Skip to main content
SpringerLink
Log in

Manifold Constrained Non-uniformly Elliptic Problems

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

We consider the problem of minimizing variational integrals defined on nonlinear Sobolev spaces of competitors taking values into the sphere. The main novelty is that the underlying energy features a non-uniformly elliptic integrand exhibiting different polynomial growth conditions and no homogeneity. We develop a few intrinsic methods aimed at proving partial regularity of minima and providing techniques for treating larger classes of similar constrained non-uniformly elliptic variational problems. To give estimates for the singular sets, we use a general family of Hausdorff type measures following the local geometry of the integrand. A suitable comparison is provided with respect to the naturally associated capacities.

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.: Regularity results for electrorheological fluids: the stationary case. C. R. Math. Acad. Sci. Paris 334, 817–822 (2002)

    Google Scholar 

  2. Baroni, P., Colombo, M., Mingione, G.: Non-autonomous functionals, borderline cases and related function classes. St. Petersburg Math. J. 27, 347–379 (2016)

    Google Scholar 

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

    Google Scholar 

  4. Baruah, D., Harjulehto, P., Hästo, P.: Capacities in generalized Orlicz spaces. J. Funct. Spaces 4, 10 (2018)

    Google Scholar 

  5. Byun, S.S., Oh, J.: Global gradient estimates for non-uniformly elliptic equations. Calc. Var. PDE 56, 46 (2017)

    Google Scholar 

  6. Cencelj, M., Radulescu, V., Repovš, D.D.: Double phase problems with variable growth. Nonlinear Anal. 177, 270–287 (2018)

    Google Scholar 

  7. Chlebicka, I., De Filippis, C.: Removable sets in non-uniformly elliptic problems. Ann. Mat. Pura Appl. (2019). https://doi.org/10.1007/s10231-019-00894-1

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  10. Colombo, M., Mingione, G.: Calderón–Zygmund estimates and non-uniformly elliptic operators. J. Funct. Anal. 270, 1416–1478 (2016)

    Google Scholar 

  11. Choquet, G.: Theory of capacities. Ann. Inst. Fourier Grenoble 5, 131–295 (1955)

    Google Scholar 

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

    Google Scholar 

  13. De Filippis, C.: On the regularity of the \(\omega \)-minima of \(\varphi \)-functionals. Nonlinear Anal. https://doi.org/10.1016/j.na.2019.02.017

  14. De Filippis, C.: Partial regularity for manifold constrained \(p(x)\)-harmonic maps. Calc. Var. PDE 58, 47 (2019). https://doi.org/10.1007/s00526-019-1483-6

    Google Scholar 

  15. De Filippis, C., Mingione, G.: A borderline case of Calderón–Zygmund estimates for non-uniformly elliptic problems. St. Petersburg Math. J., to appear

  16. De Filippis, C., Oh, J.: Regularity for multi-phase variational problems. J. Differ. Equ. 267, 1631–1670 (2019)

    Google Scholar 

  17. De Filippis, C., Palatucci, G.: Hölder regularity for nonlocal double phase equations. J. Differ. Equ. 267, 547–586 (2019)

    Google Scholar 

  18. Diening, L., Harjulehto, P., Hästo, P., Ružička, M.: Lebesgue and Sobolev Spaces with a Variable Growth Exponent. Springer Lecture Notes on Mathematics 2017. Springer, New York (2011)

    Google Scholar 

  19. Diening, L., Stroffolini, B., Verde, A.: Everywhere regularity of functionals with \(\varphi \)-growth. Manuscr. Math. 129, 449–481 (2012)

    Google Scholar 

  20. Diening, L., Stroffolini, B., Verde, A.: The \(\varphi \)-harmonic approximation and the regularity of \(\varphi \)-harmonic maps. J. Differ. Equ. 253, 1943–1958 (2012)

    Google Scholar 

  21. Duerinckx, M., Gloria, A.: Stochastic homogenization of nonconvex unbounded integral functionals with convex growth. Arch. Ration. Mech. Anal. 221, 1511–1584 (2016)

    Google Scholar 

  22. Duzaar, F., Mingione, G.: The \(p\)-harmonic approximation and the regularity of \(p\)-harmonic maps. Calc. Var. PDE 20, 235–256 (2004)

    Google Scholar 

  23. Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)

    Google Scholar 

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

    Google Scholar 

  25. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC, Boca Raton (1991)

    Google Scholar 

  26. Fonseca, I., Malý, J., Mingione, G.: Scalar minimizers with fractal singular sets. Arch. Ration. Mech. Anal. 172, 295–307 (2004)

    Google Scholar 

  27. Frehse, J.: Capacity methods in the theory of partial differential equations. Jahresber. Deutsch. Math.-Verein. 84, 1–44 (1982)

    Google Scholar 

  28. Fuchs, M.: \(p\)-harmonic obstacle problems. I: Partial regularity theory. Ann. Mat. Pura Appl. 156, 127–158 (1990)

    Google Scholar 

  29. Fuchs, M.: \(p\)-Harmonic obstacle problems. III. Boundary regularity. Ann. Mat. Pura Appl. 156, 159–180 (1990)

    Google Scholar 

  30. Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific Publishing Co., Inc., River Edge (2003)

    Google Scholar 

  31. Hardt, R.M., Lin, F.G.: Mappings minimizing the \(L^p\) norm of the gradient. Commun. Pure Appl. Math. 40(5), 555–588 (1987)

    Google Scholar 

  32. Hardt, R.M., Kinderlehrer, D., Lin, F.G.: Stable defects of minimizers of constrained variational principles. Ann. Inst. H. Poincaré Anal. Non Linèaire 5, 297–322 (1988)

    Google Scholar 

  33. Harjulehto, P., Hästö, P.: Orlicz Spaces and Generalized Orlicz Spaces. Springer, New York (2018)

    Google Scholar 

  34. Harjulehto, P., Hästö, P., Toivanen, O.: Hölder regularity of quasiminimizers under generalized growth conditions. Calc. Var. PDE 56, 22 (2017)

    Google Scholar 

  35. Hästö, P.: The maximal operator on generalized Orlicz spaces. J. Funct. Anal. 269, 4038–4048 (2015)

    Google Scholar 

  36. Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear potential theory of degenerate elliptic equations. Dover Publications Inc., Mineola. Unabridged republication of the 1993 original (2006)

  37. Hopper, C.: Partial regularity for holonomic minimizers of quasiconvex functionals. Arch. Ration. Mech. Anal. 222, 91–141 (2016)

    Google Scholar 

  38. Kristensen, J., Mingione, G.: The singular set of minima of integral functionals. Arch. Ration. Mech. Anal. 180, 331–398 (2006)

    Google Scholar 

  39. Kristensen, J., Mingione, G.: The singular set of Lipschitzian minima of multiple integrals. Arch. Ration. Mech. Anal. 184, 341–369 (2007)

    Google Scholar 

  40. Leonetti, F., Siepe, F.: Maximum principle for vector valued minimizers. J. Convex Anal. 12, 267–278 (2005)

    Google Scholar 

  41. Luckhaus, S.: Partial Hölder continuity for minima of certain energies among maps into Riemannian manifold. Indiana Univ. Math. J. 37, 349–367 (1988)

    Google Scholar 

  42. Manfredi, J.J.: Regularity of the gradient for a class of nonlinear possibly degenerate elliptic equations. Ph.D. Thesis. University of Washington, St. Louis (1986)

  43. Marcellini, P.: Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions. Arch. Ration. Mech. Anal. 105, 267–284 (1989)

    Google Scholar 

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

    Google Scholar 

  45. Mazowiecka, M., Miśkiewicz, M., Schikorra, A.: On the size of the singular set of minimizing harmonic maps into the 2-sphere in dimension four and higher. Arxiv preprint (2019)

  46. Mazya, V.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Second, Revised and Augmented Edition. Grundlehren der Mathematischen Wissenschaften, 342. Springer, Heidelberg (2011)

    Google Scholar 

  47. Mingione, G.: Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51, 355–425 (2006)

    Google Scholar 

  48. Nieminen, E.: Hausdorff measures, capacities, and Sobolev spaces with weights. Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes No. 81 (1991), 39 pp

  49. Ok, J.: Gradient estimates for elliptic equations with \(L^{p(\cdot )}\log L\) growth. Calc. Var. PDE 55, 26 (2016)

    Google Scholar 

  50. Ok, J.: Regularity of \(\omega \)-minimizers for a class of functionals with non-standard growth. Calc. Var. PDE 56, 48 (2017)

    Google Scholar 

  51. Papageorgiou, N., Radulescu, V., Repovš, D.: Double-phase problems with reaction of arbitrary growth. Z. Angew. Math. Phys. 69, 21 (2018)

    Google Scholar 

  52. Radulescu, V., Repovs, D.: Partial differential equations with variable exponents: variational methods and qualitative analysis. Chapman & Hall/CRC Monographs and Research Notes in Mathematics. CRC, New York (2015)

    Google Scholar 

  53. Ragusa, M.A., Tachikawa, A.: Boundary regularity of minimizers of \(p(x)\)-energy functionals. Ann. Inst. H. Poincaré Anal. Non Linèaire 33, 451–476 (2017)

    Google Scholar 

  54. Ragusa, M.A., Tachikawa, A.: Partial regularity of \(p(x)\)-harmonic maps. Trans. Am. Math. Soc. 365, 3329–3353 (2013)

    Google Scholar 

  55. Schoen, R., Uhlenbeck, K.: A regularity theory for harmonic maps. J. Differ. Geom. 17, 307–336 (1983)

    Google Scholar 

  56. Schoen, R., Uhlenbeck, K.: Boundary regularity and the Dirichlet problem for harmonic maps. J. Differ. Geom. 18, 253–268 (1983)

    Google Scholar 

  57. Simon, L.: Lectures on Regularity and Singularities of Harmonic Maps. Birkhäuser, Basel (1996)

    Google Scholar 

  58. Turesson, B.O.: Nonlinear Potential Theory and Weighted Sobolev Spaces. Lecture Notes in Mathematics, 1736. Springer, Berlin (2000)

    Google Scholar 

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

    Google Scholar 

  60. Ural’tseva, N.N.: Degenerate quasilinear elliptic systems. Zap. Na. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7, 184–222 (1968)

    Google Scholar 

  61. Ural’tseva, N.N., Urdaletova, A.B.: The boundedness of the gradients of generalized solutions of degenerate quasilinear non-uniformly elliptic equations. Vestnik Leningrad Univ. Math. 19 (1983) (Russian) English tranlation: 16, 263–270 (1984)

  62. Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat. 50, 675–710 (1986)

    Google Scholar 

  63. Zhikov, V.V.: On Lavrentiev’s Phenomenon. Russ. J. Math. Phys. 3, 249–269 (1995)

    Google Scholar 

  64. Zhikov, V.V.: On some variational problems. Russ. J. Math. Phys. 5, 105–116 (1997)

    Google Scholar 

Download references

Acknowledgements

This work is supported by the Engineering and Physical Sciences Research Council [EP/L015811/1]. The authors thank I. Chlebicka and J. Ok for comments on a preliminary version of the manuscript.

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX26GG, UK

    Cristiana De Filippis

  2. Dipartimento SMFI, Università di Parma, Viale delle Scienze 53/a, Campus, 43124, Parma, Italy

    Giuseppe Mingione

Authors
  1. Cristiana De Filippis
    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.

Additional information

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

De Filippis, C., Mingione, G. Manifold Constrained Non-uniformly Elliptic Problems. J Geom Anal 30, 1661–1723 (2020). https://doi.org/10.1007/s12220-019-00275-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-019-00275-3

Keywords

Search

Navigation