Annali di Matematica Pura ed Applicata (1923 -)

, Volume 194, Issue 4, pp 1025–1069 | Cite as

On Hölder continuity of solutions for a class of nonlinear elliptic systems with \(p\)-growth via weighted integral techniques

  • Miroslav Bulíček
  • Jens Frehse
  • Mark Steinhauer


We consider weak solutions of nonlinear elliptic systems in a \(W^{1,p}\)-setting which arise as Euler–Lagrange equations of certain variational integrals with pollution term, and we also consider minimizers of a variational problem. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes with respect to variations of the independent and the dependent variables. We impose new structural conditions on the nonlinearities which yield \(\fancyscript{C}^{\alpha }\)-regularity and \(\fancyscript{C}^{\alpha }\)-estimates for the solutions. These structure conditions cover variational integrals like \(\int F(\nabla u)\,\mathrm{d}x \) with potential \(F(\nabla u):=\tilde{F} (Q_1(\nabla u),\ldots , Q_N(\nabla u))\) and positive definite quadratic forms \(Q_i\) in \(\nabla u\) defined as \(Q_i(\nabla u)=\sum \nolimits _{\alpha \beta } a_i^{\alpha \beta } \nabla u^\alpha \cdot \nabla u^\beta \). A simple example consists in \({\tilde{F}}(\xi _1,\xi _2):= |\xi _1|^{\frac{p}{2}} + |\xi _2|^{\frac{p}{2}}\) or \(\tilde{F}(\xi _1,\xi _2):= |\xi _1|^{\frac{p}{4}}|\xi _2|^{\frac{p}{4}}.\) Since the quadratic forms \(Q_i\) need not to be linearly dependent, our result covers a class of nondiagonal, possibly nonmonotone elliptic systems. As a by-product, we also prove a kind of Liouville theorem. As a new analytical tool, we use a weighted integral technique with singular weights in an \(L^p\)-setting for the proof and establish a weighted hole-filling inequality in a setting where Green-function techniques are not available.


Nonlinear elliptic systems Regularity Noether equation Hölder continuity Liouville theorem 

Mathematics Subject Classification (2010)

35J60 49N60 



The authors thank the Collaborative Research Center (SFB) 611 and the Hausdorff Center for Mathematics for their support. The support of the project MORE (ERC-CZ project no. LL1202 financed by the Ministry of Education, Youth and Sports, Czech Republic) is also acknowledged. M. Bulíček is thankful to the Karel Janeček Endowment for Science and Research for its support. M. Bulíček is a researcher in the University Centre for Mathematical Modelling, Applied Analysis and Computational Mathematics (Math MAC) and a member of the Nečas Center for Mathematical Modeling.


  1. 1.
    Bulíček, M., Frehse, J.: \({\fancyscript {C}}^{\alpha }\)-regularity for a class of non-diagonal elliptic systems with p-growth. Calc. Var. Partial Differ. Equ. 43(3), 441–462 (2012)zbMATHCrossRefGoogle Scholar
  2. 2.
    Bulíček, M., Frehse, J., Steinhauer, M.: Everywhere \({\fancyscript {C}}^{\alpha }\)-estimates for a class of nonlinear elliptic systems with critical growth. Adv. Calc. Var. Ahead Print (2014)Google Scholar
  3. 3.
    Uhlenbeck, K.: Regularity for a class of non-linear elliptic systems. Acta Math. 138(3–4), 219–240 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Marcellini, P.: Everywhere regularity for a class of elliptic systems without growth conditions. Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) 23(1), 1–25 (1996)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Marcellini, P., Papi, G.: Nonlinear elliptic systems with general growth. J. Differ. Equ. 221(2), 412–443 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Nečas, J.: Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity. In: Theory of nonlinear operators (Proceedings of Fourth International Summer School, Acad. Sci., Berlin, 1975). Akademie-Verlag, Berlin (1977)Google Scholar
  7. 7.
    Šverák, V., Yan, X.: Non-Lipschitz minimizers of smooth uniformly convex functionals. Proc. Natl. Acad. Sci. USA 99(24), 15,269–15,276 (2002)CrossRefGoogle Scholar
  8. 8.
    Frehse, J.: A discontinuous solution of a mildly nonlinear elliptic system. Math. Z. 134, 229–230 (1973)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton (1983)Google Scholar
  10. 10.
    Giaquinta, M., Giusti, E.: On the regularity of the minima of variational integrals. Acta Math. 148, 31–46 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Giaquinta, M., Hildebrandt, S.: Calculus of variations. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 310. Springer, Berlin (1996). The Lagrangian formalismGoogle Scholar
  12. 12.
    Bensoussan, A., Frehse, J.: Systems of Bellman equations to stochastic differential games with discount control. Boll. Unione Mat. Ital. (9) 1(3), 663–681 (2008)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Bensoussan, A., Frehse, J.: Diagonal elliptic Bellman systems to stochastic differential games with discount control and noncompact coupling. Rend. Mat. Appl. (7) 29(1), 1–16 (2009)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Beck, L., Frehse, J.: Regular and irregular solutions for a class of elliptic systems in the critical dimension. NoDEA Nonlinear Differ. Equ. Appl. 20(3), 943–976 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Wiegner, M.: Über die Regularität schwacher Lösungen gewisser elliptischer Systeme. Manuscr. Math. 15(4), 365–384 (1975)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Wiegner, M.: A-priori Schranken für Lösungen gewisser elliptischer Systeme. Manuscr. Math. 18(3), 279–297 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Wiegner, M.: Ein optimaler Regularitätssatz für schwache Lösungen gewisser elliptischer Systeme. Math. Z. 147(1), 21–28 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Wiegner, M.: On two-dimensional elliptic systems with a one-sided condition. Math. Z. 178(4), 493–500 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Giusti, E., Miranda, M.: Un esempio di soluzioni discontinue per un problema di minimo relativo ad un integrale regolare del calcolo delle variazioni. Boll. Un. Mat. Ital. 4(1), 219–226 (1968)MathSciNetGoogle Scholar
  20. 20.
    Bildhauer, M., Fuchs, M.: Partial regularity for local minimizers of splitting-type variational integrals. Asymptot. Anal. 55(1–2), 33–47 (2007)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Breit, D.: The partial regularity for minimizers of splitting type variational integrals under general growth conditions I. The autonomous case. J. Math. Sci. (N. Y.) 166(3), 239–258 (2010). Problems in mathematical analysis. No. 45Google Scholar
  22. 22.
    Breit, D.: The partial regularity for minimizers of splitting type variational integrals under general growth conditions II. The nonautonomous case. J. Math. Sci. (N. Y.) 166(3), 259–281 (2010). Problems in mathematical analysis. No. 45zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Fusco, N., Hutchinson, J.: Partial regularity for minimisers of certain functionals having nonquadratic growth. Ann. Mat. Pura Appl. 4(155), 1–24 (1989)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Giaquinta, M.: Nonlinear elliptic systems. Theory of regularity. Boll. Un. Mat. Ital. A (5) 16(2), 259–283 (1979)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Mingione, G.: Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51(4), 355–426 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Bulíček, M.: On properties of minimizers to some variational integrals. Preprint no. 2013–012. (2012)
  27. 27.
    Evans, L. C.: Partial regularity for stationary harmonic maps into spheres. Arch. Ration. Mech. Anal. 116(2), 101–113 (1991)Google Scholar
  28. 28.
    Meier, M.: Removable singularities of harmonic maps and an application to minimal submanifolds. Indiana Univ. Math. J. 35(4), 705–726 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Pohožaev, S.I.: On the eigenfunctions of the equation \(\Delta u+\lambda f(u)=0\). Dokl. Akad. Nauk SSSR 165, 36–39 (1965)MathSciNetGoogle Scholar
  30. 30.
    Giaquinta, M., Nečas, J.: On the regularity of weak solutions to nonlinear elliptic systems via Liouville’s type property. Comment. Math. Univ. Carolin. 20(1), 111–121 (1979)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Giaquinta, M., Nečas, J.: On the regularity of weak solutions to nonlinear elliptic systems of partial differential equations. J. Reine Angew. Math. 316, 140–159 (1980)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Giaquinta, M., Nečas, J., John, O., Stará, J.: On the regularity up to the boundary for second order nonlinear elliptic systems. Pac. J. Math. 99(1), 1–17 (1982)zbMATHCrossRefGoogle Scholar
  33. 33.
    Meier, M.: Liouville theorems for nonlinear elliptic equations and systems. Manuscr. Math. 29(2–4), 207–228 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Meier, M.: Liouville theorems, partial regularity and Hölder continuity of weak solutions to quasilinear elliptic systems. Trans. Am. Math. Soc. 284(1), 371–387 (1984)zbMATHMathSciNetGoogle Scholar
  35. 35.
    Lions, P. L., Nečas, J., Netuka, I.: A Liouville theorem for nonlinear elliptic systems with isotropic nonlinearities. Comment. Math. Univ. Carolin. 23(4), 645–655 (1982)Google Scholar
  36. 36.
    Frehse, J.: On Signorini’s problem and variational problems with thin obstacles. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4(2), 343–362 (1977)zbMATHMathSciNetGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Miroslav Bulíček
    • 1
  • Jens Frehse
    • 2
  • Mark Steinhauer
    • 3
  1. 1.Faculty of Mathematics and Physics, Mathematical InstituteCharles University Sokolovská 83Praha 8Czech Republic
  2. 2.Department of applied analysis, Institute for Applied MathematicsUniversity of BonnBonnGermany
  3. 3.Mathematical InstituteUniversity of Koblenz-LandauKoblenzGermany

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