DCDS
Schauder estimates for a class of non-local elliptic equations
Hongjie Dong Doyoon Kim
Discrete & Continuous Dynamical Systems - A 2013, 33(6): 2319-2347 doi: 10.3934/dcds.2013.33.2319
We prove Schauder estimates for a class of non-local elliptic operators with kernel $K(y)=a(y)/|y|^{d+\sigma}$ and either Dini or Hölder continuous data. Here $0 < \sigma < 2$ is a constant and $a$ is a bounded measurable function, which is not necessarily to be homogeneous, regular, or symmetric. As an application, we prove that the operators give isomorphisms between the Lipschitz--Zygmund spaces $\Lambda^{\alpha+\sigma}$ and $\Lambda^\alpha$ for any $\alpha>0$. Several local estimates and an extension to operators with kernels $K(x,y)$ are also discussed.
keywords: Lévy processes. Non-local elliptic equations Schauder estimates
DCDS
Conormal derivative problems for stationary Stokes system in Sobolev spaces
Jongkeun Choi Hongjie Dong Doyoon Kim
Discrete & Continuous Dynamical Systems - A 2018, 38(5): 2349-2374 doi: 10.3934/dcds.2018097

We prove the solvability in Sobolev spaces of the conormal derivative problem for the stationary Stokes system with irregular coefficients on bounded Reifenberg flat domains. The coefficients are assumed to be merely measurable in one direction, which may differ depending on the local coordinate systems, and have small mean oscillations in the other directions. In the course of the proof, we use a local version of the Poincaré inequality on Reifenberg flat domains, the proof of which is of independent interest.

keywords: Stokes system Reifenberg flat domains measurable coefficients conormal derivative boundary condition

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