A Hölder estimate for non-uniform elliptic equations in a random medium. (January 2017) - Wang, Shiah-Sen; Yeh, Li-Ming
Abstract: Uniform regularity for second order elliptic equations in a highly heterogeneous random medium is concerned. The medium is separated by a random ensemble of simply closed interfaces into a connected sub-region with high conductivity and a disconnected subset with low conductivity. The elliptic equations, whose diffusion coefficients depend on the conductivity, have fast diffusion in the connected sub-region and slow diffusion in the disconnected subset. Without a stationary–ergodic assumption, a uniform Hölder estimate in ω, ϵ, λ for the elliptic solutions is derived, where ω is a realization of the random ensemble, ϵ ∈ ( 0, 1 ] is the length scale of the interfaces, and λ 2 ∈ ( 0, 1 ] is the conductivity ratio of the disconnected subset to the connected sub-region. Results show that if external sources are small enough in the disconnected subset, the uniform Hölder estimate in ω, ϵ, λ holds in the whole domain. If not, it holds only in the connected sub-region. Meanwhile, the elliptic solutions change rapidly in the disconnected subset.