论文标题
在具有单数或退化系数的抛物线和椭圆方程上
On parabolic and elliptic equations with singular or degenerate coefficients
论文作者
论文摘要
我们在半空间中研究差异和非差异形成抛物线和椭圆方程式，其系数为$ x_d^α$的产物，并且均匀地限制了无限制的可测量的矩阵可得出的功能，其中$α\ in（$α\ in（-1，\ infty）$。因此，系数在半空间边界附近是单数或退化。对于具有综合或Neumann边界条件的方程式，我们证明了加权Sobolev空间中解决方案的存在，独特性和规律性，而当系数仅在$ X_D $方向上可测量并且在小型圆柱体中其他平均振动中的较小振荡时，在小型calinders中的系数仅可测量。即使在系数为常数的特殊情况下，我们的结果也是新的，当$α= 0 $时，它们将减少为经典结果。
We study both divergence and non-divergence form parabolic and elliptic equations in the half space $\{x_d>0\}$ whose coefficients are the product of $x_d^α$ and uniformly nondegenerate bounded measurable matrix-valued functions, where $α\in (-1, \infty)$. As such, the coefficients are singular or degenerate near the boundary of the half space. For equations with the conormal or Neumann boundary condition, we prove the existence, uniqueness, and regularity of solutions in weighted Sobolev spaces and mixed-norm weighted Sobolev spaces when the coefficients are only measurable in the $x_d$ direction and have small mean oscillation in the other directions in small cylinders. Our results are new even in the special case when the coefficients are constants, and they are reduced to the classical results when $α=0$
