含各向異性與周期性纖維的複合介質在介面附近的先驗均勻估計

Abstract

Uniform estimates for the solutions of non-uniform elliptic equations around the interfaces are concerned. The equations describe the behavior of current (heat, velocity, etc.) in an heterogeneous medium, which consisting of anisotropic fibres embedded into an isotropic matrix. It is a ’double porosity’-type medium (see Figure 3) and, inside the fibres, there is a large contrast between the conductivity along the fibres and the conductivities in the transverse directions. Let ϵ be the conductivity ratio of the ’fibres’ to the ’matrix’ in the transverse directions. We derive a priori uniform Hölder estimate as well as uniform gradient Hölder estimate in ϵ. However, the coefficients in the estimates depend on ε^(-1). To characterize the role of ε^(-1), we precisely write down the power of ε^(-1) in the estimates.

我們關心非均勻橢圓方程在介面附近的解的均勻估計。這個方程式描述了 在異質媒介中電流 (熱、速度等) 的行為。其中，異質媒介是具各向異性的傳導 纖維嵌入的各向同性的方陣。詳細的說，我們考慮的是嵌入了具我們所關心的 各向異性傳導纖維 (通過介面方向的傳導系數與在沿纖維方向的傳導系數的比值 小於 1) 的雙面多孔類型區域 (見 Figure 3)。在通過介面的方向上，纖維對於方 陣的傳導系數比值記作 ϵ ∈ (0,1)。我們得到在介面附近的解的先驗均勻 Hölder 估計，還有在介面附近的梯度 Hölder 估計。但是在估計中的係數與 ε^(-1) 有關。 為了凸顯出 ϵ −1 的作用，我們精確的寫出在 Hölder 估計中 ε^(-1) 的次方數。