Skip to main content
Log in

A-priori gradient bound for elliptic systems under either slow or fast growth conditions

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We obtain an a-priori \(W_{{\mathrm{loc}}}^{1,\infty }\left( \Omega ; {\mathbb {R}}^{m}\right) \)-bound for weak solutions to the elliptic system

$$\begin{aligned} \text {div}A\left( x,Du\right) =\sum _{i=1}^{n}\frac{\partial }{\partial x_{i}} a_{i}^{\alpha }\left( x,Du\right) =0,\;\;\;\;\;\alpha =1,2,\ldots ,m, \end{aligned}$$

where \(\Omega \) is an open set of \({\mathbb {R}}^{n}\), \(n\ge 2\), u is a vector-valued map \(u:\Omega \subset {\mathbb {R}}^{n}\rightarrow {\mathbb {R}}^{m}\) . The vector field \(A\left( x,\xi \right) \) has a variational nature in the sense that \(A\left( x,\xi \right) =D_{\xi }f\left( x,\xi \right) \), where \( f=f\left( x,\xi \right) \) is a convex function with respect to \(\xi \in {\mathbb {R}}^{m\times n}\). In this context of vector-valued maps and systems, a classical assumption finalized to the everywhere regularity is a modulus-dependence in the energy integrand; i.e., we require that \(f\left( x,\xi \right) =g\left( x,\left| \xi \right| \right) \), where \( g\left( x,t\right) \) is convex and increasing with respect to the gradient variable \(t\in \left[ 0,\infty \right) \). We allow x-dependence, which turns out to be a relevant difference with respect to the autonomous case and not only a technical perturbation. Our assumptions allow us to consider both fast and slow growth. We consider fast growth even of exponential type; and slow growth, for instance of Orlicz-type with energy-integrands such as \(g\left( x,\left| Du\right| \right) =a(x)|Du|^{p(x)}\log (1+|Du|)\) or, when \(n=2,3\), even asymptotic linear growth with energy integrals of the type

$$\begin{aligned} \int _{\Omega }g\left( x,\left| Du\right| \right) dx\,=\int _{\Omega }\left\{ \left| Du\right| -a\left( x\right) \sqrt{\left| Du\right| }\right\} dx. \end{aligned}$$

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

Access this article

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. Baroni, P., Colombo, M., Mingione, G.: Harnack inequalities for double phase functionals. Nonlinear Anal. 121, 206–222 (2015)

    Article  MathSciNet  Google Scholar 

  2. Baroni, P., Colombo, M., Mingione, G., Nonautonomous functionals, borderline cases and related function classes, Algebra i Anal. 27, 6–50 (2015); translation in St. Petersburg Math. J. 27, 347–379 (2016)

  3. Baroni, P., Colombo, M., Mingione, G.: Regularity for general functionals with double phase. Calc. Var. Partial Differ. Equ. 57, 57–62 (2018)

    Article  MathSciNet  Google Scholar 

  4. Beck, L., Mingione, G.: Lipschitz bounds and non-uniformly ellipticity. Commun. Pure Appl. Math. 73, 944–1034 (2020)

    Article  Google Scholar 

  5. Bögelein, V., Dacorogna, B., Duzaar, F., Marcellini, P., Scheven, C.: Integral convexity and parabolic systems. SIAM J. Math. Anal.: SIMA 52, 1489–1525 (2020)

    Article  MathSciNet  Google Scholar 

  6. Bögelein, V., Duzaar, F., Marcellini, P.: Parabolic equations with \(p, q\)-growth. J. Math. Pures Appl. 100, 535–563 (2013)

    Article  MathSciNet  Google Scholar 

  7. Bögelein, V., Duzaar, F., Marcellini, P., Signoriello, S.: Nonlocal diffusion equations. J. Math. Anal. Appl. 432, 398–428 (2015)

    Article  MathSciNet  Google Scholar 

  8. Bousquet, P., Brasco, L.: Lipschitz regularity for orthotropic functionals with nonstandard growth conditions. Rev. Mat, Iberoam. (2020). https://www.ems-ph.org/JOURNALS/of_article.php?jrn=rmi&doi=1189

  9. Carozza, M., Giannetti, F., Leonetti, F., Passarelli di Napoli, A.: Pointwise bounds for minimizers of some anisotropic functionals. Nonlinear Anal. 177, 254–269 (2018)

    Article  MathSciNet  Google Scholar 

  10. Cencelja, M., Rădulescu, V., Repovš, D.: Double phase problems with variable growth. Nonlinear Anal. 177, 270–287 (2018)

    Article  MathSciNet  Google Scholar 

  11. Chlebicka, I.: A pocket guide to nonlinear differential equations in Musielak–Orlicz spaces. Nonlinear Anal. 175, 1–27 (1918)

    Article  MathSciNet  Google Scholar 

  12. Chlebicka, I., Gwiazda, P., Zatorska-Goldstein, A.: Parabolic equation in time and space dependent anisotropic Musielak–Orlicz spaces in absence of Lavrentiev’s phenomenon. Ann. I. H. Poincaré 36, 1431–1465 (2019)

    Article  MathSciNet  Google Scholar 

  13. Colombo, M., Mingione, G.: Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215, 443–496 (2015)

    Article  MathSciNet  Google Scholar 

  14. Colombo, M., Mingione, G.: Bounded minimisers of double phase variational integrals. Arch. Ration. Mech. Anal. 218, 219–273 (2015)

    Article  MathSciNet  Google Scholar 

  15. Cupini, G., Giannetti, F., Giova, R., Passarelli di Napoli, A.: Regularity results for vectorial minimizers of a class of degenerate convex integrals. J. Differ. Equ. 265, 4375–4416 (2018)

    Article  MathSciNet  Google Scholar 

  16. Cupini, G., Marcellini, P., Mascolo, E.: Regularity under sharp anisotropic general growth conditions. Discrete Contin. Dyn. Syst. Ser. B 11, 66–86 (2009)

    MathSciNet  MATH  Google Scholar 

  17. Cupini, G., Marcellini, P., Mascolo, E.: Local boundedness of solutions to quasilinear elliptic systems. Manuscr. Math. 137, 287–315 (2012)

    Article  MathSciNet  Google Scholar 

  18. Cupini, G., Marcellini, P., Mascolo, E.: Nonuniformly elliptic energy integrals with \(p, q\)-growth. Nonlinear Anal. 177, 312–324 (2018)

    Article  MathSciNet  Google Scholar 

  19. De Giorgi, E.: Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, (Italian). Boll. Un. Mat. Ital. 1, 135–137 (1968)

    MathSciNet  MATH  Google Scholar 

  20. De Filippis, C.: Higher integrability for constrained minimizers of integral functionals with \((p, q)\)-growth in low dimension. Nonlinear Anal. 170, 1–20 (2018)

    Article  MathSciNet  Google Scholar 

  21. De Filippis, C., Mingione, G.: On the regularity of minima of non-autonomous functionals. J. Geom. Anal. 30, 1584–1626 (2020)

    Article  MathSciNet  Google Scholar 

  22. De Silva, D., Savin, O.: Minimizers of convex functionals arising in random surfaces. Duke Math. J. 151, 487–532 (2010)

    Article  MathSciNet  Google Scholar 

  23. Eleuteri, M., Marcellini, P., Mascolo, E.: Lipschitz estimates for systems with ellipticity conditions at infinity. Ann. Mat. Pura Appl. 195, 1575–1603 (2016)

    Article  MathSciNet  Google Scholar 

  24. Eleuteri, M., Marcellini, P., Mascolo, E.: Lipschitz continuity for energy integrals with variable exponents. Atti. Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27, 61–87 (2016)

    Article  MathSciNet  Google Scholar 

  25. Eleuteri, M., Marcellini, P., Mascolo, E.: Regularity for scalar integrals without structure conditions. Adv. Calc. Var. (2018) (in press).https://doi.org/10.1515/acv-2017-0037

  26. Esposito, L., Leonetti, F., Mingione, G.: Sharp regularity for functionals with \((p, q)\) growth. J. Differ. Equ. 204, 5–55 (2004)

    Article  MathSciNet  Google Scholar 

  27. Giusti, E., Miranda, M.: Un esempio di soluzioni discontinue per un problema di minimo relativo ad un integrale regolare del calcolo delle variazioni (Italian). Boll Un. Mat. Ital. 1, 219–226 (1968)

    MathSciNet  MATH  Google Scholar 

  28. Harjulehto, P., Hästö, P., Toivanen, O.: Hölder regularity of quasiminimizers under generalized growth conditions. Calc. Var. Partial Differ. Equ. 56, 26 (2017)

    Article  Google Scholar 

  29. Hästö, O, Ok, J.: Maximal regularity for local minimizers of non-autonomous functionals (2019). arXiv:1902.00261 [math.AP]

  30. Marcellini, P.: Regularity and existence of solutions of elliptic equations with \(p, q\)-growth conditions. J. Differ. Equ. 90, 1–30 (1991)

    Article  MathSciNet  Google Scholar 

  31. Marcellini, P.: Everywhere regularity for a class of elliptic systems without growth conditions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23, 1–25 (1996)

    MathSciNet  MATH  Google Scholar 

  32. Marcellini, P.: Regularity under general and \(p, q\)-growth conditions. Discrete Contin. Dyn. Syst. S Ser. 13, 2009–2031 (2020)

    MathSciNet  MATH  Google Scholar 

  33. Marcellini, P.: A variational approach to parabolic equations under general and \(p, q\)-growth conditions. Nonlinear Anal. 194, (2020). https://doi.org/10.1016/j.na.2019.02.010

  34. Marcellini, P.: Growth conditions and regularity for weak solutions to nonlinear elliptic pdes. J. Math. Anal. Appl. (2020) (to appear)

  35. Marcellini, P., Papi, G.: Nonlinear elliptic systems with general growth. J. Differ. Equ. 221, 412–443 (2006)

    Article  MathSciNet  Google Scholar 

  36. Mascolo, E., Migliorini, A.: Everywhere regularity for vectorial functionals with general growth. ESAIM Control Optim. Calc. Var. 9, 399–418 (2003)

    Article  MathSciNet  Google Scholar 

  37. Mingione, G., Palatucci, G.: Developments and perspectives in nonlinear potential theory. Nonlinear Anal. 194, (2020). https://doi.org/10.1016/j.na.2019.02.006

  38. Mooney, C., Savin, O.: Some singular minimizers in low dimensions in the calculus of variations. Arch. Ration. Mech. Anal. 221, 1–22 (2016)

    Article  MathSciNet  Google Scholar 

  39. Mooney, C.: Minimizers of convex functionals with small degeneracy set. Calc. Var. Partial Differ. Equ. 59, (2020). https://doi.org/10.1007/s00526-020-1723-9

  40. Nečas, J.: Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity. Theory of nonlinear operators. In: Proceedings of Fourth International Summer School, Academic Science, Berlin 1975, pp. 197–206. Abh. Akad. Wiss. DDR Abt. Math.-Natur.-Tech., Jahrgang 1977, 1, Akademie, Berlin (1977)

  41. Rǎdulescu, V., Zhang, Q.: Double phase anisotropic variational problems and combined effects of reaction and absorption terms. J. Math. Pures Appl. 118, 159–203 (2018)

    Article  MathSciNet  Google Scholar 

  42. Šverák, V., Yan, X.: A singular minimizer of a smooth strongly convex functional in three dimensions. Calc. Var. Partial Differ. Equ. 10, 213–221 (2000)

    Article  MathSciNet  Google Scholar 

  43. Uhlenbeck, K.: Regularity for a class of non-linear elliptic systems. Acta Math. 138, 219–240 (1977)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The authors thank the referee for having carefully read the manuscript, for her/his patience, remarks and suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Paolo Marcellini.

Additional information

Communicated by A. Malchiodi.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Di Marco, T., Marcellini, P. A-priori gradient bound for elliptic systems under either slow or fast growth conditions. Calc. Var. 59, 120 (2020). https://doi.org/10.1007/s00526-020-01769-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-020-01769-7

Keywords

Mathematics Subject Classification

Navigation