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Global integral gradient bounds for quasilinear equations below or near the natural exponent

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Arkiv för Matematik

Abstract

We obtain sharp integral potential bounds for gradients of solutions to a wide class of quasilinear elliptic equations with measure data. Our estimates are global over bounded domains that satisfy a mild exterior capacitary density condition. They are obtained in Lorentz spaces whose degrees of integrability lie below or near the natural exponent of the operator involved. As a consequence, nonlinear Calderón–Zygmund type estimates below the natural exponent are also obtained for \(\mathcal{A}\)-superharmonic functions in the whole space ℝn. This answers a question raised in our earlier work (On Calderón–Zygmund theory for p- and \(\mathcal{A}\)-superharmonic functions, to appear in Calc. Var. Partial Differential Equations, DOI 10.1007/s00526-011-0478-8) and thus greatly improves the result there.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M. and Vazquez, J. L., An L 1 theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. 22 (1995), 241–273.

    MATH  Google Scholar 

  3. Boccardo, L. and Gallouët, T., Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149–169.

    Article  MATH  MathSciNet  Google Scholar 

  4. Boccardo, L. and Gallouët, T., Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations 17 (1992), 641–655.

    Article  MATH  MathSciNet  Google Scholar 

  5. Byun, S. and Wang, L., Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math. 57 (2004), 1283–1310.

    Article  MATH  MathSciNet  Google Scholar 

  6. Caffarelli, L. and Peral, I., On W 1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998), 1–21.

    Article  MATH  MathSciNet  Google Scholar 

  7. Dal Maso, G., Murat, F., Orsina, A. and Prignet, A., Renormalized solutions of elliptic equations with general measure data, Ann. Sc. Norm. Super. Pisa Cl. Sci. 28 (1999), 741–808.

    MATH  MathSciNet  Google Scholar 

  8. Dall’Aglio, A., Approximated solutions of equations with L 1 data. Applications to the H-convergence of quasilinear parabolic equations, Ann. Mat. Pura Appl. 170 (1996), 207–240.

    Article  MATH  MathSciNet  Google Scholar 

  9. Del Vecchio, T., Nonlinear elliptic equations with measure data, Potential Anal. 4 (1995), 185–203.

    Article  MATH  MathSciNet  Google Scholar 

  10. Duzaar, F. and Mingione, G., Gradient estimates via linear nonlinear potentials, J. Funct. Anal. 259 (2010), 2961–2998.

    Article  MATH  MathSciNet  Google Scholar 

  11. Duzaar, F. and Mingione, G., Gradient estimates via non-linear potentials, Amer. J. Math. 133 (2011), 1093–1149.

    Article  MATH  MathSciNet  Google Scholar 

  12. Giusti, E., Direct Methods in the Calculus of Variations, World Scientific, River Edge, NJ, 2003.

    Book  MATH  Google Scholar 

  13. Heinonen, J., Kilpeläinen, T. and Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Oxford, 1993.

    MATH  Google Scholar 

  14. Iwaniec, T., Projections onto gradient fields and L p-estimates for degenerated elliptic operators, Studia Math. 75 (1983), 293–312.

    MATH  MathSciNet  Google Scholar 

  15. Jerison, D. and Kenig, C., The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), 161–219.

    Article  MATH  MathSciNet  Google Scholar 

  16. Kilpeläinen, T. and Li, G., Estimates for p-Poisson equations, Differential Integral Equations 13 (2000), 791–800.

    MATH  MathSciNet  Google Scholar 

  17. Kilpeläinen, T. and Malý, J., Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Sc. Norm. Super. Pisa Cl. Sci. 19 (1992), 591–613.

    MATH  Google Scholar 

  18. Kilpeläinen, T. and Malý, J., The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137–161.

    Article  MATH  MathSciNet  Google Scholar 

  19. Lewis, J. L., Uniformly fat sets, Trans. Amer. Math. Soc. 308 (1988), 177–196.

    Article  MATH  MathSciNet  Google Scholar 

  20. Mengesha, T. and Phuc, N. C., Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains, J. Differential Equations 250 (2011), 2485–2507.

    Article  MATH  MathSciNet  Google Scholar 

  21. Meyers, N. G., An L p-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. 17 (1963), 189–206.

    MATH  MathSciNet  Google Scholar 

  22. Mikkonen, P., On the Wolff potential and quasilinear elliptic equations involving measures, Ann. Acad. Sci. Fenn. Math. Diss. 104 (1996), 71.

    MathSciNet  Google Scholar 

  23. Mingione, G., The Calderón–Zygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa Cl. Sci. 6 (2007), 195–261.

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. Mingione, G., Nonlinear measure data problems, Milan J. Math. 79 (2011), 429–496.

    Article  MATH  MathSciNet  Google Scholar 

  26. Muckenhoupt, B. and Wheeden, R., Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261–274.

    Article  MATH  MathSciNet  Google Scholar 

  27. Phuc, N. C., On Calderón–Zygmund theory for p- and \(\mathcal{A}\)-superharmonic functions, Calc. Var. Partial Differential Equations 46 (2013), 165–181.

    Article  MATH  MathSciNet  Google Scholar 

  28. Phuc, N. C. and Torres, M., Characterizations of the existence and removable singularities of divergence measure vector fields, Indiana Univ. Math. J. 57 (2008), 1573–1597.

    Article  MATH  MathSciNet  Google Scholar 

  29. Phuc, N. C. and Verbitsky, I. E., Quasilinear and Hessian equations of Lane–Emden type, Ann. of Math. 168 (2008), 859–914.

    Article  MATH  MathSciNet  Google Scholar 

  30. Trudinger, N. S. and Wang, X.-J., On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math. 124 (2002), 369–410.

    Article  MATH  MathSciNet  Google Scholar 

  31. Wang, L., A geometric approach to the Calderón–Zygmund estimates, Acta Math. Sin. (Engl. Ser.) 19 (2003), 381–396.

    Article  MathSciNet  Google Scholar 

  32. Ziemer, W. P., Weakly Differentiable Functions, Grad. Texts in Math. 120, Springer, New York, 1989.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, LA, 70803, U.S.A.

    Nguyen Cong Phuc

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  1. Nguyen Cong Phuc
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Correspondence to Nguyen Cong Phuc.

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Supported in part by NSF grant DMS-0901083.

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Phuc, N.C. Global integral gradient bounds for quasilinear equations below or near the natural exponent. Ark Mat 52, 329–354 (2014). https://doi.org/10.1007/s11512-012-0177-5

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  • DOI: https://doi.org/10.1007/s11512-012-0177-5

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