Abstract
We obtain conditions for the differentiability of weak solutions for a second-order uniformly elliptic equation in divergence form with a homogeneous co-normal boundary condition. The modulus of continuity for the coefficients is assumed to satisfy the square-Dini condition and the boundary is assumed to be differentiable with derivatives also having this modulus of continuity. Additional conditions for the solution to be Lipschitz continuous or differentiable at a point on the boundary depend upon the stability of a dynamical system that is derived from the coefficients of the elliptic equation.
Keywords
- Differentiability
- Lipschitz continuity
- weak solution
- elliptic equation
- divergence form
- co-normal boundary condition
- modulus of continuity
- square-Dini condition
- dynamical system
- asymptotically constant
- uniformly stable
Mathematics Subject Classification (2010)
In memory of Victor Havin
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Maz’ya, V., McOwen, R. (2018). Differentiability of Solutions to the Neumann Problem with Low-regularity Data via Dynamical Systems. In: Baranov, A., Kisliakov, S., Nikolski, N. (eds) 50 Years with Hardy Spaces. Operator Theory: Advances and Applications, vol 261. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-59078-3_18
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DOI: https://doi.org/10.1007/978-3-319-59078-3_18
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-59077-6
Online ISBN: 978-3-319-59078-3
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