×
Dong, Hongjie; Krylov, N. V.; Li, Xu

On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains. (English. Russian original) Zbl 1278.35133

St. Petersbg. Math. J. 24, No. 1, 39-69 (2013); translation from Algebra Anal. 24, No. 1, 53-94 (2012).
The paper is devoted to fully nonlinear elliptic and parabolic equations with vanishing mean oscillation coefficients in bounded domains or cylinders. The systems under consideration particularly cover parabolic Bellman’s equations. The solvability in the Sobolev spaces of the terminal-boundary value problem is proved. The solvability in \(W^{1,2}_p\), \(p>d+1\), of the corresponding elliptic boundary value problem is also obtained. The proofs of main propositions of the paper are based on the approach developed in works by the second author [Commun. Partial Differ. Equations 32, No. 3, 453–475 (2007; Zbl 1114.35079); J. Funct. Anal. 250, No. 2, 521–558 (2007; Zbl 1133.35052)].
Reviewer: Vyacheslav I. Maksimov (Ekaterinburg)

Cited in 20 Documents

MSC:

35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
35R05 PDEs with low regular coefficients and/or low regular data

Keywords:

vanishing mean oscillation; Bellman’s equations

Citations:

Zbl 1114.35079; Zbl 1133.35052
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Marco Bramanti and M. Cristina Cerutti, \?_{\?}^{1,2} solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients, Comm. Partial Differential Equations 18 (1993), no. 9-10, 1735 – 1763. · Zbl 0816.35045 · doi:10.1080/03605309308820991
[2] Luis A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2) 130 (1989), no. 1, 189 – 213. · Zbl 0692.35017 · doi:10.2307/1971480
[3] Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. · Zbl 0834.35002
[4] Filippo Chiarenza, Michele Frasca, and Placido Longo, Interior \?^{2,\?} estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat. 40 (1991), no. 1, 149 – 168. · Zbl 0772.35017
[5] Filippo Chiarenza, Michele Frasca, and Placido Longo, \?^{2,\?}-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336 (1993), no. 2, 841 – 853. · Zbl 0818.35023
[6] M. G. Crandall, M. Kocan, and A. Świȩch, \?^{\?}-theory for fully nonlinear uniformly parabolic equations, Comm. Partial Differential Equations 25 (2000), no. 11-12, 1997 – 2053. · Zbl 0973.35097 · doi:10.1080/03605300008821576
[7] Hongjie Dong and Doyoon Kim, On the \?_{\?}-solvability of higher order parabolic and elliptic systems with BMO coefficients, Arch. Ration. Mech. Anal. 199 (2011), no. 3, 889 – 941. · Zbl 1228.35113 · doi:10.1007/s00205-010-0345-3
[8] Luis Escauriaza, \?^{2,\?} a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J. 42 (1993), no. 2, 413 – 423. · Zbl 0792.35020 · doi:10.1512/iumj.1993.42.42019
[9] Doyoon Kim and N. V. Krylov, Elliptic differential equations with coefficients measurable with respect to one variable and VMO with respect to the others, SIAM J. Math. Anal. 39 (2007), no. 2, 489 – 506. · Zbl 1138.35308 · doi:10.1137/050646913
[10] Doyoon Kim and N. V. Krylov, Parabolic equations with measurable coefficients, Potential Anal. 26 (2007), no. 4, 345 – 361. · Zbl 1124.35024 · doi:10.1007/s11118-007-9042-8
[11] Нелинейные ѐллиптические и параболические уравнения второго порядка, ”Наука”, Мосцощ, 1985 (Руссиан). Н. В. Крылов, Нонлинеар еллиптиц анд параболиц ечуатионс оф тхе сецонд ордер, Матхематицс анд иц Апплицатионс (Совиет Сериес), вол. 7, Д. Реидел Публишинг Цо., Дордречт, 1987. Транслатед фром тхе Руссиан бы П. Л. Бузыцкы [П. Л. Бузыцкий]. · Zbl 0586.35002
[12] N. V. Krylov, Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differential Equations 32 (2007), no. 1-3, 453 – 475. · Zbl 1114.35079 · doi:10.1080/03605300600781626
[13] N. V. Krylov, Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms, J. Funct. Anal. 250 (2007), no. 2, 521 – 558. · Zbl 1133.35052 · doi:10.1016/j.jfa.2007.04.003
[14] N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, Graduate Studies in Mathematics, vol. 96, American Mathematical Society, Providence, RI, 2008. · Zbl 1147.35001
[15] Nicolai V. Krylov, On Bellman’s equations with VMO coefficients, Methods Appl. Anal. 17 (2010), no. 1, 105 – 121. · Zbl 1208.35052 · doi:10.4310/MAA.2010.v17.n1.a4
[16] N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 161 – 175, 239 (Russian).
[17] Линейные и квазилинейные уравнения ѐллиптического типа, Издат. ”Наука”, Мосцощ, 1964 (Руссиан). Олга А. Ладыженская анд Нина Н. Урал’цева, Линеар анд чуасилинеар еллиптиц ечуатионс, Транслатед фром тхе Руссиан бы Сцрипта Течница, Инц. Транслатион едитор: Леон Ехренпреис, Ацадемиц Пресс, Нещ Ыорк-Лондон, 1968.
[18] Линейные и квазилинейные уравнения параболического типа, Издат. ”Наука”, Мосцощ, 1967 (Руссиан). О. А. Ладыžенскаја, В. А. Солонников, анд Н. Н. Урал\(^{\приме}\)цева, Линеар анд чуасилинеар ечуатионс оф параболиц тыпе, Транслатед фром тхе Руссиан бы С. Смитх. Транслатионс оф Матхематицал Монограпхс, Вол. 23, Америцан Матхематицал Социеты, Провиденце, Р.И., 1968 (Руссиан). О. А. Ладыžенскаја, В. А. Солонников, анд Н. Н. Урал\(^{\приме}\)цева, Линейные и квазилинейные уравнения параболического типа, Издат. ”Наука”, Мосцощ, 1967 (Руссиан). О. А. Ладыžенскаја, В. А. Солонников, анд Н. Н. Урал\(^{\приме}\)цева, Линеар анд чуасилинеар ечуатионс оф параболиц тыпе, Транслатед фром тхе Руссиан бы С. Смитх. Транслатионс оф Матхематицал Монограпхс, Вол. 23, Америцан Матхематицал Социеты, Провиденце, Р.И., 1968 (Руссиан).
[19] Gary M. Lieberman, Regularized distance and its applications, Pacific J. Math. 117 (1985), no. 2, 329 – 352. · Zbl 0535.35028
[20] Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. · Zbl 0884.35001
[21] Fang-Hua Lin, Second derivative \?^{\?}-estimates for elliptic equations of nondivergent type, Proc. Amer. Math. Soc. 96 (1986), no. 3, 447 – 451. · Zbl 0599.35039
[22] Antonino Maugeri, Dian K. Palagachev, and Lubomira G. Softova, Elliptic and parabolic equations with discontinuous coefficients, Mathematical Research, vol. 109, Wiley-VCH Verlag Berlin GmbH, Berlin, 2000. · Zbl 0958.35002
[23] M. V. Safonov, Harnack’s inequality for elliptic equations and Hölder property of their solutions, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 96 (1980), 272 – 287, 312 (Russian). Boundary value problems of mathematical physics and related questions in the theory of functions, 12. · Zbl 0458.35028
[24] -, Nonlinear elliptic equations of second order, Lecture Notes Dip. Mat. Appl. “G. Sansone”, Univ. Degli Studi Firenze, 1991.
[25] -, On the boundary value problems for fully nonlinear elliptic equations of second order, Math. Res. Report No. MRR 049-94, Austral. Nat. Univ., Canberra, 1994. ”http://www.math.umn.edu/ safonov/NOTES/PDE_94/PDE.pdf”
[26] Neil S. Trudinger, Fully nonlinear, uniformly elliptic equations under natural structure conditions, Trans. Amer. Math. Soc. 278 (1983), no. 2, 751 – 769. · Zbl 0518.35036
[27] Lihe Wang, On the regularity theory of fully nonlinear parabolic equations, Bull. Amer. Math. Soc. (N.S.) 22 (1990), no. 1, 107 – 114. · Zbl 0704.35025
[28] Lihe Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math. 45 (1992), no. 1, 27 – 76. · Zbl 0832.35025 · doi:10.1002/cpa.3160450103
[29] Niki Winter, \?^{2,\?} and \?^{1,\?}-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend. 28 (2009), no. 2, 129 – 164. · Zbl 1206.35116 · doi:10.4171/ZAA/1377
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.