We show that weak solutions to conormal derivative problem for elliptic equations in divergence form are continuously differentiable up to the boundary, provided that the mean oscillations of the leading coefficients satisfy the Dini condition, the lower order coefficients satisfy certain suitable conditions, and the boundary is locally represented by a C1 function whose derivatives are Dini continuous. We also prove that strong solutions to oblique derivative problem for elliptic equations in nondivergence form are twice continuously differentiable up to the boundary if the mean oscillations of coefficients satisfy the Dini condition and the boundary is locally represented by a C1 function whose derivatives are double Dini continuous. This in particular extends a result of M.V. Safonov (Comm. Partial Differential Equations 20:1349–1367, 1995).
|Number of pages||39|
|Journal||Indiana University Mathematics Journal|
|Publication status||Published - 2020|
Bibliographical noteFunding Information:
The first author was partially supported by the National Science Foundation (agreement DMS-1600593). The third author is partially supported by the NRF (grant nos. NRF-2016R1D1A1B03931680 and NRF-20151009350).
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