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

16 Jul 2021  ·  Hongjie Dong, Tuoc Phan, Hung Vinh Tran ·

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... Under a partially VMO assumption on the coefficients, we obtain the wellposedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems. read more

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Analysis of PDEs