Simple proofs of some results of Reshetnyak
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- by Daniel Spector
- Proc. Amer. Math. Soc. 139 (2011), 1681-1690
- DOI: https://doi.org/10.1090/S0002-9939-2010-10593-2
- Published electronically: September 17, 2010
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Previous version: Original version posted September 16, 2010
Corrected version: Current version removes inaccurate historical reference in footnote 1.
Abstract:
In this paper we give simpler proofs of the classical continuity and lower semicontinuity theorems of Reshetnyak.References
- E. Acerbi and G. Dal Maso, New lower semicontinuity results for polyconvex integrals, Calc. Var. Partial Differential Equations 2 (1994), no. 3, 329–371. MR 1385074, DOI 10.1007/BF01235534
- Giovanni Alberti and Luigi Ambrosio, A geometrical approach to monotone functions in $\textbf {R}^n$, Math. Z. 230 (1999), no. 2, 259–316. MR 1676726, DOI 10.1007/PL00004691
- Luigi Ambrosio and Gianni Dal Maso, On the relaxation in $\textrm {BV}(\Omega ;\textbf {R}^m)$ of quasi-convex integrals, J. Funct. Anal. 109 (1992), no. 1, 76–97. MR 1183605, DOI 10.1016/0022-1236(92)90012-8
- 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. MR 1113814
- Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292
- Giovanni Bellettini, Guy Bouchitté, and Ilaria Fragalà, BV functions with respect to a measure and relaxation of metric integral functionals, J. Convex Anal. 6 (1999), no. 2, 349–366. MR 1736243
- Giuseppe Buttazzo and Paolo Guasoni, Shape optimization problems over classes of convex domains, J. Convex Anal. 4 (1997), no. 2, 343–351. MR 1613491
- Gerald B. Folland, Real analysis, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. Modern techniques and their applications; A Wiley-Interscience Publication. MR 1681462
- Irene Fonseca, Lower semicontinuity of surface energies, Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), no. 1-2, 99–115. MR 1149987, DOI 10.1017/S0308210500015018
- Irene Fonseca, The Wulff theorem revisited, Proc. Roy. Soc. London Ser. A 432 (1991), no. 1884, 125–145. MR 1116536, DOI 10.1098/rspa.1991.0009
- Irene Fonseca and Stefan Müller, A uniqueness proof for the Wulff theorem, Proc. Roy. Soc. Edinburgh Sect. A 119 (1991), no. 1-2, 125–136. MR 1130601, DOI 10.1017/S0308210500028365
- Irene Fonseca, Giovanni Leoni, and Jan Malý, Weak continuity and lower semicontinuity results for determinants, Arch. Ration. Mech. Anal. 178 (2005), no. 3, 411–448. MR 2196498, DOI 10.1007/s00205-005-0377-2
- Irene Fonseca and Giovanni Leoni, Modern methods in the calculus of variations: $L^p$ spaces, Springer Monographs in Mathematics, Springer, New York, 2007. MR 2341508
- C. Herring, Some Theorems on the Free Energies of Crystal Surfaces, Phys. Rev. 82 (1951), 87-93.
- Jan Kristensen and Filip Rindler, Relaxation of signed integral functionals in BV, Calc. Var. Partial Differential Equations 37 (2010), no. 1-2, 29–62. MR 2564396, DOI 10.1007/s00526-009-0250-5
- J. Kristensen, F. Rindler, Characterization of Generalized Gradient Young Measures Generated by Sequences in $W^{1,1}$ and $BV$, Arch. Rational Mech. Anal., 197, no. 2 (2010), 539–598.
- Robert Kohn, Felix Otto, Maria G. Reznikoff, and Eric Vanden-Eijnden, Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation, Comm. Pure Appl. Math. 60 (2007), no. 3, 393–438. MR 2284215, DOI 10.1002/cpa.20144
- Nam Q. Le, A gamma-convergence approach to the Cahn-Hilliard equation, Calc. Var. Partial Differential Equations 32 (2008), no. 4, 499–522. MR 2402921, DOI 10.1007/s00526-007-0150-5
- Stephan Luckhaus and Luciano Modica, The Gibbs-Thompson relation within the gradient theory of phase transitions, Arch. Rational Mech. Anal. 107 (1989), no. 1, 71–83. MR 1000224, DOI 10.1007/BF00251427
- Ju. G. Rešetnjak, The weak convergence of completely additive vector-valued set functions, Sibirsk. Mat. Ž. 9 (1968), 1386–1394 (Russian). MR 0240274
- Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
Bibliographic Information
- Daniel Spector
- Affiliation: Department of Mathematics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213-3890
- Email: dspector@andrew.cmu.edu
- Received by editor(s): February 16, 2010
- Received by editor(s) in revised form: May 18, 2010
- Published electronically: September 17, 2010
- Communicated by: Tatiana Toro
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 1681-1690
- MSC (2010): Primary 49J45; Secondary 46E27, 46G10, 28C15
- DOI: https://doi.org/10.1090/S0002-9939-2010-10593-2
- MathSciNet review: 2763757
Dedicated: To Pei Chen, whose love and support lift me daily