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On Hardy inequalities with a remainder term

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Abstract

In this paper we study some improvements of the classical Hardy inequality. We add to the right hand side of the inequality a term that depends on some Lorentz norms of u or of its gradient and we find the best values of the constants for remaining terms. In both cases we show that the problem of finding the optimal value of the constant can be reduced to a spherically symmetric situation. This result is new when the right hand side is a Lorentz norm of the gradient.

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References

  1. Adimurthi, Chauduri N., Ramaswamy M.: An improved Hardy–Sobolev inequality and its application. Proc. Am. Math. Soc. 130, 485–505 (2002)

    Article  Google Scholar 

  2. Alvino A., Lions P.L., Trombetti G.: On optimization problems with prescribed rearrangements. Nonlinear Anal. 13, 185–220 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alvino A., Trombetti G.: Sulle migliori costanti di maggiorazione per una classe di equazioni ellittiche degeneri. Ricerche Math. 27, 413–428 (1978)

    MATH  MathSciNet  Google Scholar 

  4. 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 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bennet C., Sharpley R.: Interpolation of Operators , Pure Appl. Math., xiv+469 pp. Academic Press Inc., Boston (1988)

    Google Scholar 

  6. Brezis H., Marcus M.: Hardy’s inequalities revisited. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25(4), 217–237 (1998)

    MathSciNet  Google Scholar 

  7. Brezis H., Vazquez J.L.: Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complut. Madrid. 10, 443–469 (1997)

    MATH  MathSciNet  Google Scholar 

  8. Chauduri N., Ramaswamy M.: Existence of positive solutions of some semilinear elliptic equations with singular coefficients. Proc. Soc. Edinburgh Sect. A. 131, 1275–1295 (2001)

    Article  Google Scholar 

  9. Chong, K. M., Rice, N. M.: Equimeasurable rearrangements of functions, Queen’s Papers in Pure and Applied Mathematics, no. 28. Queen’s University, Kingston (1971)

  10. Cianchi A.: On the L q-norm of functions having equidistributed gradients. Nonlinear Anal. 26, 2007–2021 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ferone A., Volpicelli R.: Some relations between pseudo-rearrangement and relative rearrangement. Nonlinear Anal. Ser. A Theory Methods 41, 855–869 (2000)

    Article  MathSciNet  Google Scholar 

  12. Ferone A., Volpicelli R.: Polar factorization and pseudo-rearrangements: applications to Pólya–Szegö type inequalities. Nonlinear Anal. 53, 929–949 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Filippas S., Maz’ja V.G., Tertikas A.: Sharp Hardy–Sobolev inequalities. C. R. Math. Acad. Sci. Paris 339, 483–486 (2004)

    MATH  MathSciNet  Google Scholar 

  14. Filippas S., Tertikas A.: Optimizing improved Hardy inequalities. J. Funct. Anal. 192, 186–233 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gazzola F., Grunau H.C., Mitiediri E.: Hardy inequalities with optimal constants and remainder terms. Trans. Am. Math. Soc. 356, 2149–2168 (2004)

    Article  MATH  Google Scholar 

  16. Ghoussoub N., Moradifam A.: On the best possible remaining term in the Hardy inequality. Proc. Natl. Acad. Sci. USA 105, 13746–13751 (2008)

    Article  MathSciNet  Google Scholar 

  17. Giarrusso E., Nunziante D.: Symmetrization in a class of first-order Hamilton–Jacobi equations. Nonlinear Anal. 8, 289–299 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  18. Giarrusso E., Nunziante D.: Comparison theorems for a class of first order Hamilton–Jacobi equations. Ann. Fac. Sci. Toulouse Math. 7, 57–73 (1985)

    MATH  MathSciNet  Google Scholar 

  19. Hardy G.H.: Notes on some points in the integral calculus. Messenger Math. 48, 107/112 (1919)

    Google Scholar 

  20. Hardy G.H., Littlewood J.E., Pólya G.: Inequalities, xii+324 pp. Cambridge University Press, Cambridge (1988)

    Google Scholar 

  21. Maz’ja V.G.: Sobolev Spaces, Transl. from the Russian by T. O. Shaposhnikova, pp xix+486. Springer, Berlin (1985)

    Google Scholar 

  22. Mossino J., Temam R.: Directional derivative of the increasing rearrangement mapping, and application to a queer differential equation in plasma physics. Duke Math. J. 48, 475–495 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  23. Pólya G., Szegö G.: Isoperimetric Inequalities in Mathematical Physics, Ann. Math. Studies, no.27, pp xvi+279. Princeton University Press, Princeton (1951)

    Google Scholar 

  24. Talenti G.: On Functions Whose Gradients have a Prescribed Rearrangement, Inequalities and Applications, 559–571, World Sci. Ser. Appl. Anal., 3. World Scientific Publication, River Edge (1994)

    Google Scholar 

  25. Vazquez J.L., Zuazua E.: The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential. J. Funct. Anal. 173, 103–153 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia, Monte S. Angelo, 80126, Naples, Italy

    Angelo Alvino & Roberta Volpicelli

  2. Dipartimento per le Tecnologie, Università degli Studi di Napoli “Parthenope”, Centro Direzionale Isola C/4, 80143, Naples, Italy

    Bruno Volzone

Authors
  1. Angelo Alvino
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  2. Roberta Volpicelli
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  3. Bruno Volzone
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Correspondence to Angelo Alvino.

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Alvino, A., Volpicelli, R. & Volzone, B. On Hardy inequalities with a remainder term. Ricerche mat. 59, 265–280 (2010). https://doi.org/10.1007/s11587-010-0086-5

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  • DOI: https://doi.org/10.1007/s11587-010-0086-5

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