Advertisement

Weighted variable exponent Sobolev estimates for elliptic equations with non-standard growth and measure data

  • The Anh Bui
  • Xuan Thinh Duong
Article

Abstract

Consider the following nonlinear elliptic equation of p(x)-Laplacian type with nonstandard growth
$$\begin{aligned} {\left\{ \begin{array}{ll} \mathrm{div}a(Du, x)=\mu &{}\quad \text {in} \quad \Omega ,\\ u=0 &{} \quad \text {on} \quad \partial \Omega , \end{array}\right. } \end{aligned}$$
where \(\Omega \) is a Reifenberg domain in \(\mathbb {R}^n\), \(\mu \) is a Radon measure defined on \(\Omega \) with finite total mass and the nonlinearity \(a: \mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathbb {R}^n\) is modeled upon the \(p(\cdot )\)-Laplacian. We prove the estimates on weighted variable exponent Lebesgue spaces for gradients of solutions to this equation in terms of Muckenhoupt–Wheeden type estimates. As a consequence, we obtain some new results such as the weighted \(L^q-L^r\) regularity (with constants \(q < r\)) and estimates on Morrey spaces for gradients of the solutions to this non-linear equation.

Keywords

Nonlinear p(x)-Laplacian type equation Measure data Reifenberg domain Weighted generalized Lebesgue spaces 

Mathematics Subject Classification

35B65 35J60 35J99 

References

  1. 1.
    Almeida, A., Harjulehto, P., Hästö, P., Lukkari, T.: Riesz and Wolff potentials and elliptic equations in variable exponent weak Lebesgue spaces. Ann. Mat. Pura Appl. (4) 194(2), 405–424 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Acerbi, E., Mingione, G.: Regularity results for stationary electro-rheological fluids. Arch. Ration. Mech. Anal. 164, 213–259 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Acerbi, E., Mingione, G.: Gradient estimates for the \(p(x)\)-Laplacean system. J. Reine Angew. Math. 584, 117–148 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baroni, P., Habermann, J.: Elliptic interpolation estimates for non-standard growth operators. Ann. Acad. Sci. Fenn. Math. 39(1), 119–162 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vázquez, J.L.: An \(L^1\)-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 22, 241–273 (1995)zbMATHGoogle Scholar
  6. 6.
    Bendahmane, M., Wittbold, P., Zimmermann, A.: Renormalized solutions for a nonlinear parabolic equation with variable exponents and \(L^1\)-data. J. Differ. Equ. 249(6), 1483–1515 (2010)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bögelein, V., Habermann, J.: Gradient estimates via non standard potentials and continuity. Ann. Acad. Sci. Fenn. Math. 35, 641–678 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Boccardo, L., Gallouët, T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149–169 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Boccardo, L., Gallouët, T.: Nonlinear elliptic equations with right-hand side measures. Commun. Partial Differ. Equ. 17(34), 641–655 (1992)MathSciNetzbMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    Bui, T.A., Duong, X.T., Le, X.T.: Regularity estimates for higher order elliptic systems on Reifenberg flat domains. J. Differ. Equ. 261(10), 5637–5669 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Byun, S.-S., Wang, L.: Elliptic equations with BMO coefficients in Reifenberg domains. Commun. Pure Appl. Math. 57(10), 1283–1310 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Byun, S.-S., Ok, J., Ryu, S.: Global gradient estimates for elliptic equations of \(p(x)\)-Laplacian type with BMO nonlinearity. J. Reine Angew. Math. 715, 1–38 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Byun, S.-S., Ok, J.: On \(W^{1,q(\cdot )}\)-estimates for elliptic equations of \(p(x)\)-Laplacian type. J. Math. Pures Appl. (9) 106(3), 512–545 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Caffarelli, L.A., Peral, I.: On \(W^{1, p}\) estimates for elliptic equations in divergence form. Commun. Pure Appl. Math. 51, 1–21 (1998)CrossRefzbMATHGoogle Scholar
  16. 16.
    Capone, C., Cruz-Uribe, D., Fiorenza, A.: The fractional maximal operator and fractional integrals on variable \(L^p\) spaces. Rev. Mat. Iberoam. 23(3), 743–770 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66, 1383–1406 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Basel (2013)CrossRefzbMATHGoogle Scholar
  19. 19.
    David, G., Toro, T.: A generalization of Reifenbergs theorem in \(\mathbb{R}^3\). Geom. Funct. Anal. 18(4), 1168–1235 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Diening, L., Ružička, M.: Calderón-Zygmund operators on generalized Lebesgue spaces \(L^{p(\cdot )}\) and problems related to fluid dynamics. J. Reine Angew. Math. 563, 197220 (2003)zbMATHGoogle Scholar
  21. 21.
    Diening, L., Harjulehto, P., Hästö, P., Ružička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Heidelberg (2011)CrossRefzbMATHGoogle Scholar
  22. 22.
    Duoandikoetxea, J.: Fourier Analysis. Graduate Studies in Mathematics, vol. 29. American Math. Soc, Providence (2000)Google Scholar
  23. 23.
    Duzaar, F., Mingione, G.: Gradient estimates via non-linear potentials. Am. J. Math. 133, 1093–1149 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Duzaar, F., Mingione, G.: Gradient estimates via linear and nonlinear potentials. J. Funct. Anal. 259(11), 2961–2998 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Duzaar, F., Mingione, G.: Gradient continuity estimates. Calc. Var. Partial Differ. Equ. 39(3–4), 379–418 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific Publishing Co., Inc., River Edge (2003)CrossRefzbMATHGoogle Scholar
  27. 27.
    Harjulehto, P., Kuusi, T., Lukkari, T., Marola, N., Parviainen, M.: Harnacks inequality for quasiminimizers with non-standard growth conditions. J. Math. Anal. Appl. 344(1), 504–520 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kilpeläinen, T., Malý, J.: Degenerate elliptic equations with measure data and nonlinear potentials. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 19, 591–613 (1992)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Kilpeläinen, T., Malý, J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172, 137–161 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kuusi, T., Mingione, G.: Linear potentials in nonlinear potential theory. Arch. Ration. Mech. Anal. 207(1), 215–246 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kuusi, T., Mingione, G.: Universal potential estimates. J. Funct. Anal. 262(10), 4205–4269 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Lieberman, G.M.: The natural generalization of the natural conditions of Ladyzenskaja and Uraltzeva for elliptic equations. Commun. Partial Differ. Equ. 16, 311–361 (1991)CrossRefzbMATHGoogle Scholar
  33. 33.
    Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Mingione, G.: Gradient potential estimates. J. Eur. Math. Soc.: JEMS 13(2), 459–486 (2011)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Mengesha, T., Phuc, N.C.: Global estimates for quasilinear elliptic equations on Reifenberg flat domains. Arch. Ration. Mech. Anal. 203(1), 189–216 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Mengesha, T., Phuc, N.C.: Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains. J. Differ. Equ. 250, 2485–2507 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Nguyen, Q.-H.: Potential estimates and quasilinear parabolic equations with measure data. arxiv:1405.2587
  38. 38.
    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
  39. 39.
    Rajagopal, K., Ružička, M.: Mathematical modelling of electro-rheological fluids. Contin. Mech. Thermodyn. 13, 59–78 (2001)CrossRefzbMATHGoogle Scholar
  40. 40.
    Reifenberg, E.: Solutions of the plateau problem for \(m\)-dimensional surfaces of varying topological type. Acta Math. 104, 1–92 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Ružička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748. Springer, Berlin (2000)zbMATHGoogle Scholar
  42. 42.
    Sanchón, M., Urbano, J.M.: Entropy solutions for the \(p(x)\)-Laplace equation. Trans. Am. Math. Soc. 361(12), 6387–6405 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Toro, T.: Doubling and flatness: geometry of measures. Not. Am. Math. Soc. 44, 1087–1094 (1997)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Trudinger, N.S., Wang, J.X.: On the weak continuity of elliptic operators and applications to potential theory. Am. J. Math. 124, 369–410 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Zhang, X., Fu, Y.: Solutions for nonlinear elliptic equations with variable growth and degenerate coercivity. Ann. Mat. Pura Appl. (4) 193(1), 133–161 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Zhang, C., Zhou, S.: Global weighted estimates for quasilinear elliptic equations with non-standard growth. J. Funct. Anal. 267(2), 605–642 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 50(4), 675–710 (1986)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsMacquarie UniversitySydneyAustralia

Personalised recommendations