# zbMATH — the first resource for mathematics

Simple proofs of some results of Reshetnyak. (English) Zbl 1215.49025
Summary: We give simpler proofs of the classical continuity and lower semicontinuity theorems of Reshetnyak.

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 46E27 Spaces of measures 46G10 Vector-valued measures and integration 28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
Full Text:
##### References:
 [1] E. Acerbi and G. Dal Maso, New lower semicontinuity results for polyconvex integrals, Calc. Var. Partial Differential Equations 2 (1994), no. 3, 329 – 371. · Zbl 0810.49014 [2] Giovanni Alberti and Luigi Ambrosio, A geometrical approach to monotone functions in \?$$^{n}$$, Math. Z. 230 (1999), no. 2, 259 – 316. · Zbl 0934.49025 [3] Luigi Ambrosio and Gianni Dal Maso, On the relaxation in \?\?(\Omega ;\?^{\?}) of quasi-convex integrals, J. Funct. Anal. 109 (1992), no. 1, 76 – 97. · Zbl 0769.49009 [4] 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. · Zbl 0662.49007 [5] 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. · Zbl 0957.49001 [6] 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. · Zbl 0959.49015 [7] Giuseppe Buttazzo and Paolo Guasoni, Shape optimization problems over classes of convex domains, J. Convex Anal. 4 (1997), no. 2, 343 – 351. · Zbl 0921.49030 [8] 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. · Zbl 0924.28001 [9] Irene Fonseca, Lower semicontinuity of surface energies, Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), no. 1-2, 99 – 115. · Zbl 0757.49013 [10] Irene Fonseca, The Wulff theorem revisited, Proc. Roy. Soc. London Ser. A 432 (1991), no. 1884, 125 – 145. · Zbl 0725.49017 [11] 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. · Zbl 0752.49019 [12] 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. · Zbl 1081.49013 [13] Irene Fonseca and Giovanni Leoni, Modern methods in the calculus of variations: \?^{\?} spaces, Springer Monographs in Mathematics, Springer, New York, 2007. · Zbl 1153.49001 [14] C. Herring, Some Theorems on the Free Energies of Crystal Surfaces, Phys. Rev. 82 (1951), 87-93. · Zbl 0042.23201 [15] Jan Kristensen and Filip Rindler, Relaxation of signed integral functionals in BV, Calc. Var. Partial Differential Equations 37 (2010), no. 1-2, 29 – 62. · Zbl 1189.49018 [16] 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. · Zbl 1245.49060 [17] 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. · Zbl 1154.35021 [18] Nam Q. Le, A gamma-convergence approach to the Cahn-Hilliard equation, Calc. Var. Partial Differential Equations 32 (2008), no. 4, 499 – 522. · Zbl 1147.35118 [19] 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. · Zbl 0681.49012 [20] Ju. G. Rešetnjak, The weak convergence of completely additive vector-valued set functions, Sibirsk. Mat. Ž. 9 (1968), 1386 – 1394 (Russian). [21] Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. · Zbl 0867.46001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.