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Transactions of the American Mathematical Society

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

 
 

 

Classical solutions of oblique derivative problem in nonsmooth domains with mean Dini coefficients


Authors: Hongjie Dong and Zongyuan Li
Journal: Trans. Amer. Math. Soc. 373 (2020), 4975-4997
MSC (2010): Primary 35J25, 35B65; Secondary 35J15
DOI: https://doi.org/10.1090/tran/8042
Published electronically: March 31, 2020
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Abstract: We consider second-order elliptic equations in nondivergence form with oblique derivative boundary conditions. We show that any strong solutions to such problems are twice continuously differentiable up to the boundary when the boundary can be locally represented by a $ C^1$ function whose first derivatives are Dini continuous and the mean oscillations of coefficients satisfy the Dini condition. This improves a recent result by Dong, Lee, and Kim. To the best of our knowledge, such a result is new even for the Poisson equation. An extension to concave fully nonlinear elliptic equations is also presented.


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Additional Information

Hongjie Dong
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02906
Email: hongjie_dong@brown.edu

Zongyuan Li
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02906
Email: zongyuan_li@brown.edu

DOI: https://doi.org/10.1090/tran/8042
Keywords: Oblique derivative problem, classical solutions, $C^{1,Dini}$ domain, $L_1$-mean Dini condition
Received by editor(s): April 12, 2019
Received by editor(s) in revised form: September 11, 2019, and November 6, 2019
Published electronically: March 31, 2020
Additional Notes: The authors were partially supported by the NSF under agreement DMS-1600593.
Article copyright: © Copyright 2020 American Mathematical Society