Boundedly nonhomogeneous elliptic and parabolic equations

N. V. Krylov

Izvestiya. Mathematics , Tome 20 (1983) no. 3, pp. 459-492.

Résumé

This paper considers elliptic equations of the form \begin{equation*} 0=F(u_{x^ix^j},u_{x^i},u,1,x) \tag{

Bibliographie

[1] Krylov N. V., “Ogranichenno neodnorodnye ellipticheskie i parabolicheskie uravneniya na ploskosti”, Matem. sbornik, 82:1 (1970), 99–110 | MR | Zbl

[2] Krylov N. V., “Ob upravlenii resheniem stokhasticheskogo integralnogo uravneniya pri nalichii vyrozhdeniya”, Izvestiya AN SSSR. Ser. matem., 36:1 (1972), 248–261 | MR | Zbl

[3] Krylov N. V., “On the control of diffusion type processes”, Proc. Second. Japan–USSR Symp. on Probab Theory, Kyoto, 1972, 126–137

[4] Krylov N. V., Upravlyaemye protsessy diffuzionnogo tipa, Nauka, M., 1977 | MR

[5] Lions P. L., “Contrôle de diffusions dans $R^N$”, Comptes Rendus Acad. Sci.(A–B), 288 (1979), 339–342 | MR | Zbl

[6] Krylov N. V., “Nekotorye novye rezultaty iz teorii upravlyaemykh diffuzionnykh protsessov”, Matem. sbornik, 109:1 (1979), 146–164 | MR | Zbl

[7] Safonov M. V., “On the control of diffusion processes in a multidimensional cylindrical domain”, Intern. Symp. Stoch. Diff. Eq. (1978, Vilnius, Abstr. Comm., In-t matem. i kibern. Lit. AN), 1978, 168–172

[8] Safonov M. V., “O zadache Dirikhle dlya uravneniya Bellmana v mnogomernoi oblasti”, Dokl. AN SSSR, 253:3 (1980), 535–540 | MR | Zbl

[9] Lions P. L., “Equations de Hamilton–Jacobi–Bellman dégénérées”, Comptes Rendus Acad. Sci. (A), 289 (1979), 329–332 | MR | Zbl

[10] Krylov N. V., “Ob upravlyaemykh diffuzionnykh protsessakh s neogranichennymi koeffitsientami”, Izvestiya AN SSSR. Ser. matem., 45:4 (1981), 734–759 | MR | Zbl

[11] Krylov N. V., “Ob upravlenii diffuzionnym protsessom do momenta pervogo vykhoda iz oblasti”, Izvestiya AN SSSR. Ser. matem., 45:5 (1981), 1029–1048 | MR | Zbl

[12] Brezis H., Evans L. C., “A variational inequality approach to the Bellman–Direchlet equation for two elliptic operators”, Arch. Rat. Mech. and Anal., 71:1 (1979), 1–13 | DOI | MR | Zbl

[13] Evans L. C., Friedman A., “Optimal stochastic switching and the Dirichlet problem for the Bellman equation”, Trans. Amer. Math. Soc., 253 (1979), 365–389 | DOI | MR | Zbl

[14] Evans L. C., Second derivative estimates for the Bellman equation, preprint

[15] Krylov N. V., “Zadacha s dvumya svobodnymi granitsami dlya ellipticheskogo uravneniya i optimalnaya ostanovka markovskogo protsessa”, Dokl. AN SSSR, 194:6 (1970), 1263–1265 | Zbl

[16] Krylov N. V., Safonov M. V., “Nekotoroe svoistvo reshenii parabolicheskikh uravnenii s izmerimymi koeffitsientami”, Izvestiya AN SSSR. Ser. matem., 44:1 (1980), 161–175 | MR | Zbl

[17] Evans L. C., Classical solutions of fully nonlinear convex second order elliptic equations, preprint | MR

[18] Evans L. C., Classical solutions of the Hamilton–Jacobi–Bellman equaton for uniformly elliptic operators, preprint | MR

[19] Ivanov A. V., “Apriornye otsenki dlya reshenii nelineinykh ellipticheskikh uravnenii vtorogo poryadka”, Zap. nauchn. seminarov Leningr. otd. Matem. in-ta AN SSSR, 59, 1976, 31–59 | Zbl

[20] Schulz F., “Innere Abschätzungen für Lösungen nicht-linearer elliptischer Differentialgleichungen zweiter Ordnung in $n$ Variablen”, Math. Z., 169 (1979), 13–22 | DOI | MR | Zbl

[21] Pogorelov A. V., Mnogomernaya problema Minkovskogo, Nauka, M., 1975 | MR | Zbl

[22] Cheng S. Y., Yau S. T., “On the regularity of the Monge–Ampère equation det $(\partial^2u/\partial x_i\partial x_j)=F(x,u)$”, Comm. Pure and Appl. Math., 30:1 (1977), 41–68 | DOI | MR | Zbl

[23] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[24] Fridman A., Uravneniya s chastnymi proizvodnymi parabolicheskogo tipa, Mir, M., 1968

[25] Agmon S., Duglis A., Nirenberg L., Otsenki vblizi granitsy reshenii ellipticheskikh uravnenii v chastnykh proizvodnykh pri obschikh granichnykh usloviyakh. I, IL, M., 1962

[26] Krylov N. V., “O predelnom perekhode v vyrozhdennykh uravneniyakh Bellmana. II”, Matem. sbornik, 107:1 (1978), 56–68 | MR | Zbl

[27] Krylov N. V., “Neskolko otsenok iz teorii stokhasticheskogo integrala”, Teoriya veroyatn. i ee primenen., 18:1 (1973), 56–65 | MR | Zbl

[28] Krylov N. V., “O edinstvennosti resheniya uravneniya Bellmana”, Izvestiya AN SSSR. Ser. matem., 35:6 (1971), 1377–1388 | MR | Zbl

[29] Ivanov A. V., “Apriornye otsenki proizvodnykh reshenii nelineinykh uravnenii vtorogo poryadka na granitse oblasti”, Zap. nauchn. seminarov Leningr. otd. Matem. in-ta AN SSSR, 69, 1977, 65–76 | Zbl