S., Maz’ya, V.G., Tertikas, A.: Critical Hardy–Sobolev inequalities. J. Math. Pures Appl. 87, 37–56 (2007)

  14. 14.

    Filippas, S., Psaradakis, G.: The Hardy–Morrey and Hardy–John–Nirenberg inequalities involving distance to the boundary. J. Differ. Equ. 261, 3107–3136 (2016)

  15. 15.

    Frank, R.L., Loss, M.: Hardy–Sobolev–Maz’ya inequalities for arbitrary domains. J. Math. Pures Appl. 97, 39–54 (2011)

  16. 16.

    Giga, Y., Pisante, G.: On representation of boundary integrals involving the mean curvature for mean-convex domains. In: Geometric Partial Differential Equations. CRM Series, vol. 15, Norm. edn, pp. 171–187 (2013)

  17. 17.

    Gromov, M.: Sign and geometric meaning of curvature. Rend. Semin. Mat. Fis. Milano 61, 9–123 (1991)

  18. 18.

    Kufner, A., John, O., Fučík, S.: Function Spaces. Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis. Noordhoff International Publishing, Leyden, Academia, Prague (1977)

  19. 19.

    Lewis, R.T., Li, J., Li, Y.-Y.: A geometric characterization of a sharp Hardy inequality. J. Funct. Anal. 262, 3159–3185 (2012)

  20. 20.

    Lindqvist, P.: On the equation \((|\nabla u|^{p-2}\nabla u)+\lambda |u|^{p-2}u=0\). Proc. Am. Math. Soc. 109, 157–164 (1990)

  21. 21.

    Maz’ya, V.G.: Sobolev Spaces. Translated from Russian by Tatyana Shaposhnikova. Springer Ser. Soviet Math. Springer, Berlin (1985)

  22. 22.

    Maz’ya, V.G., Shaposhnikova, T.: A collection of sharp dilation invariant integral inequalities for differentiable functions. In: Sobolev Spaces in Mathematics I, Int. Math. Ser. (N. Y.), vol. 8, pp. 223–247. Springer (2009)

  23. 23.

    Mercaldo, A., Sano, M., Takahashi, F.: Finsler Hardy inequalities. arXiv:1806.04901v2

  24. 24.

    Nguyen, V.H.: Sharp weighted Sobolev and Gagliardo–Nirenberg inequalities on half spaces via mass transport and consequences. Proc. Lond. Math. Soc. 111, 127–138 (2015)

  25. 25.

    Ohta, S.-I.: Uniform convexity and smoothness, and their applications in Finsler geometry. Math. Ann. 343, 669–699 (2009)

  26. 26.

    Psaradakis, G.: \(L^1\) Hardy inequalities with weights. J. Geom. Anal. 23, 1703–1728 (2013)

  27. 27.

    Rockafellar, R.T.: Convex Analysis. Princeton Math. Ser., vol. 28. Princeton University Press, Princeton (1970)

  28. 28.

    Schneider, R.: Convex Bodies: the Brunn–Minkowski Theory (2nd expanded edition). Encyclopedia Math. Appl., vol. 151. Cambridge University Press, Cambridge (2014)

  29. 29.

    Van Schaftingen, J.: Anisotropic symmetrization. Ann. Inst. H. Poincaré Anal. Non Linéaire 23, 539–565 (2006)

Download references

Acknowledgements

G. di Blasio and G. Pisante are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the IstitutoNazionale di Alta Matematica (INdAM) whose support through the GNAMPA Project 2019 Pogram (U-UFMBAZ-2019-000477 11-03-2019) is gratefully acknowledged. G Pisante and G. Psaradakis also acknowledge the INdAM-GNAMPA for the Visiting Professor Program 2018 (U-FMBAZ-2018-001525 18-12-2018). G. Psaradakis was supported in part from Università degli Studi della Campania “Luigi Vanvitelli” (D.R. 0950-2017) through a visiting researcher position. The authors are grateful to both referees whose remarks and suggestions helped us to considerably enhance the initial version of the article.

Author information

Correspondence to Giovanni Pisante.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Y. Giga.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

di Blasio, G., Pisante, G. & Psaradakis, G. A weighted anisotropic Sobolev type inequality and its applications to Hardy inequalities. Math. Ann. (2020) doi:10.1007/s00208-019-01930-4

Download citation

Mathematics Subject Classification

  • 53C60
  • 58J60
  • 26D10
  • 46E35