Good-\(\lambda \) and Muckenhoupt–Wheeden type bounds in quasilinear measure datum problems, with applications

Abstract

Weighted good-\(\lambda \) type inequalities and Muckenhoupt–Wheeden type bounds are obtained for gradients of solutions to a class of quasilinear elliptic equations with measure data. Such results are obtained globally over sufficiently flat domains in \(\mathbb {R}^n\) in the sense of Reifenberg. The principal operator here is modeled after the p-Laplacian, where for the first time singular case \(\frac{3n-2}{2n-1}<p\le 2-\frac{1}{n}\) is considered. Those bounds lead to useful compactness criteria for solution sets of quasilinear elliptic equations with measure data. As an application, sharp existence results and sharp bounds on the size of removable singular sets are deduced for a quasilinear Riccati type equation having a gradient source term with linear or super-linear power growth.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Springer, Berlin (1996)

    Google Scholar 

  2. 2.

    Adimurthi, K., Phuc, N.C.: Quasilinear equations with natural growth in the gradients in spaces of Sobolev multipliers. Calc. Var. Partial Differ. Equ. 57, 74 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Adimurthi, K., Phuc, N.C.: Nonlinear equations with gradient natural growth and distributional data, with applications to a Schrödinger type equation. J. Lond. Math. Soc. arXiv:1804.09612 (to appear)

  4. 4.

    Alvino, A., Lions, P.-L., Trombetti, G.: Comparison results for elliptic and parabolic equations via Schwarz symmetrization. Ann. Inst. H. Poincaré Anal. Non Linéaire 7, 37–65 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Benilan, P., Boccardo, L., Gallouet, T., Gariepy, R., Pierre, M., Vazquez, J.L.: An \(L^1\) theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa (IV) 22, 241–273 (1995)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Bensoussan, A., Boccardo, L., Murat, F.: On a nonlinear partial differential equation having natural growth terms and unbounded solution. Ann. Inst. H. Poincaré Anal. Non Linéaire. 5, 347–364 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  7. 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. 80, 90–124 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Boccardo, L., Gallouët, T., Orsina, L.: Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data. Ann. Inst. H. Poincaré Anal. Non Linéaire 13, 539–551 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Boccardo, L., Murat, F., Puel, J.-P.: Existence of bounded solutions for nonlinear elliptic unilateral problems. Ann. Mat. Pura Appl. 152, 183–196 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Boccardo, L., Murat, F., Puel, J.-P.: \(L^\infty \) estimate for some nonlinear elliptic partial differential equations and application to an existence result. SIAM J. Math. Anal. 23, 326–333 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Bidaut-Veron, M.F., Garcia-Huidobro, M., Veron, L.: Remarks on some quasilinear equations with gradient terms and measure data. Recent trends in nonlinear partial differential equations. II. Stationary problems, Contemp. Math., vol. 595. American Mathematical Society, Providence, RI, pp. 31–53 (2013)

  12. 12.

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

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Byun, S.-S., Wang, L.: Elliptic equations with BMO nonlinearity in Reifenberg domains. Adv. Math. 219, 1937–1971 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Caffarelli, L., Peral, I.: On \(W^{1, p}\) estimates for elliptic equations in divergence form. Commun. Pure Appl. Math. 51, 1–21 (1998)

    Article  MATH  Google Scholar 

  15. 15.

    Cho, K., Choe, H.-J.: Nonlinear degenerate elliptic partial differential equations with critical growth conditions on the gradient. Proc. Am. Math. Soc. 123, 3789–3796 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Dal Maso, G., Murat, F., Orsina, L., Prignet, A.: Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Super. Pisa (IV) 28, 741–808 (1999)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Della Pietra, F.: Existence results for non-uniformly elliptic equations with general growth in the gradient. Differ. Integral Equations 21, 821–836 (2008)

    MathSciNet  MATH  Google Scholar 

  18. 18.

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

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

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

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Giusti, E.: Direct Methods in the Calculus of Variations. World Scientic Publishing Co. Inc, River Edge (2003)

    Google Scholar 

  21. 21.

    Ferone, V., Messano, B.: Comparison and existence results for classes of nonlinear elliptic equations with general growth in the gradient. Adv. Nonlinear Stud. 7, 31–46 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Ferone, V., Murat, F.: Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small. Nonlinear Anal. 42, 1309–1326 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Ferone, V., Murat, F.: Nonlinear elliptic equations with natural growth in the gradient and source terms in Lorentz Spaces. J. Differ. Equations 256, 577–608 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Ferone, V., Posteraro, M.R., Rakotoson, J.M.: \(L^\infty \)-estimates for nonlinear elliptic problems with \(p\)-growth in the gradient. J. Inequal. Appl. 3, 109–125 (1999)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Frazier, M., Verbitsky, I.E.: Positive solutions to Schrödinger’s equation and the exponential integrability of the balayage (2015). arXiv:1509.09005 (Preprint)

  26. 26.

    Fukushima, M., Sato, K., Taniguchi, S.: On the closable part of pre-Dirichlet forms and the fine support of the underlying measures. Osaka J. Math. 28, 517–535 (1991)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Education, Inc., Upper Saddle River, pp. xii+931 (2004)

  28. 28.

    Grenon, N.: Existence and comparison results for quasilinear elliptic equations with critical growth in the gradient. J. Differ. Equations 171, 1–23 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Grenon, N., Trombetti, C.: Existence results for a class of nonlinear elliptic problems with \(p\)-growth in the gradient. Nonlinear Anal. 52, 931–942 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Grenon, N., Murat, F., Porretta, A.: Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms. C. R. Math. Acad. Sci. Paris 342, 23–28 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Grenon, N., Murat, F., Porretta, A.: A priori estimates and existence for elliptic equations with gradient dependent terms. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13, 137–205 (2014)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Hajlasz, P., Martio, O.: Traces of Sobolev functions on fractal type sets and characterization of extension domains. J. Funct. Anal. 143, 221–246 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Hansson, K., Maz’ya, V.G., Verbitsky, I.E.: Criteria of solvability for multidimensional Riccati equations. Ark. Mat. 37, 87–120 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Jaye, B., Maz’ya, V.G., Verbitsky, I.E.: Existence and regularity of positive solutions of elliptic equations of Schrdinger type. J. Anal. Math. 118, 577–621 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Jaye, B., Maz’ya, V.G., Verbitsky, I.E.: Quasilinear elliptic equations and weighted Sobolev–Poincaré inequalities with distributional weights. Adv. Math. 232, 513–542 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Jones, P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147, 71–88 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  37. 37.

    Kenig, C., Toro, T.: Free boundary regularity for harmonic measures and the Poisson kernel. Ann. Math. 150, 367–454 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  38. 38.

    Kenig, C., Toro, T.: Poisson kernel characterization of Reifenberg flat chord arc domains. Ann. Sci. École Norm. Sup. (4) 36, 323–401 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Kuusi, T., Mingione, G.: Linear potentials in nonlinear potential theory. Arch. Ration. Mech. Anal. 207, 215–246 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Kuusi, T., Mingione, G.: Universal potential estimates. J. Funct. Anal. 262, 4205–4269 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Maz’ya, V.G., Verbitsky, E.I.: Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers. Ark. Mat. 33, 81–115 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  42. 42.

    Mengesha, T., Phuc, N.C.: Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains. J. Differ. Equations 250, 1485–2507 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Messano, B.: Symmetrization results for classes of nonlinear elliptic equations with \(q\)-growth in the gradient. Nonlinear Anal. 64, 2688–2703 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    Meyer, P.-A.: Sur le lemme de la Valle Poussin et un théorème de Bismut, (French) Séminaire de Probabilités, XII (Univ. Strasbourg, Strasbourg, 1976/1977), Lecture Notes in Math., vol. 649. Springer, Berlin, pp. 770–774 (1978)

  45. 45.

    Mingione, G.: The Calderón–Zygmund theory for elliptic problems with measure data. Ann. Scu. Norm. Sup. Pisa Cl. Sci. (5) 6, 195–261 (2007)

    MATH  Google Scholar 

  46. 46.

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

    MathSciNet  Article  MATH  Google Scholar 

  47. 47.

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

    MathSciNet  Article  MATH  Google Scholar 

  48. 48.

    Porretta, A., Segura de León, S.: Nonlinear elliptic equations having a gradient term with natural growth. J. Math. Pures Appl. 85, 465–492 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  49. 49.

    Nguyen, Q.-H.: Potential estimates and quasilinear parabolic equations with measure data. arXiv:1405.2587v2 (submitted for publication)

  50. 50.

    Nguyen, Q.-H.: Global estimates for quasilinear parabolic equations on Reifenberg flat domains and its applications to Riccati type parabolic equations with distributional data. Calc. Var. Partial Differ. Equations 54, 3927–3948 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  51. 51.

    Phuc, N.C.: Quasilinear Riccati type equations with super-critical exponents. Commun. Partial Differ. Equations 35, 1958–1981 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  52. 52.

    Phuc, N.C.: Erratum to: Quasilinear Riccati type equations with super-critical exponents. Commun. Partial Differ. Equations 42, 1335–1341 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  53. 53.

    Phuc, N.C.: Global integral gradient bounds for quasilinear equations below or near the natural exponent. Ark. Mat. 52, 329–354 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  54. 54.

    Phuc, N.C.: Nonlinear Muckenhoupt–Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations. Adv. Math. 250, 387–419 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  55. 55.

    Phuc, N.C.: Corrigendum to: Nonlinear Muckenhoupt–Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations. Adv. Math. 328, 1353–1359 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  56. 56.

    Phuc, N.C.: Morrey global bounds and quasilinear Riccati type equations below the natural exponent. J. Math. Pures Appl. 102, 99–123 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  57. 57.

    Reifenberg, E.: Solutions of the plateau problem for \(m\)-dimensional surfaces of varying topological type. Acta Math. 104, 1–92 (1960)

    MathSciNet  Article  MATH  Google Scholar 

  58. 58.

    Sarason, D.: Functions of vanishing mean oscillation. Trans. Am. Math. Soc. 207, 391–405 (1975)

    MathSciNet  Article  MATH  Google Scholar 

  59. 59.

    Toro, T.: Doubling and flatness: geometry of measures. Notices Am. Math. Soc. 44, 1087–1094 (1997)

    MathSciNet  MATH  Google Scholar 

  60. 60.

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

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors kindly thank the anonymous referee for his/her comments that help improve the quality of the paper. Q.-H. Nguyen is supported by the Centro De Giorgi, Scuola Normale Superiore, Pisa, Italy. N. C. Phuc is supported in part by Simons Foundation, award number 426071.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Quoc-Hung Nguyen.

Additional information

Communicated by Loukas Grafakos.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Nguyen, QH., Phuc, N.C. Good-\(\lambda \) and Muckenhoupt–Wheeden type bounds in quasilinear measure datum problems, with applications. Math. Ann. 374, 67–98 (2019). https://doi.org/10.1007/s00208-018-1744-2

Download citation

Mathematics Subject Classification

  • Primary 35J60
  • 35J61
  • 35J62
  • Secondary 35J75
  • 42B37