Lp gradient estimate for elliptic equations with high-contrast conductivities in Rn
Lp gradient estimate for elliptic equations with high-contrast conductivities in Rn
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Rn 中具有高对比度电导率的椭圆方程的 Lp 梯度估计
DOI：
10.1016/j.jde.2016.03.027
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发表时间：
2016
影响因子：
2.4
通讯作者：
L. Yeh
中科院分区：
文献类型：
--
作者：
L. Yeh
研究了Rn中具有高对比度电导率的椭圆型方程解的一致估计。空间域由一个周期连通子区域和一个周期不连通矩阵块子集组成。椭圆方程在连通子区域扩散快，在不连通子区域扩散慢。设φ ∈（0，1]为每个矩阵块的直径，ω 2∈（0，1]为不连通矩阵块子集与连通子区域的电导率比。证明了连通子区域中椭圆解的W1，p范数在ε，ω中一致有界;当ε≤ ω时，整个空间中椭圆解的Lp范数在ε，ω中一致有界;穿孔域中椭圆解的W1，p范数在ε，ω中一致有界。然而，在连通子区域上解的二阶导数的Lp范数在ω，ω中可能不是一致有界的.
Uniform estimate for the solutions of elliptic equations with high-contrast conductivities in R n is concerned. The space domain consists of a periodic connected sub-region and a periodic disconnected matrix block subset. The elliptic equations have fast diffusion in the connected sub-region and slow diffusion in the disconnected subset. Suppose ϵ∈(0, 1] is the diameter of each matrix block and ω 2∈(0, 1] is the conductivity ratio of the disconnected matrix block subset to the connected sub-region. It is proved that the W 1, p norm of the elliptic solutions in the connected sub-region is bounded uniformly in ϵ, ω; when ϵ≤ ω, the L p norm of the elliptic solutions in the whole space is bounded uniformly in ϵ, ω; the W 1, p norm of the elliptic solutions in perforated domains is bounded uniformly in ϵ. However, the L p norm of the second order derivatives of the solutions in the connected sub-region may not be bounded uniformly in ϵ, ω.