Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Interpretation of Lavrentiev phenomenon by relaxation: the higher order case

Author: Marino Belloni
Journal: Trans. Amer. Math. Soc. 347 (1995), 2011-2023
MSC: Primary 49J45; Secondary 49J05
MathSciNet review: 1290714
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider integral functionals of the calculus of variations of the form

$\displaystyle F(u) = \int\limits_0^1 {f(x,u,u', \ldots ,{u^{(n)}})dx} $

defined for $ u \in {W^{n,\infty }}(0,1)$, and we show that the relaxed functional $ F$ with respect to the weak $ W_{{\text{loc}}}^{n,1}(0,1)$ convergence can be written as

$\displaystyle \overline F (u) = \int\limits_0^1 {f(x,u,u', \ldots ,{u^{(n)}})dx + L(u),} $

where the additional term $ L(u)$, the Lavrentiev Gap, is explicitly identified in terms of $ F$.

References [Enhancements On Off] (What's this?)

  • [BM1] J. M. Ball and V. J. Mizel, Singular minimizers for regular one-dimensional problems in the calculus of variations, Bull. Amer. Math. Soc. (N.S.) 11 (1984), 143-146. MR 741726 (86f:49004)
  • [BM2] -, One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation, Arch. Rational Mech. Anal. 90 (1985), 325-388. MR 801585 (86k:49002)
  • [Be] M. Belloni, Ph.D. Thesis (in preparation).
  • [B] G. Buttazzo, Semicontinuity, relaxation and integral representation in the calculus of variations, Pitman Res. Notes Math. Ser., vol. 203, Longman, Harlow, 1989. MR 1020296 (91c:49002)
  • [BuM] G. Buttazzo and V. J. Mizel, Interpretation of the Lavrentiev phenomenon by relaxation, J. Funct. Anal. 110 (1992), 434-460. MR 1194993 (93i:49004)
  • [CA] L. Cesari and T. S. Angell, On the Lavrentiev phenomenon, Calcolo 22 (1985), 17-29. MR 817037 (87f:49023)
  • [C] C. W. Cheng, The Lavrentiev phenomenon and its applications in nonlinear elasticity, Ph.D. Thesis, Carnegie Mellon Univ., Pittsburgh, 1993.
  • [CM] C. W. Cheng and V. J. Mizel, On the Lavrentiev phenomenon for autonomous second order integrands, Arch. Rational Tech. Anal. 126 (1994), 21-34. MR 1268047 (95b:49013)
  • [CV] F. H. Clarke and R. B. Vinter, Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Amer. Math. Soc. 291 (1985), 73-98. MR 779053 (86h:49020)
  • [HM1] A. C. Henricher and V. J. Mizel, The Lavrentiev phenomenon for invariant variational problems, Arch. Rational Mech. Anal. 102 (1988), 57-93. MR 938384 (90a:49020)
  • [HM2] -, A new example of the Lavrentiev phenomenon, SIAM J. Control Optim. 26 (1988), 1490-1503. MR 969341 (89i:49002)
  • [L] M. Lavrentiev, Sur quelques problèmes du calcul des variations, Ann. Mat. Pura Appl. 4 (1926), 107-124.
  • [M] B. Manià, Sopra un esempio di Lavrentieff, Boll. Un. Mat. Ital. 13 (1934), 146-153.
  • [MM] M. Marcus and V. J. Mizel, Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rational Mech. Anal. 45 (1972), 294-320. MR 0338765 (49:3529)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 49J45, 49J05

Retrieve articles in all journals with MSC: 49J45, 49J05

Additional Information

Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society