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Gradient estimates for nonlinear elliptic equations with first order terms

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Abstract

We study existence and Lorentz regularity of distributional solutions to elliptic equations with measurable coefficients and either a convection or a drift first order term. The presence of such a term makes the problem not coercive. The main tools are pointwise estimates of the rearrangements of both the solution and its gradient.

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References

  1. Alvino, A., Ferone, V., Trombetti, G.: Estimates for the gradient of solutions of nonlinear elliptic equations with \(L^1 (\Omega )\) data. Ann. Mat. Pura Appl. 178, 129–142 (2000)

    Article  MathSciNet  Google Scholar 

  2. Alvino, A., Trombetti, G.: Sulle migliori costanti di maggiorazione per una classe di equazioni ellitiche degeneri. Ric. Mat. 27, 413–428 (1978)

    MATH  Google Scholar 

  3. Amann, H.: Ordinary differential equations: an introduction to nonlinear analysis. De Gruyter Studies in Mathematics, Walter de Gruyter & Co (1990)

  4. Benilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vazquez, J.L.: An \(L^1 (\Omega )\) theory of existence and uniqueness of nonlinear elliptic equations. Ann. Sc. Norm. Sup. Pisa, Cl. Sci. 22, 240–273 (1995)

    MATH  Google Scholar 

  5. Bennett, C.: Intermediate spaces and the class \(L \, \text{ log }^+ L\). Ark. Mat. 11, 215–228 (1973)

    Article  MathSciNet  Google Scholar 

  6. Betta, M.F., Ferone, V., Mercaldo, A.: Regularity for solutions of nonlinear elliptic equations. Bull. Sci. Math. 118, 539–567 (1994)

    MathSciNet  MATH  Google Scholar 

  7. Betta, M.F., Mercaldo, A., Murat, F., Porzio, M.M.: Existence of renormalized solutions to nonlinear elliptic equations with a lower-order term and right-hand side a measure. J. Math. Pures Appl. 82, 90–124 (2003)

    Article  MathSciNet  Google Scholar 

  8. Boccardo, L.: Some nonlinear Dirichlet problems in \(L^{1}(\Omega )\) involving lower order terms in divergence form, progress in elliptic and parabolic partial differential equations (Capri, 1994) Pitman Res. Notes Math. Ser. 350, 43–57 (1996)

    Google Scholar 

  9. Boccardo, L.: Marcinkiewicz estimates for solutions of some elliptic problems with nonregular data. Ann. Mat. Pura Appl. 188, 591–601 (2009)

    Article  MathSciNet  Google Scholar 

  10. Boccardo, L.: Some developments on Dirichlet problems with discontinuous coefficients. Boll. Unione Mat. Ital. 9, 285–297 (2009)

    MathSciNet  MATH  Google Scholar 

  11. Boccardo, L.: Dirichlet problems with singular convection terms and applications. J. Differ Equ. 258, 2290–2314 (2015)

    Article  MathSciNet  Google Scholar 

  12. Boccardo, L.: Two semilinear Dirichlet problems "almost" in duality. Boll. Unione Mat. Ital. 12, 349–356 (2018)

    Article  MathSciNet  Google Scholar 

  13. Boccardo, L., Buccheri, S., Cirmi, G.R.: Two linear noncoercive Dirichlet problems in duality. Milan J. Math. 86, 97–104 (2018)

    Article  MathSciNet  Google Scholar 

  14. Boccardo, L., Gallouet, T.: Nonlinear elliptic equations with right hand side measures. Commun. Part. Diff. Equ. 17, 641–655 (1992)

    Article  MathSciNet  Google Scholar 

  15. Boccardo, L., Gallouet, T.: \(W^{1,1}_0\) solutions in some borderline cases of Calderon-Zygmund theory. J. Differ. Equ. 253, 2698–2714 (2012)

    Article  Google Scholar 

  16. Boccardo, L., Murat, F.: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. Theory Methods Appl. 19, 581–597 (1992)

    Article  MathSciNet  Google Scholar 

  17. Boccardo, L., Orsina, L.: Very singular solutions for linear Dirichlet problems with singular convection terms. Nonlinear Anal. Theory Methods Appl. (2019). https://doi.org/10.1016/j.na.2019.01.027

    Article  MATH  Google Scholar 

  18. Bottaro, G., Marina, M.E.: Problema di Dirichlet per equazioni ellittiche di tipo variazionale su insiemi non limitati. Boll. Unione Mat. Ital. 8, 46–56 (1973)

    MathSciNet  MATH  Google Scholar 

  19. Buccheri, S.: The Bottaro-Marina slice method for distributional solutions to elliptic equations with drift term. Nonlinear Anal. 194, 111437 (2020)

    Article  MathSciNet  Google Scholar 

  20. Del Vecchio, T., Posteraro, M.R.: An existence result for nonlinear and noncoercive problems. Nonlinear Anal. Theory Methods Appl. 3, 191–206 (1998)

    Article  MathSciNet  Google Scholar 

  21. Droniou, J.: Non-coercive linear elliptic problems. Potential Anal. 17, 181–203 (2002)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Edu. Inc., Upper Saddle River (2004)

    MATH  Google Scholar 

  24. Kilpeläinen, T., Li, G.: Estimates for \(p\)-Poisson equations. Differ. Integral Equ. 13, 791–800 (2000)

    MathSciNet  MATH  Google Scholar 

  25. Ladyzenskaya, O., Ural’tseva, N.: Linear and Quasilinear Elliptic Equations, Translated by Scripta Tecnica. Accademic Press, New York (1968)

    Google Scholar 

  26. Leray, J., Lions, J.L.: Quelques résultats de Višik sur les problèmes elliptiques semi-linéaires par les méthodes de Minty et Browder. Bull. Soc. Math. 93, 97–107 (1965)

    MATH  Google Scholar 

  27. Mingione, G.: Gradient estimates below the duality exponent. Math. Ann. 346, 571–627 (2010)

    Article  MathSciNet  Google Scholar 

  28. Oklander, E.T.: Interpolación, espacios de Lorentz y teorema de Marcinkiewicz. Universidad de Buenos Aires, Buenos Aires (1965)

    MATH  Google Scholar 

  29. Porretta, A.: Some remarks on the regularity of solutions for a class of elliptic equations with measure data. Huston J. Math. 26, 183–213 (2000)

    MathSciNet  MATH  Google Scholar 

  30. Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier 15, 189–258 (1965)

    Article  MathSciNet  Google Scholar 

  31. Talenti, G.: Elliptic equations and rearrangements. Ann. Sc. Norm. Sup. Pisa. Cl. Sci. 3, 697–718 (1976)

    MathSciNet  MATH  Google Scholar 

  32. Trudinger, N.S.: Linear elliptic operator with measurable coefficients. Ann. Sc. Norm. Sup. Pisa. Cl. Sci. 27, 65–308 (1973)

    MathSciNet  Google Scholar 

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Acknowledgements

The author is grateful to the anonymous referee for the careful reading of the manuscript and for all the comments and corrections. The author has been partially supported by: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (PNPD/CAPES-UnB-Brazil), Grants 88887.363582/2019-00.

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  1. Departamento de Matemática, UNB - Brasília, DF, 70297-400, Brasil

    Stefano Buccheri

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  1. Stefano Buccheri
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Correspondence to Stefano Buccheri.

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Buccheri, S. Gradient estimates for nonlinear elliptic equations with first order terms. manuscripta math. 165, 191–225 (2021). https://doi.org/10.1007/s00229-020-01210-5

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