各项异性椭圆方程基本解的存在性

The existence of fundamental solution for anisotropic elliptic equation

证明了右端可测的各项异性椭圆方程基本解的存在性,其中应用了各项异性Sobolev空间和Lebesgue空间.首先得到近似方程的解,然后通过对这些解的子列取极限,得到原方程的解.关键是要有一个近似函数空间以及近似方程的先验估计.最后运用Vitali定理证明了原方程基本解的存在性,推广和改进了已有方程.

This paper proves the existence of fundamental solution for anisotropic equation-Δ-pu - Δ-qu = 0with measure valued right hand-side.Anisotropic Sobolev space and weak Lebesgue space are involved in thefunctional setting.First,we get the solutions of approximate equations.Then by taking limit of the sequence ofthese solutions,we get the solution of the original equation.The point is to have a approximate function space anddo a priori estimate for the approximate equations.Final,the Vitali’s theorem is applied for the existence of fundamentalsolution for anisotropic equation-Δ-pu-Δ-qu = 0.Extend and improve the existing equation-Δp-u = 0.