Archive for Rational Mechanics and Analysis

, Volume 95, Issue 3, pp 227–252 | Cite as

Quasiconvexity and partial regularity in the calculus of variations

  • Lawrence C. Evans
Article

Abstract

We prove partial regularity of minimizers of certain functionals in the calculus of variations, under the principal assumption that the integrands be uniformly strictly quasiconvex. This is of interest since quasiconvexity is known in many circumstances to be necessary and sufficient for the weak sequential lower semicontinuity of these functionals on appropriate Sobolev spaces. Examples covered by the regularity theory include functionals with integrands which are convex in the determinants of various submatrices of the gradient matrix.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. Acerbi & N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal. 86 (1984), 125–145.Google Scholar
  2. 2.
    W. K. Allard & F. J. Almgren, Jr., An introduction to regularity theory for parametric elliptic variational problems, in Proc. Symposia Pure Math. XXIII (1973), American Math. Soc., 231–260.Google Scholar
  3. 3.
    F. J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. Math. 87 (1968), 321–391.Google Scholar
  4. 4.
    J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337–403.Google Scholar
  5. 5.
    J. M. Ball, Strict convexity, strong ellipticity, and regularity in the calculus of variations, Math. Proc. Camb. Phil. Soc. 87 (1980), 501–513.Google Scholar
  6. 6.
    E. Bombieri, Regularity theory for almost minimal currents, Arch. Rational Mech. Anal. 78 (1982), 99–130.Google Scholar
  7. 7.
    H. Federer, Geometric Measure Theory, Springer, New York, 1969.Google Scholar
  8. 8.
    M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton U. Press, Princeton, 1983.Google Scholar
  9. 9.
    M. Giaquinta & E. Giusti, Nonlinear elliptic systems with quadratic growth, Manu. Math. 24 (1978), 323–349.Google Scholar
  10. 10.
    M. Giaquinta & E. Giusti, On the regularity of the minima of variational integrals, Acta. Math. 148 (1982), 31–46.Google Scholar
  11. 11.
    E. Giusti & M. Miranda, Sulla regolarità delle soluzioni deboli di una classe di sistemi ellittici quasi-lineari, Arch. Rational Mech. Anal. 31 (1968), 173–184.Google Scholar
  12. 12.
    R. J. Knops & C. A. Stuart, Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity, Arch. Rational Mech. Anal. 86 (1984), 233–249.Google Scholar
  13. 13.
    N. Meyers, Quasiconvexity and lower semicontinuity of multiple variational integrals of any order, Trans. Am. Math. Soc. 119 (1965), 125–149.Google Scholar
  14. 14.
    C. B. Morrey, Jr., Quasiconvexity and the lower semicontinuity of multiple integrals, Pac. J. Math. 2 (1952), 25–53.MathSciNetGoogle Scholar
  15. 15.
    C. B. Morrey, Jr., Multiple Integrals in the Calculus of Variations, Springer, New York, 1966.Google Scholar
  16. 16.
    C. B. Morrey, Jr., Partial regularity results for nonlinear elliptic systems, J. Math. and Mech. 17 (1968), 649–670.Google Scholar

Copyright information

© Springer-Verlag GmbH & Co 1986

Authors and Affiliations

  • Lawrence C. Evans
    • 1
  1. 1.Department of MathematicsUniversity of MarylandCollege Park

Personalised recommendations