Second-order elliptic and parabolic equations with $B(\mathbb {R}^{2}, VMO)$ coefficients
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- by Hongjie Dong and N. V. Krylov PDF
- Trans. Amer. Math. Soc. 362 (2010), 6477-6494 Request permission
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
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Additional Information
- Hongjie Dong
- Affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912
- MR Author ID: 761067
- ORCID: 0000-0003-2258-3537
- Email: Hongjie_Dong@brown.edu
- N. V. Krylov
- Affiliation: Department of Mathematics, 127 Vincent Hall, University of Minnesota, Minneapolis, Minnesota 55455
- MR Author ID: 189683
- Email: krylov@math.umn.edu
- 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 - © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2678983