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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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.
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.
Boccardo, L. and Gallouët, T., Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149–169.
Boccardo, L. and Gallouët, T., Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations 17 (1992), 641–655.
Byun, S. and Wang, L., Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math. 57 (2004), 1283–1310.
Caffarelli, L. and Peral, I., On W 1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998), 1–21.
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.
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.
Del Vecchio, T., Nonlinear elliptic equations with measure data, Potential Anal. 4 (1995), 185–203.
Duzaar, F. and Mingione, G., Gradient estimates via linear nonlinear potentials, J. Funct. Anal. 259 (2010), 2961–2998.
Duzaar, F. and Mingione, G., Gradient estimates via non-linear potentials, Amer. J. Math. 133 (2011), 1093–1149.
Giusti, E., Direct Methods in the Calculus of Variations, World Scientific, River Edge, NJ, 2003.
Heinonen, J., Kilpeläinen, T. and Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Oxford, 1993.
Iwaniec, T., Projections onto gradient fields and L p-estimates for degenerated elliptic operators, Studia Math. 75 (1983), 293–312.
Jerison, D. and Kenig, C., The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), 161–219.
Kilpeläinen, T. and Li, G., Estimates for p-Poisson equations, Differential Integral Equations 13 (2000), 791–800.
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.
Kilpeläinen, T. and Malý, J., The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137–161.
Lewis, J. L., Uniformly fat sets, Trans. Amer. Math. Soc. 308 (1988), 177–196.
Mengesha, T. and Phuc, N. C., Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains, J. Differential Equations 250 (2011), 2485–2507.
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.
Mikkonen, P., On the Wolff potential and quasilinear elliptic equations involving measures, Ann. Acad. Sci. Fenn. Math. Diss. 104 (1996), 71.
Mingione, G., The Calderón–Zygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa Cl. Sci. 6 (2007), 195–261.
Mingione, G., Gradient estimates below the duality exponent, Math. Ann. 346 (2010), 571–627.
Mingione, G., Nonlinear measure data problems, Milan J. Math. 79 (2011), 429–496.
Muckenhoupt, B. and Wheeden, R., Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261–274.
Phuc, N. C., On Calderón–Zygmund theory for p- and \(\mathcal{A}\)-superharmonic functions, Calc. Var. Partial Differential Equations 46 (2013), 165–181.
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.
Phuc, N. C. and Verbitsky, I. E., Quasilinear and Hessian equations of Lane–Emden type, Ann. of Math. 168 (2008), 859–914.
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.
Wang, L., A geometric approach to the Calderón–Zygmund estimates, Acta Math. Sin. (Engl. Ser.) 19 (2003), 381–396.
Ziemer, W. P., Weakly Differentiable Functions, Grad. Texts in Math. 120, Springer, New York, 1989.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported in part by NSF grant DMS-0901083.
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11512-012-0177-5