Skip to main content
SpringerLink
Log in

Borderline gradient continuity for nonlinear parabolic systems

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We consider the evolutionary \(p\)-Laplacean system

$$\begin{aligned} \partial _t u-\triangle _p u=F,\qquad p > \frac{2n}{n+2} \end{aligned}$$

in cylindrical domains of \( \mathbb R^{n}\times \mathbb R\), and prove the continuity of the spatial gradient \(Du\) under the Lorentz space assumption \(F\in L(n+2,1)\). When \(F\) is time independent the condition improves in \(F \in L(n,1)\). This is the limiting case of a result of DiBenedetto claiming that \(Du\) is Hölder continuous when \(F \in L^{q}\) for \(q>n+2\). At the same time, this is the natural nonlinear parabolic analog of a linear result of Stein, claiming the gradient continuity of solutions to the linear elliptic system \(\triangle u \in L(n,1)\) is continuous. New potential estimates are derived and moreover suitable nonlinear potentials are used to describe fine properties of solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Acerbi, E., Mingione, G.: Gradient estimates for a class of parabolic systems. Duke Math. J. 136, 285–320 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baroni, P.: New contributions to Nonlinear Calderón-Zygmund theory. Ph. D. Thesis, Scuola Normale Superiore, Pisa (2012)

  3. Boccardo, L., Dall’Aglio, A., Gallouët, T., Orsina, L.: Nonlinear parabolic equations with measure data. J. Funct. Anal. 147, 237–258 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cianchi, A.: Maximizing the \(L^\infty \) norm of the gradient of solutions to the Poisson equation. J. Geom. Anal. 2, 499–515 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cianchi, A., Maz’ya, V.: Global Lipschitz regularity for a class of quasilinear equations. Commun. PDE 36, 100–133 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cianchi, A., Maz’ya, V.: Global boundedness of the gradient for a class of nonlinear elliptic systems. Arch. Ration. Mech. Anal. 212, 129–177 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. DiBenedetto, E.: Degenerate Parabolic Equations. Universitext. Springer, New York (1993)

    Book  Google Scholar 

  8. DiBenedetto, E., Friedman, A.: Hölder estimates for nonlinear degenerate parabolic systems. J. Reine Angew. Math. (Crelles J.) 357, 1–22 (1985)

    MathSciNet  MATH  Google Scholar 

  9. DiBenedetto, E., Gianazza, U., Vespri, V.: Harnack estimates for quasi-linear degenerate parabolic differential equations. Acta Math. 200, 181–209 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. DiBenedetto, E., Gianazza, U., Vespri, V.: Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations. Ann. Scu. Norm. Sup. Pisa Cl. Sci. (V) 9, 385–422 (2010)

  11. Duzaar, F. Mingione, G.: Local Lipschitz regularity for degenerate elliptic systems. Ann. Inst. H. Poincaré Anal. Non Linèaire 27, 1361–1396 (2010)

  12. Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Edu. Inc., Upper Saddle River (2004)

    MATH  Google Scholar 

  13. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Reprint of the 1952 edition. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1988)

  14. Hamburger, C.: Regularity of differential forms minimizing degenerate elliptic functionals. J. Reine Angew. Math. (Crelles J.) 431, 7–64 (1992)

    MathSciNet  MATH  Google Scholar 

  15. Havin, M., Mazya, V.G.: A nonlinear potential theory. Russ. Math. Surveys 27, 71–148 (1972)

    Google Scholar 

  16. Hedberg, L.I., Wolff, T.: Thin sets in nonlinear potential theory. Ann. Inst. Fourier (Grenoble) 33, 161–187 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jaye, B., Verbitsky, I.: Local and global behaviour of solutions to nonlinear equations with natural growth terms. Arch. Ration. Mech. Anal. 204, 627–681 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jin, T., Mazya, V., Van Schaftingen, J.: Pathological solutions to elliptic problems in divergence form with continuous coefficients. C. R. Acad. Sci. Paris Ser. I 347, 773–778 (2009)

    Article  MATH  Google Scholar 

  19. Kilpeläinen, T., Malý, J.: Degenerate elliptic equations with measure data and nonlinear potentials. Ann. Scu. Norm. Sup. Pisa Cl. Sci. (V) 19, 591–613 (1992)

    MATH  Google Scholar 

  20. Kilpeläinen, T., Malý, J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172, 137–161 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kinnunen, J., Lewis, J.L.: Higher integrability for parabolic systems of \(p\)-Laplacian type. Duke Math. J. 102, 253–271 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kinnunen, J., Lewis, J.L.: Very weak solutions of parabolic systems of \(p\)-Laplacian type. Ark. Mat. 40, 105–132 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kinnunen, J., Lukkari, T., Parviainen, M.: An existence result for superparabolic functions. J. Funct. Anal. 258, 713–728 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kuusi, T., Mingione, G.: Potential estimates and gradient boundedness for nonlinear parabolic systems. Rev. Mat. Iberoamericana 28, 535–576 (2012)

    MathSciNet  MATH  Google Scholar 

  25. Kuusi, T., Mingione, G.: The Wolff gradient bound for degenerate parabolic equations. J. Eur. Math. Soc. 16, 835–892 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kuusi, T., Mingione, G.: Gradient regularity for nonlinear parabolic equations. Ann. Scu. Norm. Sup. Pisa Cl. Sci. (V) 12, 755–822 (2013)

    MathSciNet  MATH  Google Scholar 

  27. Kuusi, T., Mingione, G.: Riesz potentials and nonlinear parabolic equations. Arch. Ration. Mech. Anal. 212, 727–780 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kuusi, T., Mingione, G.: New perturbation methods for nonlinear parabolic problems. J. Math. Pures Appl. (IX) 98, 390–427 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kuusi, T., Mingione, G.: Guide to nonlinear potential estimates. Bull. Math. Sci. 4, 1–82 (2014)

    Article  MathSciNet  Google Scholar 

  30. Kuusi, T., Mingione, G.: A nonlinear stein theorem. Calc Var. PDE (2014, to appear)

  31. Manfredi, J.J.: Regularity for minima of functionals with \(p\)-growth. J. Differ. Equ. 76, 203–212 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  32. Maz’ya, V., McOwen, R.: Differentiability of solutions to second-order elliptic equations via dynamical systems. J. Differ. Equ. 250, 1137–1168 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Phuc, N.C., Verbitsky, I.E.: Quasilinear and Hessian equations of Lane–Emden type. Ann. Math. (II) 168, 859–914 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Phuc, N.C., Verbitsky, I.E.: Singular quasilinear and Hessian equations and inequalities. J. Funct. Anal. 256, 1875–1906 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Math. Ser., vol. 32. Princeton University Press, Princeton (1971)

  36. Stein, E.M.: Editor’s note: the differentiability of functions in \({\mathbb{R}}^n\). Ann. Math. (II) 113, 383–385 (1981)

    MATH  Google Scholar 

  37. Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Math. Series, vol. 43. Princeton University Press, Princeton (1993)

  38. Urbano, J.M.: The method of intrinsic scaling. A systematic approach to regularity for degenerate and singular PDEs. Lecture Notes in Mathematics, vol. 1930. Springer, Berlin (2008)

Download references

Acknowledgments

The authors are supported by the ERC grant 207573 “Vectorial Problems” and by the Academy of Finland project “Regularity theory for nonlinear parabolic partial differential equations”. The authors also thank Paolo Baroni for remarks on a preliminary version of the paper and the institute Mittag-Leffer for hospitality in the frame of the program “Evolutionary problems”. Last but not least we thank the referee for the careful reading of the manuscript, and the suggestions, leading to a better presentation.

Author information

Authors and Affiliations

  1. Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 111000, 00076 , Aalto, Finland

    Tuomo Kuusi

  2. Dipartimento di Matematica e Informatica, Università di Parma, Parco Area delle Scienze 53/a, Campus, 43124 , Parma, Italy

    Giuseppe Mingione

Authors
  1. Tuomo Kuusi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giuseppe Mingione
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giuseppe Mingione.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kuusi, T., Mingione, G. Borderline gradient continuity for nonlinear parabolic systems. Math. Ann. 360, 937–993 (2014). https://doi.org/10.1007/s00208-014-1055-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-014-1055-1

Search

Navigation