Degenerate linear parabolic equations in divergence form on the upper half space

Abstract:

We study a class of second-order degenerate linear parabolic equations in divergence form in $(-\infty , T) \times {\mathbb {R}}^d_+$ with homogeneous Dirichlet boundary condition on $(-\infty , T) \times \partial {\mathbb {R}}^d_+$, where ${\mathbb {R}}^d_+ = \{x \in {\mathbb {R}}^d: x_d>0\}$ and $T\in {(-\infty , \infty ]}$ is given. The coefficient matrices of the equations are the product of $\mu (x_d)$ and bounded uniformly elliptic matrices, where $\mu (x_d)$ behaves like $x_d^\alpha$ for some given $\alpha \in (0,2)$, which are degenerate on the boundary $\{x_d=0\}$ of the domain. Our main motivation comes from the analysis of degenerate viscous Hamilton-Jacobi equations. Under a partially VMO assumption on the coefficients, we obtain the well-posedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems.