Elliptic problems with nonlocal boundary conditions and Feller semigroups

P. L. Gurevich

Contemporary Mathematics. Fundamental Directions, Partial differential equations, Tome 38 (2010), pp. 3-173.

Résumé

This monograph is devoted to the following interrelated problems: the solvability and smoothness of elliptic linear equations with nonlocal boundary conditions and the existence of Feller semigroups that appear in the theory of multidimensional diffusion processes.

Bibliographie

[1] Agranovich M. S., Vishik M. I., “Ellipticheskie zadachi s parametrom i parabolicheskie zadachi obschego vida”, Usp. mat. nauk, 19:3 (1964), 53–161 | MR | Zbl

[2] Akhiezer N. I., Glazman I. M., Teoriya lineinykh operatorov v gilbertovom prostranstve, Nauka, M., 1966 | MR | Zbl

[3] Bitsadze A. V., “K teorii nelokalnykh kraevykh zadach”, Dokl. AN SSSR, 277:1 (1984), 17–19 | MR | Zbl

[4] Bitsadze A. V., “Ob odnom klasse uslovno razreshimykh nelokalnykh kraevykh zadach dlya garmonicheskikh funktsii”, Dokl. AN SSSR, 280:3 (1985), 521–524 | MR | Zbl

[5] Bitsadze A. V., Samarskii A. A., “O nekotorykh prosteishikh obobscheniyakh lineinykh ellipticheskikh kraevykh zadach”, Dokl. AN SSSR, 185:4 (1969), 739–740 | Zbl

[6] Venttsel A. D., “O granichnykh usloviyakh dlya mnogomernykh diffuzionnykh protsessov”, Teor. veroyatn. i ee primeneniya, 4:2 (1959), 172–185 | Zbl

[7] Vishik M. I., “Ob obschikh kraevykh zadachakh dlya ellipticheskikh differentsialnykh uravnenii”, Tr. Mosk. Mat. ob-va, 1, 1952, 187–246 | Zbl

[8] Volevich L. R., “Razreshimost kraevykh zadach dlya obschikh ellipticheskikh sistem”, Mat. sb., 68(110):3 (1965), 373–416 | MR | Zbl

[9] Galakhov E. I., “O dostatochnykh usloviyakh suschestvovaniya polugrupp Fellera”, Mat. zametki, 60:3 (1996), 442–444 | MR | Zbl

[10] Galakhov E. I., Skubachevskii A. L., “O szhimayuschikh neotritsatelnykh polugruppakh s nelokalnymi usloviyami”, Matem. sb., 189:1 (1998), 45–78 | MR | Zbl

[11] Gilbarg D., Trudinger M., Ellipticheskie differentsialnye uravneniya vtorogo poryadka, Nauka, M., 1989 | Zbl

[12] Gokhberg I. Ts., Sigal E. I., “Operatornoe obobschenie teoremy o logarifmicheskom vychete i teoremy Rushe”, Matem. sb., 84(126):4 (1971), 607–629 | MR | Zbl

[13] Gurevich P. L., “Asimptotika reshenii nelokalnykh ellipticheskikh zadach v ploskikh uglakh”, Tr. sem. im. I. G. Petrovskogo, 23, 2003, 93–126 | Zbl

[14] Gurevich P. L., “Nelokalnye ellipticheskie zadachi s nelineinymi preobrazovaniyami peremennykh vblizi tochek sopryazheniya”, Izv. RAN. Ser. matem., 67:6 (2003), 71–110 | MR | Zbl

[15] Gurevich P. L., “O gladkosti obobschennykh reshenii nelokalnykh ellipticheskikh zadach na ploskosti”, Dokl. AN, 398:3 (2004), 295–299 | MR

[16] Gurevich P. L., “Obobschennye resheniya nelokalnykh ellipticheskikh zadach”, Matem. zametki, 77:5 (2005), 665–682 | MR | Zbl

[17] Gurevich P. L., “Ob ustoichivosti indeksa neogranichennykh nelokalnykh operatorov v prostranstvakh Soboleva”, Trudy MIAN, 255, 2006, 116–135 | MR

[18] Gurevich P. L., “O neustoichivosti indeksa nekotorykh nelokalnykh ellipticheskikh zadach”, Trudy seminara im. I. G. Petrovskogo, 26, 2007, 179–194

[19] Gurevich P. L., “Neogranichennye vozmuscheniya dvumernykh diffuzionnykh protsessov s nelokalnymi kraevymi usloviyami”, Dokl. AN, 417:4 (2007), 451–455 | MR | Zbl

[20] Gurevich P. L., “O suschestvovanii polugruppy Fellera s atomarnoi meroi v nelokalnom kraevom uslovii”, Trudy MIAN, 260, 2008, 164–179 | MR

[21] Gurevich P. L., “Ogranichennye vozmuscheniya dvumernykh diffuzionnykh protsessov s nelokalnymi usloviyami vblizi granitsy”, Matem. zametki, 83:2 (2008), 181–198 | MR | Zbl

[22] Gurevich P. L., “O nesuschestvovanii polugrupp Fellera v netransversalnom sluchae”, Uspekhi matem. nauk, 63:3 (2008), 159–160 | MR | Zbl

[23] Gurevich P. L., Skubachevskii A. L., “O fredgolmovoi i odnoznachnoi razreshimosti nelokalnykh ellipticheskikh zadach v mnogomernykh oblastyakh”, Trudy Mosk. matem. obsch., 68, 2007, 288–373 | MR | Zbl

[24] Gurevich P. L., Eger V., Skubachevskii A. L., “O suschestvovanii periodicheskikh reshenii nekotorykh nelineinykh zadach termokontrolya”, Dokl. AN, 418:2 (2008), 151–154 | MR | Zbl

[25] Guschin A. K., “Uslovie kompaktnosti odnogo klassa operatorov i ego primenenie k issledovaniyu razreshimosti nelokalnykh zadach dlya ellipticheskikh uravnenii”, Matem. sb., 193:5 (2002), 17–36 | MR | Zbl

[26] Guschin A. K., Mikhailov V. P., “O razreshimosti nelokalnykh zadach dlya ellipticheskogo uravneniya vtorogo poryadka”, Matem. sb., 185:1 (1994), 121–160 | MR | Zbl

[27] Guschin A. K., Mikhailov V. P., “O nepreryvnosti reshenii odnogo klassa nelokalnykh zadach dlya ellipticheskogo uravneniya”, Matem. sb., 186:2 (1995), 37–58 | MR | Zbl

[28] Zhitarashu N. V., Eidelman S. D., “O nelokalnykh granichnykh zadachakh dlya ellipticheskikh uravnenii”, Matem. issled., 6:2(20) (1971), 63–73 | MR | Zbl

[29] Ilin V. A., “Neobkhodimye i dostatochnye usloviya bazisnosti Rissa kornevykh vektorov razryvnykh operatorov vtorogo poryadka”, Differents. uravn., 22:12 (1986), 2059–2071 | MR

[30] Ilin V. A., Moiseev E. I., “Apriornaya otsenka resheniya zadachi, sopryazhennoi k nelokalnoi kraevoi zadache pervogo roda”, Differents. uravneniya, 24:5 (1988), 795–804 | MR

[31] Ilin V. A., Moiseev E. I., “Dvumernaya nelokalnaya kraevaya zadacha dlya operatora Puassona v differentsialnoi i raznostnoi traktovkakh”, Matem. model., 2:8 (1990), 139–156 | MR | Zbl

[32] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | Zbl

[33] Kishkis K. Yu., “Ob indekse zadachi Bitsadze–Samarskogo dlya garmonicheskikh funktsii”, Differents. uravneniya, 24:1 (1988), 105–110 | MR | Zbl

[34] Kishkis K. Yu., “K teorii nelokalnykh zadach dlya uravneniya Laplasa”, Differents. uravneniya, 25:1 (1989), 59–64 | MR | Zbl

[35] Kovaleva O. A., Skubachevskii A. L., “Razreshimost nelokalnykh ellipticheskikh zadach v prostranstvakh s vesom”, Matem. zametki, 67:6 (2000), 882–898 | MR | Zbl

[36] Kondratev V. A., “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi ili uglovymi tochkami”, Tr. Mosk. matem. obsch., 16, 1967, 209–292

[37] Kondratev V. A., “Osobennosti reshenii zadachi Dirikhle dlya ellipticheskikh uravnenii vtorogo poryadka v okrestnosti rebra”, Differents. uravneniya, 13:11 (1977), 2026–2032 | MR

[38] Krein S. G., Lineinye uravneniya v banakhovom prostranstve, Nauka, M., 1971 | MR

[39] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971 | Zbl

[40] Mazya V. G., Plamenevskii B. A., “$L_p$-otsenki reshenii ellipticheskikh kraevykh zadach v oblastyakh s rebrami”, Trudy Mosk. matem. obsch., 37, 1978, 49–93 | MR

[41] Mitidieri E., Pokhozhaev S. I., “Teoremy tipa Liuvillya dlya nekotorykh nelineinykh nelokalnykh zadach”, Dokl. AN, 399:6 (2004), 732–736 | MR

[42] Moiseev E. I., “O spektralnykh kharakteristikakh odnoi nelokalnoi kraevoi zadachi”, Differents. uravneniya, 30:5 (1994), 864–872 | MR | Zbl

[43] Moiseev E. I., “Ob otsutstvii svoistva bazisnosti u sistemy kornevykh funktsii odnoi nelokalnoi kraevoi zadachi”, Differents. uravneniya, 30:12 (1994), 2082–2093 | MR | Zbl

[44] Muskhelishvili N. I., Singulyarnye integralnye uravneniya, Fizmatgiz, M., 1962

[45] Mustafin M. A., “Summiruemost metodom Abelya ryadov po kornevym vektoram nelokalnykh ellipticheskikh zadach”, Differents. uravneniya, 26:1 (1990), 167–168 | MR | Zbl

[46] Nazarov S. A., Plamenevskii B. A., Ellipticheskie zadachi v oblastyakh s kusochno gladkoi granitsei, Nauka, M., 1991

[47] Onanov G. G., Skubachevskii A. L., “Differentsialnye uravneniya s otklonyayuschimisya argumentami v statsionarnykh zadachakh mekhaniki deformiruemogo tela”, Prikladnaya mekhanika, 15:5 (1979), 39–47 | MR | Zbl

[48] Paneyakh B. P., “O nekotorykh nelokalnykh kraevykh zadachakh dlya lineinykh differentsialnykh operatorov”, Matem. zametki, 35:3 (1984), 425–434 | MR | Zbl

[49] Pod'yapolskii V. V., “Polnota i bazisnost po Abelyu sistemy kornevykh funktsii odnoi nelokalnoi zadachi”, Differents. uravneniya, 35:4 (1999), 568–569 | MR

[50] Pod'yapolskii V. V., Skubachevskii A. L., “O glavnom chlene spektralnoi asimptotiki nelokalnykh ellipticheskikh zadach”, Konferentsiya, posvyaschennaya 75-letiyu L. D. Kudryavtseva, Tezisy dokladov, Moskva, 1998, 157

[51] Riss F., Sekefalvi-Nad B., Lektsii po fukntsionalnomu analizu, Inostrannaya literatura, M., 1954

[52] Roitberg Ya. A., Sheftel Z. G., “Ob odnom klasse obschikh nelokalnykh ellipticheskikh zadach”, Dokl. AN SSSR, 192:3 (1970), 511–513 | MR

[53] Roitberg Ya. A., Sheftel Z. G., “Nelokalnye zadachi dlya ellipticheskikh uravnenii i sistem”, Sib. matem. zhurn., 13:1 (1972), 165–181 | MR | Zbl

[54] Samarskii A. A., “O nekotorykh problemakh teorii differentsialnykh uravnenii”, Differents. uravneniya, 16:11 (1980), 1925–1935 | MR

[55] Skubachevskii A. L., “O nekotorykh nelokalnykh ellipticheskikh kraevykh zadachakh”, Differents. uravneniya, 18:9 (1982), 1590–1599 | MR

[56] Skubachevskii A. L., “Nelokalnye ellipticheskie zadachi s parametrom”, Matem. sb., 121(163):2 (1983), 201–210 | MR | Zbl

[57] Skubachevskii A. L., “Ellipticheskie zadachi A. V. Bitsadze, A. A. Samarskogo”, Dokl. AN SSSR, 278:4 (1984), 813–816 | MR

[58] Skubachevskii A. L., “Razreshimost ellipticheskikh zadach s kraevymi usloviyami tipa Bitsadze–Samarskogo”, Differents. uravneniya, 21:4 (1985), 701–706 | MR

[59] Skubachevskii A. L., “Ellipticheskie zadachi s nelokalnymi usloviyami vblizi granitsy”, Matem. sb., 129(171):2 (1986), 279–302 | MR | Zbl

[60] Skubachevskii A. L., “O nekotorykh zadachakh dlya mnogomernykh diffuzionnykh protsessov”, Dokl. Akad. nauk SSSR, 307:2 (1989), 287–291 | MR

[61] Skubachevskii A. L., “O sobstvennykh znacheniyakh i sobstvennykh funktsiyakh nekotorykh nelokalnykh kraevykh zadach”, Differents. uravneniya, 25:1 (1989), 127–136 | MR

[62] Skubachevskii A. L., “Modelnye nelokalnye zadachi dlya ellipticheskikh uravnenii v dvugrannykh uglakh”, Differents. uravneniya, 26:1 (1990), 120–131 | MR

[63] Skubachevskii A. L., “O metode srezayuschikh funktsii v teorii nelokalnykh zadach”, Differents. uravneniya, 27:1 (1991), 128–139 | MR

[64] Skubachevskii A. L., “O polugruppakh Fellera dlya mnogomernykh diffuzionnykh protsessov”, Dokl. AN, 341:2 (1995), 173–176 | MR

[65] Skubachevskii A. L., “O razreshimosti nelokalnykh zadach dlya ellipticheskikh sistem v beskonechnykh uglakh”, Dokl. AN, 412:3 (2007), 317–320 | MR

[66] Skubachevskii A. L., “Neklassicheskie kraevye zadachi. I”, Sovrem. mat. Fundam. napravl., 26, 2007, 3–132 | MR | Zbl

[67] Skubachevskii A. L., “Neklassicheskie kraevye zadachi. II”, Sovrem. mat. Fundam. napravl., 33, 2009, 3–179 | MR

[68] Skubachevskii A. L., “Asimptotika reshenii nelokalnykh ellipticheskikh zadach”, Trudy MIAN, 269, 2010, 225–241 | MR | Zbl

[69] Soldatov A. P., “Zadacha Bitsadze–Samarskogo dlya funktsii, analiticheskikh po Duglisu”, Differents. uravneniya, 41:3 (2005), 396–407 | MR | Zbl

[70] Stein I., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR | Zbl

[71] Tamarkin Ya. D., O nekotorykh obschikh zadachakh teorii obyknovennykh lineinykh differentsialnykh uravnenii, Petrograd, 1917 | Zbl

[72] Shkalikov A. A., “O bazisnosti sobstvennykh funktsii obyknovennykh differentsialnykh operatorov s integralnymi kraevymi usloviyami”, Vestn. MGU. Ser. matem. i mekh., 1982, no. 6, 12–21 | MR | Zbl

[73] Amann H., “Feedback stabilization of linear and semilinear parabolic systems”, Lect. Notes Pure Appl. Math., 116, 1989, 21–57 | MR | Zbl

[74] Antonevich A., Lebedev A., Functional Differential Equations. I. $C^*$-theory, Longman Scientific Technical, Harlow, 1994 | MR | Zbl

[75] Beals R., “Nonlocal elliptic boundary value problems”, Bull. Amer. Math. Soc., 70:5 (1964), 693–696 | DOI | MR | Zbl

[76] Browder F., “Non-local elliptic boundary value problems”, Amer. J. Math., 86 (1964), 735–750 | DOI | MR | Zbl

[77] Bensoussan A., Lions J.-L., Impulse Control and Quasi-Variational Inequalities, Gauthier-Villars, Paris, 1984 | MR

[78] Bony J. M., Courrege P., Priouret P., “Semi-groups de Feller sur une variété à bord compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum”, Ann. Inst. Fourier (Grenoble), 18 (1968), 369–521 | MR | Zbl

[79] Carleman T., “Sur la théorie des equations integrales et ses applications”, Verhandlungen des Internat. Math. Kongr. Zürich, 1 (1932), 138–151 | Zbl

[80] Colli P., Grasselli M., Sprekels J., “Automatic control via thermostats of a hyperbolic Stefan problem with memory”, Appl. Math. Optim., 39 (1999), 229–255 | DOI | MR | Zbl

[81] Feller W., “The parabolic differential equations and the associated semi-groups of transformations”, Ann. Math., 55 (1952), 468–519 | DOI | MR | Zbl

[82] Feller W., “Diffusion processes in one dimension”, Trans. Amer. Math. Soc., 77 (1954), 1–30 | DOI | MR

[83] Galakhov E. I., “Multidimensional diffusion processes with nonlocal conditions”, Funct. Differ. Equ., 8:1–2 (2001), 225–238 | MR | Zbl

[84] Galakhov E. I., Skubachevskii A. L., “On Feller semigroups generated by elliptic operators with integro-differential boundary conditions”, J. Differential Equations, 176 (2001), 315–355 | DOI | MR | Zbl

[85] Garroni M. G., Menaldi J. L., Second order elliptic integro-differential problems, Chapman Hall/CRC, London–New York–Washington, D.C., 2002 | MR | Zbl

[86] Gurevich P. L., “Nonlocal problems for elliptic equations in dihedral angles and the Green formula”, Mitteilungen aus dem Mathem. Seminar Giessen, 247, Math. Inst. Univ., Giessen, Germany, 2001, 1–74 | MR | Zbl

[87] Gurevich P. L., “Asymptotics of solutions for nonlocal elliptic problems in plane bounded domains”, Funct. Diff. Equ., 10:1–2 (2003), 175–214 | MR | Zbl

[88] Gurevich P. L., “Solvability of nonlocal elliptic problems in Sobolev spaces, I”, Russ. J. Math. Phys., 10:4 (2003), 436–466 | MR | Zbl

[89] Gurevich P. L., “Solvability of nonlocal elliptic problems in Sobolev spaces, II”, Russ. J. Math. Phys., 11:1 (2004), 1–44 | MR | Zbl

[90] Gurevich P. L., “The consistency conditions and the smoothness of generalized solutions of nonlocal elliptic problems”, Adv. Differential Equations, 11:3 (2006), 305–360 | MR | Zbl

[91] Gurevich P. L., “Smoothness of generalized solutions for higher-order elliptic equations with nonlocal boundary conditions”, J. Differential Equations, 245 (2008), 1323–1355 | DOI | MR | Zbl

[92] Ishikawa Y., “A remark on the existence of a diffusion process with non-local boundary conditions”, J. Math. Soc. Japan, 42 (1990), 171–184 | DOI | MR | Zbl

[93] Krall A. M., “The development of general differential and general differential-boundary systems”, Rocky Mountain J. Math., 5 (1975), 493–542 | DOI | MR | Zbl

[94] Onanov G. G., Tsvetkov E. L., “On the minimum of the energy functional with respect to functions with deviating argument in a stationary problem of elasticity theory”, Russ. J. Math. Phys., 3:4 (1995), 491–500 | MR | Zbl

[95] Picone M., “Equazione integrale traducente il più generale problema lineare per le equazioni differenziali lineari ordinarie di qualsivoglia ordine”, Academia nazionale dei Lincei. Atti dei Conveg., 15 (1932), 942–948 | Zbl

[96] Roitberg Ya. A., Elliptic boundary value problems in the spaces of distributions, Kluwer Academic Publishers, Dordrecht–Boston–London, 1996 | MR | Zbl

[97] Sato K., Ueno T., “Multi-dimensional diffusion and the Markov process on the boundary”, J. Math. Kyoto Univ., 4 (1965), 529–605 | MR | Zbl

[98] Schechter M., “Nonlocal elliptic boundary value problems”, Ann. Scuola Norm. Sup. Pisa (3), 20:2 (1966), 421–441 | MR | Zbl

[99] Shamin R. V., “Nonlocal parabolic problems with the support of nonlocal terms inside a domain”, Funct. Differ. Equ., 10:1–2 (2003), 307–314 | MR | Zbl

[100] Skubachevskii A. L., “On the stability of index of nonlocal elliptic problems”, J. Math. Anal. Appl., 160:2 (1991), 323–341 | DOI | MR | Zbl

[101] Skubachevskii A. L., “Nonlocal elliptic problems and multidimensional diffusion processes”, Russian J. Mathematical Physics, 3 (1995), 327–360 | MR

[102] Skubachevskii A. L., Elliptic functional differential equations and applications, Birkhäuser, Basel–Boston–Berlin, 1997 | MR | Zbl

[103] Skubachevskii A. L., “Regularity of solutions for some nonlocal elliptic problem”, Russ. J. Math. Phys., 8 (2001), 365–374 | MR | Zbl

[104] Sommerfeld A., “Ein Beitrag zur hydrodinamischen Erklärung der turbulenten Flussigkeitsbewegungen”, Proc. Intern. Congr., Reale Accad. Lincei, Math Roma, 1909, 116–124

[105] Taira K., Diffusion processes and partial differential equations, Academic Press, New York–London, 1988 | MR | Zbl

[106] Taira K., “On the exiestense of Feller semigroups with boundary conditions”, Mem. Amer. Math. Soc., 99:475 (1992), 1–65 | MR

[107] Taira K., Semigroups, boundary value problems and Markov processes, Springer-Verlag, Berlin, 2004 | MR | Zbl

[108] Watanabe S., “Construction of diffusion processes with Wentzell's boundary conditions by means of Poisson point processes of Brownian excursions”, Banach Center Publ., 5 (1979), 255–271 | MR | Zbl