Mathematical problems in the theory of phase transitions in continuum mechanics
Algebra i analiz, Tome 29 (2017) no. 5, pp. 111-178.

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V. G. Osmolovskiǐ. Mathematical problems in the theory of phase transitions in continuum mechanics. Algebra i analiz, Tome 29 (2017) no. 5, pp. 111-178. http://geodesic.mathdoc.fr/item/AA_2017_29_5_a4/

[1] Burago Yu. D., Zalgaller V. A., Geometricheskie neravenstva, Nauka, L., 1980 | MR

[2] Buttatso D., Dzhakvinta M., Gildebrandt S., Odnomernye variatsionnye zadachi, Nauchn. kn., Novosibirsk, 2002

[3] Grinfeld M. A., Metody mekhaniki sploshnykh sred v teorii fazovykh prevraschenii, Nauka, M., 1990 | MR

[4] Gelfand I. M., Fomin S. V., Variatsionnoe ischislenie, Gos. izd. fiz.-mat. lit., M., 1961 | MR

[5] Demyanov A. V., “Relaksatsiya i $\Gamma$-skhodimost funktsionalov energii dlya dvukhfazovoi uprugoi sredy”, Probl. mat. anal., 30, 2005, 17–30 | Zbl

[6] Dzhusti E., Minimalnye poverkhnosti i funktsii ogranichennoi variatsii, Mir, M., 1989 | MR

[7] Eremeev V. A., Freidin A. B., Sharipova L. L., “Ob ustoichivosti ravnovesiya dvukhfazovykh uprugikh tel”, Prikl. mat. i mekh., 71:1 (2007), 66–92 | MR | Zbl

[8] Zhikov V. V., “Voprosy skhodimosti, dvoistvennosti i usredneniya dlya funktsionalov variatsionnogo ischisleniya”, Izv. AN SSSR. Ser. mat., 47:5 (1983), 961–998 | MR | Zbl

[9] Zarubin V. S., Kuvykin G. N., Matematicheskie modeli termomekhaniki, Fizmatlit, M., 2002

[10] Kristensen R., Vvedenie v mekhaniku kompozitov, Mir, M., 1982

[11] Kronover R. M., Fraktaly i khaos v dinamicheskikh sistemakh, Postmarket, M., 2000

[12] Kucher V. A., Osmolovskii V. G., “Vychislenie vtoroi variatsii dlya funktsionala energii dvukhfazovoi sredy”, Probl. mat. anal., 22, 2001, 41–73 | Zbl

[13] Landau L. D., Lifshits E. M., Teoreticheskaya fizika, v. 7, Teoriya uprugosti, Nauka, M., 1965

[14] Mikhailov A. S., Mikhailov V. S., “Fazovye perekhody v mnogofazovykh sredakh”, Probl. mat. anal., 20, 2000, 120–169

[15] Mikhailov A. S., Mikhailov V. S., “Zamechaniya k teoreme o pokrytiyakh”, Probl. mat. anal., 24, 2002, 147–155

[16] Osmolovskii V. G., Variatsionnaya zadacha o fazovykh perekhodakh v mekhanike sploshnoi sredy, Izd-vo SPbGU, SPb, 2000

[17] Osmolovskii V. G., “Suschestvovanie sostoyanii ravnovesiya v odnomernoi zadache o fazovykh perekhodakh”, Vestn. S.-Peterburg. un-ta. Ser. 1, 2006, no. 3, 54–65

[18] Osmolovskii V. G., “Ustoichivost odnofazovykh sostoyanii ravnovesiya dlya dvukhfazovoi uprugoi sredy pri nulevom koeffitsiente poverkhnostnogo natyazheniya. Odnomernyi sluchai”, Probl. mat. anal., 32, 2006, 3–19

[19] Osmolovskii V. G., “Tochnye resheniya zadachi o fazovykh perekhodakh v odnomernom modelnom sluchae”, Vestn. S.-Peterburg. un-ta. Ser. 1, 2007, no. 3, 42–48

[20] Osmolovskii V. G., “Odnomernaya zadacha o fazovykh perekhodakh v mekhanike sploshnykh sred pri nalichii mikroneodnorodnostei”, Probl. mat. anal., 46, 2010, 105–115

[21] Osmolovskii V. G., “Teorema suschestvovaniya i slabaya forma uravnenii Lagranzha dlya variatsionnoi zadachi teorii fazovykh prevraschenii”, Sib. mat. zh., 35:4 (1994), 835–846 | MR | Zbl

[22] Osmolovskii V. G., “Teorema suschestvovaniya i tochnye resheniya v variatsionnoi zadache o vysokotemperaturnykh fazovykh perekhodakh pri nulevom koeffitsiente poverkhnostnogo natyazheniya”, Probl. mat. anal., 15, 1995, 201–212

[23] Osmolovskii V. G., “Neobkhodimye usloviya ekstremuma v variatsionnoi zadache o fazovykh perekhodakh s neodnorodnymi granichnymi usloviyami”, Probl. mat. anal., 22, 2001, 160–178

[24] Osmolovskii V. G., “Kriterii slaboi polunepreryvnosti snizu funktsionala energii dvukhfazovoi uprugoi sredy”, Probl. mat. anal., 26, 2003, 215–254 | Zbl

[25] Osmolovskii V. G., “Tochnye resheniya variatsionnoi zadachi teorii fazovykh perekhodov mekhaniki sploshnykh sred”, Probl. mat. anal., 27, 2004, 171–205 | Zbl

[26] Osmolovskii V. G., “Zavisimost sostoyanii ravnovesiya dvukhfazovoi uprugoi sredy ot temperatury pri nulevom koeffitsiente poverkhnostnogo natyazheniya”, Probl. mat. anal., 28, 2004, 98–114

[27] Osmolovskii V. G., “Suschestvovanie temperatur fazovykh perekhodov dlya neodnorodnoi anizotropnoi dvukhfazovoi uprugoi sredy”, Probl. mat. anal., 31, 2005, 59–66 | Zbl

[28] Osmolovskii V. G., “O temperaturakh fazovykh perekhodov v variatsionnoi zadache teorii uprugosti dvukhfazovykh sred”, Probl. mat. anal., 41, 2009, 37–47

[29] Osmolovskii V. G., “Vozmuschenie kriticheskoi tochki funktsionala energii dvukhfazovoi uprugoi sredy. I. Vozmuschenie kontaktnoi zadachi v prostranstve Soboleva”, Probl. mat. anal., 43, 2009, 89–112

[30] Osmolovskii V. G., “Variatsionnaya zadacha o fazovykh perekhodakh pri nulevom koeffitsiente poverkhnostnogo natyazheniya”, Algebra i analiz, 22:6 (2010), 214–234 | MR | Zbl

[31] Osmolovskii V. G., “Vychislenie entropii v zadache termouprugosti dlya dvukhfazovykh uprugikh sred”, Probl. mat. anal., 56, 2011, 115–127

[32] Osmolovskii V. G., “Mnogomernaya zadacha o fazovykh perekhodakh v mekhanike sploshnykh sred pri nalichii mikroneodnorodstei”, Probl. mat. anal., 64, 2012, 93–100

[33] Osmolovskii V. G., “Nezavisimost temperatur fazovykh perekhodov ot oblasti, zanimaemoi dvukhfazovoi uprugoi sredoi”, Probl. mat. anal., 66, 2012, 147–151 | Zbl

[34] Osmolovskii V. G., “Ustoichivost regulyarnykh potentsialnykh kriticheskikh tochek funktsionala energii dvukhfazovoi uprugoi sredy”, Probl. mat. anal., 78, 2015, 141–148

[35] Osmolovskii V. G., “Temperatury fazovykh perekhodov i kvazivypuklaya obolochka dlya funktsionala energii dvukhfazovoi uprugoi sredy s anizotropnymi tenzorami ostatochnoi deformatsii”, Probl. mat. anal., 77, 2014, 119–128

[36] Osmolovskii V. G., “Kvazistatsionarnaya zadacha o dvizhenii mezhfazovykh granits v teorii fazovykh perekhodov mekhaniki sploshnykh sred”, Probl. mat. anal., 73, 2013, 115–123

[37] Osmolovskii V. G., “Zavisimost temperatury fazovykh perekhodov ot razmerov oblasti”, Zap. nauch. semin. POMI, 310, 2004, 98–113 | MR | Zbl

[38] Osmolovskii V. G., “Zavisimost sostoyanii ravnovesiya dvukhfazovoi uprugoi sredy ot temperatury pri polozhitelnom koeffitsiente poverkhnostnogo natyazheniya”, Zap. nauch. semin. POMI, 318, 2004, 220–232 | MR | Zbl

[39] Osmolovskii V. G., “Ob opredelenii koeffitsienta poverkhnostnogo natyazheniya v mekhanike dvukhfazovykh uprugikh sred”, Probl. mat. anal., 30, 2005, 61–68 | Zbl

[40] Osmolovskii V. G., “O mnozhestve reshenii variatsionnoi zadachi o fazovykh perekhodakh v mekhanike sploshnykh sred”, Probl. mat. anal., 35, 2007, 110–119

[41] Osmolovskii V. G., “Izoperimetricheskoe neravenstvo i sostoyaniya ravnovesiya dlya dvukhfazovoi sredy”, Probl. mat. anal., 36, 2007, 81–88

[42] Osmolovskii V. G., “Zavisimost ob'ëma ravnovesnoi fazy ot temperatury v zadache o fazovykh perekhodakh v mekhanike sploshnykh sred”, Probl. mat. anal., 37, 2008, 73–82 | Zbl

[43] Osmolovskii V. G., “Rol odnofazovykh sostoyanii ravnovesiya dlya dvukhfazovoi uprugoi sredy”, Probl. mat. anal., 50, 2010, 77–86 | Zbl

[44] Osmolovskii V. G., “Povedenie ploschadi granitsy razdela faz v zadachakh o fazovykh perekhodakh pri stremlenii k nulyu koeffitsienta poverkhnostnogo natyazheniya”, Probl. mat. anal., 62, 2012, 101–108 | Zbl

[45] Osmolovskii V. G., “Kvazivypuklaya obolochka dlya odnorodnoi izotropnoi fvukhfazovoi uprugoi sredy i resheniya iskhodnoi i relaksirovannoi zadach”, Probl. mat. anal., 70, 2013, 161–170 | Zbl

[46] Osmolovskii V. G., “Opisanie mnozhestva vsekh reshenii relaksirovannoi zadachi dlya odnorodnoi izotropnoi dvukhfazovoi uprugoi sredy”, Probl. mat. anal., 72, 2013, 147–155 | Zbl

[47] Osmolovskii V. G., Matematicheskie voprosy teorii fazovykh perekhodov v mekhanike sploshnykh sred, St. Petersburg Math. Soc. Preprint 2014-04, http://www.mathsoc.spb.ru/preprint/2014/index.html

[48] Syarle F., Matematicheskaya teoriya uprugosti, Mir, M., 1992

[49] Fikera G., Teoremy suschestvovaniya v teorii uprugosti, Nauka, M., 1974

[50] Fridman A., Variatsionnye printsipy i zadachi so svobodnymi poverkhnostyami, Nauka, M., 1990 | MR

[51] Evans L. K., Gariepi R. F., Teoriya mery i tonkie svoistva funktsii, Nauchn. kn., Novosibirsk, 2002

[52] Evans L. K., Metody slaboi skhodimosti dlya nelineinykh uravnenii s chastnymi proizvodnymi, Tamara Rozhkovskaya, Novosibirsk, 2006

[53] Allaire G., Lods V., “Minimizers for double-well problem with affine boundary conditions”, Proc. Roy. Soc. Edinburg Sect. A, 129:3 (1999), 439–466 | DOI | MR | Zbl

[54] Allaire G., Shape optimization by the homigenization methods, Appl. Math. Sci., 146, Springer-Verlag, Berlin, 2002 | DOI | MR

[55] Bildhauer M., Fuchs M., Osmolovskii V., “The effect of a surface energy term on the distribution of phases in an elastic medium with a two-well elastic potential”, Math. Methods Appl. Sci., 25:2 (2002), 149–178 | DOI | MR | Zbl

[56] Bildhauer M., Fuchs M., Osmolovskii V. G., “The effect of a penalty term involving higher order derivatives on the distributions of phases in an elastic medium with a two-well elastic potemtial”, Math. Meth. Appl. Sci., 25:4 (2002), 289–308 | DOI | MR | Zbl

[57] Dacorogna B., Direct methods in the calculus of variations, Appl. Math. Sci., 78, Springer-Verlag, Berlin, 1989 | DOI | MR | Zbl

[58] Dacorogna B., Pisante G., Ribeiro A. M., “On non quasiconvex problems of the calculus of variations”, Discrete Contin. Dyn. Syst., 13:4 (2005), 961–983 | DOI | MR | Zbl

[59] Dolzmann G., Variational methods for crystalline microstructure – analysis and computation, Lecture Notes in Math., 1803, Springer-Verlag, Berlin, 2003 | DOI | MR | Zbl

[60] Fu Y. B., Freidin A. B., “Characterization and stability of two-phase piecewice-homogeneous deformations”, Proc. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460:2051 (2004), 3065–3094 | DOI | MR | Zbl

[61] Giusti E., Direct methods in the calculus of variations, World Sci. Publ. Co., River Edge, NJ, 2003 | MR | Zbl

[62] Hashin Z., “The elastic moduli of heterogeneuos materials”, Trans. ASME Ser. E, J. Appl. Mech., 29 (1962), 143–159 | DOI | MR

[63] Müller S., Variational models for microstructure and phase transitions, Lecture Notes, v. 2, Max-Planck-Inst. Math., 1998 | MR

[64] Müller S., Microstructure, phase transitions and geometry, Preprint no. 3, Max-Planc-Inst. Math., 1997 | MR

[65] Osmolovski V. G., “The phase transition in mechanic of continuum media for big loading”, Math. Nachr., 177 (1996), 233–250 | DOI | MR | Zbl

[66] Tamanini I., Regularity results for almost minimal oriented hypersurface in $R^N$, Quaderni Dipart. Mat. Univ. Lecce, 1984