Minimal currents, geodesics, and relaxation of variational integrals on mappings of bounded variation
Patricio Aviles and Yoshikazu Giga
Source: Duke Math. J. Volume 67, Number 3
(1992), 517-538.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077294536
Mathematical Reviews number (MathSciNet): MR1181311
Zentralblatt MATH identifier: 0769.49010
Digital Object Identifier: doi:10.1215/S0012-7094-92-06720-2
References
[AMT] L. Ambrosio, S. Mortola, and V. M. Tortorelli, Functionals with linear growth defined on vector valued BV functions, J. Math. Pures Appl. (9) 70 (1991), no. 3, 269–323.
Mathematical Reviews (MathSciNet): MR92j:49004
Zentralblatt MATH: 0662.49007
[AG1] P. Aviles and Y. Giga, Variational integrals on mappings of bounded variation and their lower semicontinuity, Arch. Rational Mech. Anal. 115 (1991), no. 3, 201–255.
Mathematical Reviews (MathSciNet): MR92d:49019
Zentralblatt MATH: 0737.49011
Digital Object Identifier: doi:10.1007/BF00380769
[AG2] P. Aviles and Y. Giga, Minimal currents and relaxation of variational integrals on mappings of bounded variation, Proc. Japan Acad. Ser. A Math. Sci. 66 (1990), no. 3, 68–71.
Mathematical Reviews (MathSciNet): MR91c:49065
Zentralblatt MATH: 0706.49003
Digital Object Identifier: doi:10.3792/pjaa.66.68
Project Euclid: euclid.pja/1195512558
[Ba] S. Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), no. 2, 67–90.
Mathematical Reviews (MathSciNet): MR91j:76005
Zentralblatt MATH: 0702.49009
[BBC] F. Bethuel, H. Brezis, and J.-M. Coron, Relaxed energies for harmonic maps, Variational methods (Paris, 1988) eds. H. Berestycki, J. M. Coron, and I. Ekeland, Progr. Nonlinear Differential Equations Appl., vol. 4, Birkhäuser Boston, Boston, MA, 1990, pp. 37–52.
Mathematical Reviews (MathSciNet): MR94a:58046
Zentralblatt MATH: 0793.58011
[Bo] M. E. Bogovski, Solution of the first boundary value problem for the equation of continuity of an incompressible medium, Soviet Math. Dokl. 20 (1979), 1094–1098.
Zentralblatt MATH: 0499.35022
Mathematical Reviews (MathSciNet): MR553920
[BS] W. Borchers and H. Sohr, On the equations $\rm rot\,\bf v=\bf g$ and $\rm div\,\bf u=f$ with zero boundary conditions, Hokkaido Math. J. 19 (1990), no. 1, 67–87.
Mathematical Reviews (MathSciNet): MR91h:35248
Zentralblatt MATH: 0719.35014
[D] B. Dacorogna, Quasiconvexity and relaxation of nonconvex problems in the calculus of variations, J. Funct. Anal. 46 (1982), no. 1, 102–118.
Mathematical Reviews (MathSciNet): MR83g:49025
Zentralblatt MATH: 0547.49003
Digital Object Identifier: doi:10.1016/0022-1236(82)90046-5
[DM] G. Dal Maso, Integral representation on $\rm BV(\Omega )$ of $\Gamma$-limits of variational integrals, Manuscripta Math. 30 (1979/80), no. 4, 387–416.
Mathematical Reviews (MathSciNet): MR81i:28007
Zentralblatt MATH: 0474.49021
Digital Object Identifier: doi:10.1007/BF01301259
[DG] E. De Giorgi, $G$-operators and $\Gamma$-convergence, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) eds. Z. Ciesielski and C. Olech, PWN, Warsaw, 1984, pp. 1175–1191.
Mathematical Reviews (MathSciNet): MR87e:49033
Zentralblatt MATH: 0568.35025
[F1] H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.
Mathematical Reviews (MathSciNet): MR41:1976
Zentralblatt MATH: 0176.00801
[F2] H. Federer, Real flat chains, cochains and variational problems, Indiana Univ. Math. J. 24 (1974/75), 351–407.
Mathematical Reviews (MathSciNet): MR50:1095
Zentralblatt MATH: 0289.49044
Digital Object Identifier: doi:10.1512/iumj.1974.24.24031
[FR] I. Fonseca and P. Rybka, Relaxation of multiple integrals in the space $BV(\Omega,\mathbbR^p)$, preprint.
[FT] I. Fonseca and L. Tartar, The gradient theory of phase transitions for systems with two potential wells, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), no. 1-2, 89–102.
Mathematical Reviews (MathSciNet): MR90b:49021
Zentralblatt MATH: 0676.49005
[GMS1]1 M. Giaquinta, G. Modica, and J. Souček, Functionals with linear growth in the calculus of variations. I, Comment. Math. Univ. Carolin. 20 (1979), no. 1, 143–156.
Mathematical Reviews (MathSciNet): MR80b:35047
Zentralblatt MATH: 0409.49006
[GMS1]2 M. Giaquinta, G. Modica, and J. Souček, Functionals with linear growth in the calculus of variations. II, Comment. Math. Univ. Carolin. 20 (1979), no. 1, 157–172.
Mathematical Reviews (MathSciNet): MR80b:35047
Zentralblatt MATH: 0409.49007
[GMS2] M. Giaquinta, G. Modica, and J. Souček, The Dirichlet energy of mappings with values into the sphere, Manuscripta Math. 65 (1989), no. 4, 489–507.
Mathematical Reviews (MathSciNet): MR90i:49053
Zentralblatt MATH: 0678.49006
Digital Object Identifier: doi:10.1007/BF01172794
[Giu] E. Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984.
Mathematical Reviews (MathSciNet): MR87a:58041
Zentralblatt MATH: 0545.49018
[GS] C. Goffman and J. Serrin, Sublinear functions of measures and variational integrals, Duke Math. J. 31 (1964), 159–178.
Mathematical Reviews (MathSciNet): MR29:206
Zentralblatt MATH: 0123.09804
Digital Object Identifier: doi:10.1215/S0012-7094-64-03115-1
Project Euclid: euclid.dmj/1077375086
[I] M. Iri, Network flow, transportation and scheduling, Mathematics in Science and Engineering, Vol. 57, Academic Press, New York, 1969.
Mathematical Reviews (MathSciNet): MR43:7197
Zentralblatt MATH: 0281.90032
[JK] R. James and D. Kinderlehrer, Theory of diffusionless phase transitions, PDEs and continuum models of phase transitions (Nice, 1988) eds. M. Rascle, D. Serre, and M. Slemord, Lecture Notes in Phys., vol. 344, Springer, Berlin, 1989, pp. 51–84.
Mathematical Reviews (MathSciNet): MR91e:73011
Zentralblatt MATH: 0991.74504
Digital Object Identifier: doi:10.1007/BFb0024935
[M1] L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal. 98 (1987), no. 2, 123–142.
Mathematical Reviews (MathSciNet): MR88f:76038
Zentralblatt MATH: 0616.76004
Digital Object Identifier: doi:10.1007/BF00251230
[Mor] F. Morgan, Area-minimizing currents bounded by higher multiples of curves, Rend. Circ. Mat. Palermo (2) 33 (1984), no. 1, 37–46.
Mathematical Reviews (MathSciNet): MR85k:49091
Zentralblatt MATH: 0541.49018
Digital Object Identifier: doi:10.1007/BF02844410
[Re] Ju. G. Rešetnjak, The weak convergence of completely additive vector-valued set functions, Sibirsk. Mat. Ž. 9 (1968), 1386–1394, Siberian Math. J. 9, (1986), 1039–1045.
Mathematical Reviews (MathSciNet): MR39:1623
[Ro] R. T. Rockafellar, Dual problems of Lagrange for arcs of bounded variation, Calculus of variations and control theory (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975; dedicated to Laurence Chisholm Young on the occasion of his 70th birthday) ed. D. L. Russel, Academic Press, New York, 1976, 155–192. Publ. Math. Res. Center Univ. Wisconsin, No. 36.
Mathematical Reviews (MathSciNet): MR58:2534
Zentralblatt MATH: 0352.49005
[Se1] J. Serrin, A new definition of the integral for nonparametric problems in the calculus of variations, Acta Math. 102 (1959), 23–32.
Mathematical Reviews (MathSciNet): MR21:7459b
Zentralblatt MATH: 0089.08601
Digital Object Identifier: doi:10.1007/BF02559566
[Se2] J. Serrin, On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc. 101 (1961), 139–167.
Mathematical Reviews (MathSciNet): MR25:1466
Zentralblatt MATH: 0102.04601
Digital Object Identifier: doi:10.2307/1993416
JSTOR: links.jstor.org
[Si] L. Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University Centre for Mathematical Analysis, Canberra, 1983.
Mathematical Reviews (MathSciNet): MR87a:49001
Zentralblatt MATH: 0546.49019
[St1] P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal. 101 (1988), no. 3, 209–260.
Mathematical Reviews (MathSciNet): MR89h:49007
Zentralblatt MATH: 0647.49021
Digital Object Identifier: doi:10.1007/BF00253122
[St2] P. Sternberg, Vector-valued local minimizers of nonconvex variational problems, Rocky Mountain J. Math. 21 (1991), no. 2, 799–807.
Mathematical Reviews (MathSciNet): MR92e:49016
Zentralblatt MATH: 0737.49009
Digital Object Identifier: doi:10.1216/rmjm/1181072968
Project Euclid: euclid.rmjm/1181072968
[W] B. White, The least area bounded by multiples of a curve, Proc. Amer. Math. Soc. 90 (1984), no. 2, 230–232.
Mathematical Reviews (MathSciNet): MR85h:49081
Zentralblatt MATH: 0529.49023
Digital Object Identifier: doi:10.2307/2045345
JSTOR: links.jstor.org
[Y] L. C. Young, Some extremal questions for simplicial complexes. IV. The algebraic and the geometric resultant, and application of variational methods, Rend. Circ. Mat. Palermo (2) 12 (1963), 200–210.
Mathematical Reviews (MathSciNet): MR29:5987
Zentralblatt MATH: 0121.08701
Digital Object Identifier: doi:10.1007/BF02843965
Duke Mathematical Journal
