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Transactions of the American Mathematical Society
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
Published electronically: August 3, 2010
MathSciNet review: 2678983
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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$.


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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: http://dx.doi.org/10.1090/S0002-9947-2010-05215-8
PII: S 0002-9947(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.