@article{AIHPC_2008__25_4_743_0, author = {Zhang, Kewei}, title = {Compensated convexity and its applications}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, volume = {25}, year = {2008}, pages = {743-771}, doi = {10.1016/j.anihpc.2007.08.001}, mrnumber = {2436792}, zbl = {pre05306979}, language = {en}, url = {http://dml.mathdoc.fr/item/AIHPC_2008__25_4_743_0} }
Zhang, Kewei. Compensated convexity and its applications. Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008) pp. 743-771. doi : 10.1016/j.anihpc.2007.08.001. http://gdmltest.u-ga.fr/item/AIHPC_2008__25_4_743_0/
[1] Approximations and regularization of arbitrary functions in Hilbert spaces by the Lasry-Lions method, Anal. Non-Lin. H. Poincaré Inst. 10 (1993) 289-312. | Numdam | MR 1230710 | Zbl 0780.41021
, ,[2] Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal. 86 (1984) 125-145. | MR 751305 | Zbl 0565.49010
, ,[3] Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman, London, 1984. | MR 773850 | Zbl 0561.49012
,[4] Penalty and barrier methods: a unified framework, SIAM J. Optim. 10 (1999) 211-230. | MR 1742321 | Zbl 0953.90045
,[5] Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977) 337-403. | MR 475169 | Zbl 0368.73040
,[6] Constrained Optimization and Lagrangian Multiplier Methods, Academic Press, 1982. | Zbl 0572.90067
,[7] Restrictions on microstructures, Proc. Royal Soc. Edinb. A 124 (1994) 843-878. | MR 1303758 | Zbl 0808.73063
, , , ,[8] What is the subdifferential of the closed convex hull of a function, SIAM J. Math. Anal. 27 (1996) 1661-1679. | MR 1416513 | Zbl 0876.49018
, ,[9] Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal. 100 (1987) 13-52. | MR 906132 | Zbl 0629.49020
, ,[10] Proposed experimental tests of a theory of fine microstructures and the two-well problem, Phil. Royal Soc. Lond. A 338 (1992) 389-450. | Zbl 0758.73009
, ,[11] Regularity of quasi-convex envelopes, Calc. Var. Partial Differ. Equ. 11 (2000) 333-359. | MR 1808126 | Zbl 0972.49024
, , ,[12] A transformation for extracting new descriptors of shape, in: (Ed.), Prop. Symp. Models for the Perception of Speech and Visual Form, MIT Press, 1967, pp. 362-380.
,[13] A smoothing technique for nondifferentiable optimization problems, in: Optimization, Lect. Notes in Math., vol. 1405, Springer, 1989, pp. 1-11. | MR 1036540 | Zbl 0683.90078
, ,[14] Mathematical theory of medial axis transform, Pacific J. Math. 181 (1997) 57-88. | MR 1491036 | Zbl 0885.53004
, , ,[15] Distance Geometry and Molecular Conformation, John Wiley & Sons, 1988. | MR 975025
, ,[16] Smooth convex approximation to the maximum eigenvalue function, J. Global Optim. 30 (2004) 253-270. | MR 2115703 | Zbl 1066.90081
, , , ,[17] Direct Methods in the Calculus of Variations, Springer, 1989. | MR 990890 | Zbl 0703.49001
,[18] Optimal use of the translation method and relaxations of variational problems, Comm. Pure Appl. Math. 44 (1991) 643-678. | MR 1109375 | Zbl 0733.49018
,[19] Bounds and extreme microstructures for two-component composites: a unified treatment based on the translation method, Proc. Royal Soc. Lond. A 452 (1996) 919-944. | MR 1383293 | Zbl 0892.73030
,[20] On the smoothness of convex envelopes, Trans. AMS 322 (1990) 691-709. | MR 986024 | Zbl 0712.49010
, ,[21] Fundamentals of Convex Analysis, Springer, 2001. | MR 1865628 | Zbl 0998.49001
, ,[22] The relaxation of a double-well energy, Cont. Mech. Therm. 3 (1991) 981-1000. | MR 1122017 | Zbl 0825.73029
,[23] Differentiability of convex envelopes, C. R. Acad. Sci., Paris, Sr. I, Math. 333 (2001) 725-728. | MR 1868942 | Zbl 1053.49013
, ,[24] On bounding the effective conductivity of anisotropic composites, in: , , , (Eds.), Homogenization and Effective Moduli of Materials and Media, Springer, New York, 1986, pp. 97-125. | MR 859413 | Zbl 0631.73012
, ,[25] Convex Sets and Their Applications, John Wiley & Sons, 1982. | MR 655598 | Zbl 0492.52001
,[26] Exact estimates of the conductivity of composites formed two isotropically conducting media taken in prescribed proportion, Proc. Royal Soc. Edinb. A 99 (1984) 71-87. | MR 781086 | Zbl 0564.73079
, ,[27] A remark on regularization in Hilbert spaces, Israel J. Math. 55 (1986) 257-266. | MR 876394 | Zbl 0631.49018
, ,[28] On characterizing the set of possible effective tensors of composites: the variational method and the translation method, Comm. Pure Appl. Math. XLIII (1990) 63-125. | MR 1024190 | Zbl 0751.73041
,[29] Hamilton-Jacobi equations and distance functions on Riemannian manifolds, Appl. Math. Optim. 47 (2002) 1-25. | MR 1941909 | Zbl 1048.49021
, ,[30] Multiple Integrals in the Calculus of Variations, Springer, 1966. | MR 2492985 | Zbl 0142.38701
,[31] Proximité dualité dans un espace Hilbertien, Bull. Soc. Math. Fr. 93 (1965) 273-299. | Numdam | MR 201952 | Zbl 0136.12101
,[32] Fonctionnelles convexes, Lecture Notes, Collège de France, 1967.
,[33] Bounds for the effective conductivity of two-dimensional composites made of isotropic phases in prescribed volume fraction: the weighted translation method, Proc. Royal Soc. Edinb. A 125 (1995) 1219-1239. | MR 1363001 | Zbl 0852.35016
,[34] Smooth minimization of non-smooth functions, Math. Program. Ser. A 103 (2005) 127-152. | MR 2166537 | Zbl 1079.90102
,[35] Elastic materials with two preferred states, Q. J. Mech. Appl. Math. 44 (1991) 1-15. | MR 1093392 | Zbl 0735.73032
,[36] Shape interrogation for computer aided design and manufacturing, Springer, 2002. | MR 1891533 | Zbl 1035.65016
, ,[37] H.-D. Qi, P. Tseng, Analysis of piecewise smooth functions and almost smooth functions, 21 (2002) 45-66.
[38] Minimax theorems and conjugate saddle functions, Math. Scand. 14 (1964) 151-173. | MR 175037 | Zbl 0127.28309
,[39] Convex Analysis, Princeton Univ. Press, 1966. | MR 1451876 | Zbl 0193.18401
,[40] Image Analysis and Mathematical Morphology, Academic Press, 1982. | MR 753649 | Zbl 0565.92001
,[41] Solving continuous min-max problems by an iterative entropic regularization method, J. Optim. Theory Appl. 121 (2004) 597-612. | MR 2084345 | Zbl 1107.90037
, ,[42] On the problem of two wells, in: Microstructure and Phase Transition, IMA Vol. Math. Appl., vol. 54, Springer, New York, 1993, pp. 183-189. | MR 1320537 | Zbl 0797.73079
,[43] Compensated compactness and applications to partial differential equations, in: (Ed.), Nonlinear Analysis and Mechanics, Heriot-Watt Symp. IV, Pitman, 1979. | MR 584398 | Zbl 0437.35004
,[44] Estimations fine des coefficients homogénéisés, in: (Ed.), Ennio de Giorgi's Colloquium, Pitman, 1985, pp. 168-187. | MR 909716 | Zbl 0586.35004
,[45] Nouvelles applications des parametres continus a la theorie des formes quadratiques, J. Reine Angew. Math. 134 (1908) 198-287. | JFM 39.0274.01
,[46] Cut locus and medial axis in global shape interrogation and representation, ftp://ftp.gdv.uni-hannover.de/www/paper/ReportMIT93.pdf, MIT, Dept. Ocean Engineering, Design Laboratory Memorandum, 1993, no. 92-2.
,[47] Functional Analysis, Springer, 1971.
,[48] Energy minimizers in nonlinear elasticity and the implicit function theorem, Arch. Rational Mech. Anal. 114 (1991) 95-117. | MR 1094432 | Zbl 0734.73009
,[49] On some quasi-convex functions with linear growth, J. Convex Anal. 5 (1998) 133-146. | MR 1649465 | Zbl 0915.49008
,[50] On various semi-convex hulls in the calculus of variations, Cal. Var. PDEs 6 (1998) 143-160. | MR 1606473 | Zbl 0896.49005
,[51] On various semi-convex relaxations of the squared-distance function, Proc. Royal Soc. Edinb. A 129 (1999) 1309-1323. | MR 1728527 | Zbl 0936.49009
,[52] A two-well structure and intrinsic mountain pass points, Calc. Var. PDEs 13 (2001) 231-264. | MR 1861099 | Zbl 0997.49013
,[53] Maximal extension for linear spaces of real matrices with large rank, Proc. Royal Soc. Edinb. A 131 (2001) 1481-1491. | MR 1869646 | Zbl 0998.15019
,[54] An elementary derivation of the generalized Kohn-Strang relaxation formulae, J. Convex Anal. 9 (2002) 269-285. | MR 1917400 | Zbl 1004.49011
,[55] On conditions for equality of relaxations in the calculus of variations, J. Nonl. Convex Anal. 3 (2002) 145-154. | MR 1924763 | Zbl 1035.49013
,[56] On separation of gradient Young measures, Calc. Var. PDEs 17 (2003) 85-103. | MR 1979117 | Zbl 1036.49025
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