论文标题
具有单数或退化系数的抛物线和椭圆方程：Dirichlet问题
Parabolic and elliptic equations with singular or degenerate coefficients: the Dirichlet problem
论文作者
论文摘要
我们考虑了上半空间中的一类椭圆形和抛物线方程的问题，$ \ mathbb {r}^d _+$，该系数是$ x_d^α，α\ in（ - \ infty，1），$和一个有界均匀的椭圆形矩阵的产物。因此，系数在边界附近是单数或退化，$ \ {x_d = 0 \} $，它们可能无法局部集成。这项工作的新颖性在于，我们发现了索博莱夫空间中解决方案的存在，独特性和规律性的适当权重。这些结果似乎是同类产品中的第一个，即使系数是恒定的，也是新的。它们也很容易扩展到方程系统。
We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space $\mathbb{R}^d_+$, where the coefficients are the product of $x_d^α, α\in (-\infty, 1),$ and a bounded uniformly elliptic matrix of coefficients. Thus, the coefficients are singular or degenerate near the boundary $\{x_d =0\}$ and they may not locally integrable. The novelty of the work is that we find proper weights under which the existence, uniqueness, and regularity of solutions in Sobolev spaces are established. These results appear to be the first of their kind and are new even if the coefficients are constant. They are also readily extended to systems of equations.
