Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Second-order elliptic and parabolic equations with $ B(\mathbb{R}^{2}, VMO)$ coefficients


Authors: Hongjie Dong and N. V. Krylov
Journal: Trans. Amer. Math. Soc. 362 (2010), 6477-6494
MSC (2000): Primary 35K10, 35K20, 35J15
DOI: https://doi.org/10.1090/S0002-9947-2010-05215-8
Published electronically: August 3, 2010
MathSciNet review: 2678983
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The solvability in Sobolev spaces $ W^{1,2}_p$ is proved for nondivergence form second-order parabolic equations for $ p>2$ close to 2. The leading coefficients are assumed to be measurable in the time variable and two coordinates of space variables, and almost VMO (vanishing mean oscillation) with respect to the other coordinates. This implies the $ W^{2}_p$-solvability for the same $ p$ of nondivergence form elliptic equations with leading coefficients measurable in two coordinates and VMO in the others. Under slightly different assumptions, we also obtain the solvability results when $ p=2$.


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

  • 1. 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)
  • 2. S. Campanato, Un risultato relativo ad equazioni ellittiche del secondo ordine di tipo non variazionale (Italian), Ann. Scuola Norm. Sup. Pisa (3) 21 (1967), 701-707. MR 0224996 (37:595)
  • 3. 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)
  • 4. -, $ 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)
  • 5. G. Chiti, A $ W^{2,2}$ bound for a class of elliptic equations in nondivergence form with rough coefficients, Invent. Math. 33 (1976), no. 1, 55-60. MR 0404848 (53:8648)
  • 6. Hongjie Dong and Doyoon Kim, Parabolic and elliptic systems with VMO coefficients, Methods Appl. Anal. 16 (2009), no. 3, 305-388.
  • 7. Doyoon Kim, Parabolic equations with measurable coefficients II, J. Math. Anal. Appl. 334 (2007), no. 1, 534-548. MR 2332574 (2008f:35160)
  • 8. -, Elliptic and parabolic equations with measurable coefficients in $ L_p$-spaces with mixed norms, Methods Appl. Anal., 15 (2008), no. 4, 437-468. MR 2550072
  • 9. -, Parabolic equations with partially BMO coefficients and boundary value problems in Sobolev spaces with mixed norms, Potential Anal. 33 (2010), no. 1, 17-46.
  • 10. 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. MR 2338417 (2008j:35031)
  • 11. -, Parabolic equations with measurable coefficients, Potential Anal. 26 (2007), no. 4, 345-361. MR 2300337 (2008f:35161)
  • 12. N. V. Krylov, On equations of minimax type in the theory of elliptic and parabolic equations in the plane, Matematicheski Sbornik 81, no. 1 (1970), 3-22 in Russian; English translation in Math. USSR Sbornik 10 (1970), 1-20. MR 0255954 (41:614)
  • 13. -, ``Nonlinear elliptic and parabolic equations of second order'', Nauka, Moscow, 1985, in Russian; English translation, Reidel, Dordrecht, 1987.
  • 14. -, Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differential Equations 32 (2007), no. 1-3, 453-475.MR 2304157 (2008a:35125)
  • 15. -, Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms, J. Funct. Anal. 250 (2007), no. 2, 521-558. MR 2352490 (2008f:35164)
  • 16. -, Second-order elliptic equations with variably partially VMO coefficients, J. Funct. Anal. 257 (2009), no. 6, 1695-1712. MR 2540989
  • 17. -, ``Lectures on elliptic and parabolic equations in Sobolev spaces'', Graduate Studies in Math., Vol. 96, Amer. Math. Soc., Providence, RI, 2008.MR 2435520 (2009k:35001)
  • 18. O. A. Ladyzhenskaya and N. N. Ural'tseva. Linei0nye i kvazilinei0nye uravneniya èllipticheskogo tipa (Russian) [Linear and quasilinear equations of elliptic type] Second edition, revised. Izdat. ``Nauka'', Moscow, 1973. 576 pp. MR 0509265 (58:23009)
  • 19. A. Lorenzi, On elliptic equations with piecewise constant coefficients, Applicable Anal. 2 (1972), 79-96. MR 0296490 (45:5550)
  • 20. -, On elliptic equations with piecewise constant coefficients. II, Ann. Scuola Norm. Sup. Pisa (3) 26 (1972), 839-870. MR 0382836 (52:3718)
  • 21. G. Lieberman, ``Second order parabolic differential equations'', World Scientific, Singapore-New Jersey-London-Hong Kong, 1996. MR 1465184 (98k:35003)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35K10, 35K20, 35J15

Retrieve articles in all journals with MSC (2000): 35K10, 35K20, 35J15


Additional Information

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

N. V. Krylov
Affiliation: Department of Mathematics, 127 Vincent Hall, University of Minnesota, Minneapolis, Minnesota 55455
Email: krylov@math.umn.edu

DOI: https://doi.org/10.1090/S0002-9947-2010-05215-8
Keywords: Second-order elliptic and parabolic equations, vanishing mean oscillation, VMO coefficients, Sobolev spaces
Received by editor(s): October 15, 2008
Published electronically: August 3, 2010
Additional Notes: The work of the first author was partially supported by NSF Grant DMS-0635607 from IAS and NSF Grant DMS-0800129.
The work of the second author was partially supported by NSF Grant DMS-0653121
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society