Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Parabolic equations with variably partially VMO coefficients


Author: H. Dong
Original publication: Algebra i Analiz, tom 23 (2011), nomer 3.
Journal: St. Petersburg Math. J. 23 (2012), 521-539
MSC (2010): Primary 35K15, 35R05
DOI: https://doi.org/10.1090/S1061-0022-2012-01206-9
Published electronically: March 2, 2012
MathSciNet review: 2896169
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The $ W^{1,2}_{p}$-solvability of second-order parabolic equations in nondivergence form in the whole space is proved for $ p\in (1,\infty )$. The leading coefficients are assumed to be measurable in one spatial direction and have vanishing mean oscillation (VMO) in the orthogonal directions and the time variable in each small parabolic cylinder with direction allowed to depend on the cylinder. This extends a recent result by Krylov for elliptic equations. The novelty in the current paper is that the restriction $ p>2$ is removed.


References [Enhancements On Off] (What's this?)

  • 1. E. Acerbi and G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J. 136 (2007), no. 2, 285-320. MR 2286632 (2007k:35211)
  • 2. S. Byun and L. Wang, $ L^p$-estimates for general nonlinear elliptic equations, Indiana Univ. Math. J. 56 (2007), no. 6, 3193-3221. MR 2375715 (2008m:35131)
  • 3. S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623-727; II , ibid., 17 (1964), 35-92. MR 0125307 (23:A2610); MR 0162050 (28:5252)
  • 4. M. Bramanti and M. Cerutti, $ W_p^{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. MR 1239929 (94j:35180)
  • 5. F. Chiarenza, M. Frasca, and P. Longo, Interior $ W^{2,p}$ estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat. 40 (1991), no. 1, 149-168. MR 1191890 (93k:35051)
  • 6. -, $ W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336 (1993), no. 2, 841-853. MR 1088476 (93f:35232)
  • 7. H. Dong, Solvability of parabolic equations in divergence form with partially BMO coefficients, J. Funct. Anal. 258 (2010), no. 7, 2145-2172. MR 2584743 (2011b:35187)
  • 8. H. Dong and D. Kim, Elliptic equations in divergence form with partially BMO coefficients, Arch. Rational Mech. Anal. 196 (2010), no. 1, 25-70. MR 2601069 (2011b:35086)
  • 9. H. Dong and N. V. Krylov, Second-order elliptic and parabolic equations with $ B(\mathbb{R}^{2}, VMO)$ coefficients, Trans. Amer. Math. Soc. 362 (2010), no. 12, 6477-6494. MR 2678983
  • 10. G. Di Fazio, $ L^p$ estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Un. Mat. Ital. A (7) 10 (1996), no. 2, 409-420. MR 1405255 (97e:35034)
  • 11. D. Kim, Parabolic equations with measurable coefficients. II, J. Math. Anal. Appl. 334 (2007), no. 1, 534-548. MR 2332574 (2008f:35160)
  • 12. -, Elliptic and parabolic equations with measurable coefficients in $ L_p$-spaces with mixed norms, Methods Appl. Anal. 15 (2008), no. 4, 437-467. MR 2550072 (2010j:35566)
  • 13. -, Parabolic equations with partially BMO coefficients and boundary value problems in Sobolev spaces with mixed norms, Potential Anal. 33 (2010), no. 1, 17-46. MR 2644213 (2011h:35120)
  • 14. D. 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. MR 2338417 (2008j:35031)
  • 15. -, Parabolic equations with measurable coefficients, Potential Anal. 26 (2007), no. 4, 345-361. MR 2300337 (2008f:35161)
  • 16. N. V. Krylov, Nonlinear elliptic and parabolic equations of the second order, Nauka, Moscow, 1985; English transl., Math. Appl. (Soviet Series), vol. 7, D. Reidel Publ. Co., Dordrecht, 1987. MR 0815513 (87h:35002); MR 0901759 (88d:35005)
  • 17. -, Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differential Equations 32 (2007), no. 1-3, 453-475. MR 2304157 (2008a:35125)
  • 18. -, Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms, J. Funct. Anal. 250 (2007), no. 2, 521-558. MR 2352490 (2008f:35164)
  • 19. -, Second-order elliptic equations with variably partially VMO coefficients, J. Funct. Anal. 257 (2009), no. 6, 1695-1712. MR 2540989 (2010j:35096)
  • 20. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva, Linear and quasilinear equations of parabolic type, Nauka, Moscow, 1967; English transl., Transl. Math. Monogr., vol. 23, Amer. Math. Soc., Providence, RI, 1967. MR 0241822 (39:3159b)
  • 21. G. Lieberman, Second order parabolic differential equations, World Sci. Publ. Co., Inc., River Edge, NJ, 1996. MR 1465184 (98k:35003)
  • 22. -, A mostly elementary proof of Morrey space estimates for elliptic and parabolic equations with VMO coefficients, J. Funct. Anal. 201 (2003), no. 2, 457-479. MR 1986696 (2004b:35025)
  • 23. L. Softova and P. Weidemaier, Quasilinear parabolic problems in spaces of maximal regularity, J. Nonlinear Convex Anal. 7 (2006), 529-540. MR 2287547 (2008k:35251)
  • 24. P. Weidemaier, Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $ L_p$-norm, Electron. Res. Announc. Amer. Math. Soc. 8 (2002), 47-51. MR 1945779 (2003k:35088)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 35K15, 35R05

Retrieve articles in all journals with MSC (2010): 35K15, 35R05


Additional Information

H. Dong
Affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912
Email: Hongjie_Dong@brown.edu

DOI: https://doi.org/10.1090/S1061-0022-2012-01206-9
Keywords: Second-order equations, vanishing mean oscillation, partially VMO coefficients, Sobolev spaces
Received by editor(s): June 20, 2010
Published electronically: March 2, 2012
Additional Notes: Partially supported by NSF Grant DMS-0635607 from IAS and NSF Grant DMS-0800129
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society