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Abstract We consider a nonlinear Neumann problem, with periodic oscillation in the elliptic operator and on the boundary condition. Our focus is on problems posed in half-spaces, but with general normal directions that may not be parallel to the directions of periodicity. As the frequency of the oscillation grows, quantitative homogenization results are derived. When the homogenized operator is rotation-invariant, we prove the Hölder continuity of the homogenized boundary data. While we follow the outline of Choi and Kim ( Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data , Journal de Mathématiques Pures et Appliquées 102 (2014), no. 2, 419–448), new challenges arise due to the presence of tangential derivatives on the boundary condition in our problem. In addition, we improve and optimize the rate of convergence within our approach. Our results appear to be new even for the linear oblique problem.
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This content will become publicly available on December 1, 2025
Analysis and approximation of elliptic problems with Uhlenbeck structure in convex polytopes
We prove the well posedness in weighted Sobolev spaces of certain linear and nonlinear elliptic boundary value problems posed on convex domains and under singular forcing. It is assumed that the weights belong to the Muckenhoupt class with ). We also propose and analyze a convergent finite element discretization for the nonlinear elliptic boundary value problems mentioned above. As an instrumental result, we prove that the discretization of certain linear problems are well posed in weighted spaces.
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- Award ID(s):
- 2206252
- PAR ID:
- 10628093
- Publisher / Repository:
- ScienceDirect
- Date Published:
- Journal Name:
- Journal of Differential Equations
- Volume:
- 412
- Issue:
- C
- ISSN:
- 0022-0396
- Page Range / eLocation ID:
- 250 to 271
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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