On conormal and oblique derivative problem for elliptic equations with dini mean oscillation coefficients

Hongjie Dong; Jihoon Lee; Seick Kim

doi:10.1512/iumj.2020.69.8028

On conormal and oblique derivative problem for elliptic equations with dini mean oscillation coefficients

Hongjie Dong, Jihoon Lee, Seick Kim

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

Abstract

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).

Original language	English
Pages (from-to)	1815-1853
Number of pages	39
Journal	Indiana University Mathematics Journal
Volume	69
Issue number	6
DOIs	https://doi.org/10.1512/iumj.2020.69.8028
Publication status	Published - 2020

Bibliographical note

Funding 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).

Publisher Copyright:
© 2020 Department of Mathematics, Indiana University. All rights reserved.

All Science Journal Classification (ASJC) codes

Mathematics(all)

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10.1512/iumj.2020.69.8028

Cite this

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abstract = "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).",

author = "Hongjie Dong and Jihoon Lee and Seick Kim",

note = "Funding 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). Publisher Copyright: {\textcopyright} 2020 Department of Mathematics, Indiana University. All rights reserved.",

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AU - Dong, Hongjie

AU - Lee, Jihoon

AU - Kim, Seick

N1 - Funding 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). Publisher Copyright: © 2020 Department of Mathematics, Indiana University. All rights reserved.

PY - 2020

Y1 - 2020

N2 - 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).

AB - 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).

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On conormal and oblique derivative problem for elliptic equations with dini mean oscillation coefficients

Abstract

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