Existence of very weak solutions to elliptic systems of p-Laplacian type

Article

Abstract

We study vector valued solutions to non-linear elliptic partial differential equations with p-growth. Existence of a solution is shown in case the right hand side is the divergence of a function which is only q integrable, where q is strictly below but close to the duality exponent \(p'\). It implies that possibly degenerate operators of p-Laplacian type are well posed in a larger class then the natural space of existence. The key novelty here is a refined a priori estimate, that recovers a duality relation between the right hand side and the solution in terms of weighted Lebesgue spaces.

Mathematics Subject Classification

35D99 35J57 35J60 35A01 

References

  1. 1.
    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. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22(2), 241–273 (1995)Google Scholar
  2. 2.
    Breit, D., Diening, L., Schwarzacher, S.: Solenoidal lipschitz truncation for parabolic pdes. Math. Models Methods Appl. Sci. 53(14), 2671–2700 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bulíček, M.: On continuity properties of monotone operators beyond the natural domain of definition. Manuscr. Math. 138(3–4), 287–298 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bulíček, M., Diening, L., Schwarzacher, S.: Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems. Anal. PDE (2016). (Acceptted)Google Scholar
  5. 5.
    Caffarelli, L.A., Peral, I.: On \(W^{1, p}\) estimates for elliptic equations in divergence form. Commun. Pure Appl. Math. 51(1), 1–21 (1998)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    DiBenedetto, E., Manfredi, J.: On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems. Am. J. Math. 115(5), 1107–1134 (1993)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Diening, L.: Lipschitz truncation, discrete Sobolev spaces. In: Function Spaces and Inequalities: Lectures from Spring School on Analysis Paseky, pp. 1–23. Matfyzpress, Prague (2013)Google Scholar
  8. 8.
    Diening, L., Fröschl, S.: Extensions in spaces with variable exponents—the half space. In: Harmonic Analysis and Nonlinear Partial Differential Equations, RIMS Kôkyûroku Bessatsu, B22, pp. 71–92. Res. Inst. Math. Sci. (RIMS), Kyoto (2010)Google Scholar
  9. 9.
    Diening, L., Kaplický, P., Schwarzacher, S.: BMO estimates for the \(p\)-Laplacian. Nonlinear Anal. 75(2), 637–650 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Diening, L., R\(\mathop {\rm u}\limits ^\circ \)žička, M., Wolf, J.: Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9(1), 1–46 (2010)Google Scholar
  11. 11.
    Giaquinta, M., Giusti, E.: On the regularity of the minima of variational integrals. Acta Math. 148, 31–46 (1982)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Greco, L., Iwaniec, T., Sbordone, C.: Variational integrals of nearly linear growth. Differ. Integral Equ. 10(4), 687–716 (1997)MathSciNetMATHGoogle Scholar
  13. 13.
    Iwaniec, T.: Projections onto gradient fields and \(L^{p}\)-estimates for degenerated elliptic operators. Stud. Math. 75(3), 293–312 (1983)MathSciNetMATHGoogle Scholar
  14. 14.
    Iwaniec, T.: \(p\)-Harmonic tensors and quasiregular mappings. Ann. Math. (2) 136(3), 589–624 (1992)Google Scholar
  15. 15.
    Kinnunen, J., Lewis, J.L.: Very weak solutions of parabolic systems of \(p\)-Laplacian type. Ark. Mat. 40(1), 105–132 (2002)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kristensen, J., Mingione, G.: The singular set of minima of integral functionals. Arch. Ration. Mech. Anal. 180(3), 331–398 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lewis, J.L.: On very weak solutions of certain elliptic systems. Commun. Partial Differ. Equ. 18(9–10), 1515–1537 (1993)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Mingione, G.: Nonlinear aspects of Calderón–Zygmund theory. Jahresber. Dtsch. Math.-Ver. 112(3), 159–191 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Nečas, J.: Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity. In: Theory of Nonlinear Operators (Proc. Fourth Internat. Summer School, Acad. Sci., Berlin, 1975). Akademie, Berlin (1977)Google Scholar
  20. 20.
    Serrin, J.: Isolated singularities of solutions of quasi-linear equations. Acta Math. 113, 219–240 (1965)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Torchinsky, A.: Real-variable methods in harmonic analysis. In: Pure and Applied Mathematics, vol. 123. Academic Press Inc, Orlando (1986)Google Scholar
  22. 22.
    Turesson, B.O.: Nonlinear potential theory and weighted Sobolev spaces. In: Lecture Notes in Mathematics, vol. 1736. Springer, Berlin (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Mathematical Institute, Faculty of Mathematics and PhysicsCharles University in PraguePragueCzech Republic
  2. 2.Department of Mathematical Analysis, Faculty of Mathematics and PhysicsCharles University in PraguePragueCzech Republic

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