Singular-degenerate parabolic systems with the conormal boundary condition on the upper half space

Abstract

We prove the well-posedness and regularity of solutions in mixed-norm weighted Sobolev spaces for a class of second-order parabolic and elliptic systems in divergence form in the half-space $\mathbb{R}^d_+ = \{x_d > 0\}$ subject to the conormal boundary condition. Our work extends results previously available for scalar equations to the case of systems of equations. The leading coefficients are the product of $x_d^{\alpha}$ and bounded non-degenerate matrices, where $\alpha \in (-1,\infty)$. The leading coefficients are assumed to be merely measurable in the $x_d$ variable, and to have small mean oscillations in small cylinders with respect to the other variables. If the parameter $\alpha>0$, the lower-order coefficients are allowed to blow-up near the boundary. Our results readily generalize to infinite-dimensional equations in general real and complex Hilbert spaces.