Maximum principles for boundary-degenerate linear parabolic differential operators

We develop weak and strong maximum principles for boundary-degenerate, linear, parabolic, second-order partial differential operators, $Lu := -u_t-\tr(aD^2u)-\langle b, Du\rangle + cu$, with \emph{partial} Dirichlet boundary conditions. The coefficient, $a(t,x)$, is assumed to vanish along a non-empty open subset, ${\text{mydirac}}_0!\text{sQ}$, called the \emph{degenerate boundary portion}, of the parabolic boundary, $\text{mydirac}!\text{sQ}$, of the domain $\text{sQ}\subset {\text{RR}}^{d+1}$, while $a(t,x)$ may be non-zero at points in the \emph{non-degenerate boundary portion}, $\mydirac_1!\sQ := \mydirac!\sQ\less\bar{\mydirac_0!\sQ}$\mathrm{.}Pointsin$\mydirac_0!\sQ$ play the same role as those in the interior of the domain, $\text{sQ}$, and only the non-degenerate boundary portion, ${\text{mydirac}}_1!\text{sQ}$, is required for boundary comparisons. We also develop comparison principles and a priori maximum principle estimates for solutions to boundary value and obstacle problems defined by boundary-degenerate parabolic operators, again where only the non-degenerate boundary portion, ${\text{mydirac}}_1!\text{sQ}$, is required for boundary comparisons. Our results complement those in our previous articles [arXiv1204.6613, arXiv:1305.5098].