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Conormal problem of higher-order parabolic systems with time irregular coefficients


Authors: Hongjie Dong and Hong Zhang
Journal: Trans. Amer. Math. Soc. 368 (2016), 7413-7460
MSC (2010): Primary 35K52, 35J58, 35B45, 35R05
DOI: https://doi.org/10.1090/tran/6605
Published electronically: November 16, 2015
MathSciNet review: 3471096
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Abstract: The paper is a comprehensive study of $ L_p$ and Schauder estimates for higher-order divergence type parabolic systems with discontinuous coefficients on a half space and cylindrical domains with the conormal derivative boundary conditions. For the $ L_p$ estimates, we assume that the leading coefficients are only bounded and measurable in the $ t$ variable and have vanishing mean oscillations (VMO$ _x$) with respect to $ x$. We also prove the Schauder estimates in two situations: the coefficients are Hölder continuous only in the $ x$ variable; the coefficients are Hölder continuous in the $ t$ variable as well on the lateral boundary.


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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

Hong Zhang
Affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912
Email: Hong_Zhang@brown.edu

DOI: https://doi.org/10.1090/tran/6605
Received by editor(s): January 13, 2014
Received by editor(s) in revised form: September 11, 2014, and October 15, 2014
Published electronically: November 16, 2015
Additional Notes: The first author was partially supported by the NSF under agreement DMS-1056737.
The second author was partially supported by the NSF under agreement DMS-1056737.
Article copyright: © Copyright 2015 American Mathematical Society