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Transactions of the American Mathematical Society

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Nonconvex variational problems with general singular perturbations


Author: Nicholas C. Owen
Journal: Trans. Amer. Math. Soc. 310 (1988), 393-404
MSC: Primary 49A50; Secondary 49F10, 58E15, 73C60, 73K05
MathSciNet review: 965760
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Abstract: We study the effect of a general singular perturbation on a nonconvex variational problem with infinitely many solutions. Using a scaling argument and the theory of $ \Gamma $-convergence of nonlinear functionals, we show that if the solutions of the perturbed problem converge in $ {L^1}$ as the perturbation parameter goes to zero, then the limit function satisfies a classical minimal surface problem.


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  • [1] Stuart S. Antman, Nonuniqueness of equilibrium states for bars in tension, J. Math. Anal. Appl. 44 (1973), 333–349. MR 0334637
  • [2] H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 773850
  • [3] Jack Carr, Morton E. Gurtin, and Marshall Slemrod, Structured phase transitions on a finite interval, Arch. Rational Mech. Anal. 86 (1984), no. 4, 317–351. MR 759767, 10.1007/BF00280031
  • [4] Ennio De Giorgi, Convergence problems for functionals and operators, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978) Pitagora, Bologna, 1979, pp. 131–188. MR 533166
  • [5] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
  • [6] E. Guisti, Minimal surfaces and functions of bounded variation, Birkhäuser, 1984.
  • [7] Morton E. Gurtin, On a theory of phase transitions with interfacial energy, Arch. Rational Mech. Anal. 87 (1985), no. 3, 187–212. MR 768066, 10.1007/BF00250724
  • [8] -, Some results and conjectures in the gradient theory of phase transitions (to appear).
  • [9] Luciano Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal. 98 (1987), no. 2, 123–142. MR 866718, 10.1007/BF00251230
  • [10] L. Modica and S. Mortola, Un esempio de $ \Gamma $-convergenza de una famiglia de funzionali ellittici, Boll. Un. Mat. Ital. A14 (1977).
  • [11] Nicholas Owen, Existence and stability of necking deformations for nonlinearly elastic rods, Arch. Rational Mech. Anal. 98 (1987), no. 4, 357–383. MR 872752, 10.1007/BF00276914
  • [12] Peter Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal. 101 (1988), no. 3, 209–260. MR 930124, 10.1007/BF00253122

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0965760-9
Article copyright: © Copyright 1988 American Mathematical Society