Annali di Matematica Pura ed Applicata

, Volume 184, Issue 4, pp 421–448 | Cite as

Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth



We prove a small excess regularity theorem for almost minimizers of a quasi-convex variational integral of subquadratic growth. The proof is direct, and it yields an optimal modulus of continuity for the derivative of the almost minimizer. The result is new for general almost minimizers, and in the case of absolute minimizers it considerably simplifies the existing proof.


quasi-convexity almost minimizers partial regularity subquadratic growth 


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  1. 1.
    Acerbi, E., Fusco, N.: A regularity theorem for minimizers of quasiconvex variational integrals. Arch. Ration. Mech. Anal. 99, 261–281 (1987)Google Scholar
  2. 2.
    Allard, W.K.: On the first variation of a varifold. Ann. Math. (2) 95, 225–254 (1972)Google Scholar
  3. 3.
    Almgren, F.J.: Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure. Ann. Math. (2) 87, 321–391 (1968)Google Scholar
  4. 4.
    Almgren, F.J.: Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Am. Math. Soc. 165, (1976)Google Scholar
  5. 5.
    Anzelotti, G.: On the C1,α-regularity of ω-minima of quadratic functionals. Boll. Unione Mat. Ital. C (6) 2, 195–212 (1983)Google Scholar
  6. 6.
    Bombieri, E.: Regularity theory for almost minimal currents. Arch. Ration. Mech. Anal. 7, 99–130 (1982)Google Scholar
  7. 7.
    Campanato, S.: Proprietà di una famiglia di spazi funzionali. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 18, 137–160 (1964)Google Scholar
  8. 8.
    Campanato, S.: Equazioni ellitichi del IIe ordine e spazi \(\mathcal{L}^{2,\lambda}\). Ann. Mat. Pura Appl., IV. Ser. 69, 321–381 (1965)Google Scholar
  9. 9.
    Conti, S., Fonseca, I., Leoni, G.: A Γ–Convergence Result for the Two–Gradient Theory of Phase Transitions. Commun. Pure Appl. Math. 165, 141–164 (1998)Google Scholar
  10. 10.
    Carozza, M., Passarelli di Napoli, A.: A regularity theorem for minimisers of quasiconvex integrals: the case 1<p<2*. Proc. R. Soc. Edinb., Sect. A, Math. 126, 1181–1199 (1996)Google Scholar
  11. 11.
    Carozza, M., Fusco, N., Mingione, G.: Partial Regularity of Minimizers of Quasiconvex Integrals with Subquadratic Growth. Ann. Mat. Pura Appl., IV. Ser. 175, 141–164 (1998)Google Scholar
  12. 12.
    Dacorogna, B.: Direct Methods in the Calculus of Variations. Berlin, Heidelberg, New York: Springer 1989Google Scholar
  13. 13.
    De Giorgi, E.: Frontiere orientate di misura minima. Seminaro Mat. Scuola Norm. Sup. Pisa, 1–56 (1961)Google Scholar
  14. 14.
    Duzaar, F., Gastel, A.: Nonlinear elliptic systems with Dini continuous coefficients. Arch. Math. 78, 58–73 (2002)Google Scholar
  15. 15.
    Duzaar, F., Gastel, A., Grotowski, J.F.: Partial regularity for almost minimizers of quasiconvex functionals. SIAM J. Math. Anal. 32, 665–687 (2000)Google Scholar
  16. 16.
    Duzaar, F., Grotowski, J.F.: Optimal interior partial regularity for nonlinear elliptic systems: the method of A-harmonic approximation. Manuscr. Math. 103, 267–298 (2000)Google Scholar
  17. 17.
    Duzaar, F., Kronz, M.: Regularity of ω–minimizers of quasi–convex variational integrals with polynomial growth. Differ. Geom. Appl. 17, 139–152 (2002)Google Scholar
  18. 18.
    Duzaar, F., Steffen, K.: Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals. J. Reine Angew. Math. 546, 73–138 (2002)Google Scholar
  19. 19.
    Evans, L.C.: Quasiconvexitity and Partial Regularity in the Calculus of Variations. Arch. Ration. Mech. Anal. 95, 227–252 (1986)Google Scholar
  20. 20.
    Federer, H.: Geometric Measure Theory. Berlin, Heidelberg, New York: Springer 1969Google Scholar
  21. 21.
    Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton: Princeton University Press 1983Google Scholar
  22. 22.
    Giaquinta, M., Modica, G.: Partial regularity of minimizers of quasiconvex integrals. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 185–208 (1986)Google Scholar
  23. 23.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Berlin, Heidelberg, New York: Springer 1983Google Scholar
  24. 24.
    Giusti, E., Miranda, M.: Sulla regolarità delle soluzioni deboli di una classe di sistemi ellittici quasi-lineari. Arch. Ration. Mech. Anal. 31, 173–184 (1968)Google Scholar
  25. 25.
    Grotowski, J.F.: Boundary regularity for nonlinear elliptic systems. Calc. Var. Partial Differ. Equ. 15, 353–388 (2002)Google Scholar
  26. 26.
    Grotowski, J.F.: Boundary regularity for quasilinear elliptic systems. Commun. Partial Differ. Equations 27, 2491–2512 (2002)Google Scholar
  27. 27.
    Kronz, M.: Partial Regularity Results for Minimizers of Quasiconvex Functionals of Higher Order. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 19, 81–112 (2002)Google Scholar
  28. 28.
    Morrey, C.B.: Quasi-convexity and the lower semicontinuity of multiple integral. Pac. J. Math. 2, 25–53 (1952)Google Scholar
  29. 29.
    Morrey, C.B.: Multiple integrals in the calculus of variations. Berlin, Heidelberg, New York: Springer 1966Google Scholar
  30. 30.
    Simon, L.: Lectures on Geometric Measure Theory. Canberra: Australian National University Press 1983Google Scholar
  31. 31.
    Sverak, V.: Quasiconvex functions with subquadratic growth. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 433, 723–725 (1991)Google Scholar
  32. 32.
    Ziemer, W.P.: Weakly differentiable functions. Berlin, Heidelberg, New York: Springer 1989Google Scholar

Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  • Frank Duzaar
    • 1
  • Joseph F. Grotowski
    • 2
  • Manfred Kronz
    • 1
  1. 1.Mathematisches InstitutUniversität Erlangen-NürnbergErlangenGermany
  2. 2.Department of Mathematics, City College of New YorkCity University of New YorkNew YorkUSA

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