Advertisement

Mathematische Annalen

, Volume 374, Issue 1–2, pp 67–98 | Cite as

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

  • Quoc-Hung NguyenEmail author
  • Nguyen Cong Phuc
Article

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.

Mathematics Subject Classification

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

Notes

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.

References

  1. 1.
    Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Springer, Berlin (1996)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)Google Scholar
  12. 12.
    Byun, S.-S., Wang, L.: Elliptic equations with BMO coefficients in Reifenberg domains. Commun. Pure Appl. Math. 57, 1283–1310 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Byun, S.-S., Wang, L.: Elliptic equations with BMO nonlinearity in Reifenberg domains. Adv. Math. 219, 1937–1971 (2008)MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Duzaar, F., Mingione, G.: Gradient estimates via non-linear potentials. Am. J. Math. 133, 1093–1149 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Duzaar, F., Mingione, G.: Gradient estimates via linear and nonlinear potentials. J. Funct. Anal. 259, 2961–2998 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Giusti, E.: Direct Methods in the Calculus of Variations. World Scientic Publishing Co. Inc, River Edge (2003)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Education, Inc., Upper Saddle River, pp. xii+931 (2004)Google Scholar
  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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Jones, P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147, 71–88 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Kenig, C., Toro, T.: Free boundary regularity for harmonic measures and the Poisson kernel. Ann. Math. 150, 367–454 (1999)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Kuusi, T., Mingione, G.: Linear potentials in nonlinear potential theory. Arch. Ration. Mech. Anal. 207, 215–246 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Kuusi, T., Mingione, G.: Universal potential estimates. J. Funct. Anal. 262, 4205–4269 (2012)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)Google Scholar
  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)zbMATHGoogle Scholar
  46. 46.
    Mingione, G.: Gradient estimates below the duality exponent. Math. Ann. 346, 571–627 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Muckenhoupt, B., Wheeden, R.: Weighted norm inequalities for fractional integrals. Trans. Am. Math. Soc. 192, 261–274 (1974)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Phuc, N.C.: Quasilinear Riccati type equations with super-critical exponents. Commun. Partial Differ. Equations 35, 1958–1981 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Phuc, N.C.: Erratum to: Quasilinear Riccati type equations with super-critical exponents. Commun. Partial Differ. Equations 42, 1335–1341 (2017)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Reifenberg, E.: Solutions of the plateau problem for \(m\)-dimensional surfaces of varying topological type. Acta Math. 104, 1–92 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Sarason, D.: Functions of vanishing mean oscillation. Trans. Am. Math. Soc. 207, 391–405 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Toro, T.: Doubling and flatness: geometry of measures. Notices Am. Math. Soc. 44, 1087–1094 (1997)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Wang, L.: A geometric approach to the Calderoń–Zygmund estimates. Acta Math. Sin. (Engl. Ser.) 19, 381–396 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Scuola Normale Superiore, Centro Ennio de GiorgiPisaItaly
  2. 2.Department of MathematicsLouisiana State UniversityBaton RougeUSA

Personalised recommendations