Regularity of boundary points under temporal accessibility conditions

Determine whether boundary points $(x_0,t_0)$ for which $T_0<t_0\leq t$ for every $t\in\Omega_t^{x_0}$ are regular for the evolutionary equation involving the nonlocal infinity Laplacian, and characterize what occurs when $d(t_0,\Omega_t^{x_0})=0$.

Background

The paper studies viscosity solutions of the evolutionary equation $u_t=(\mathcal{L}_{\infty})^{\alpha}u$ and uses Perron solutions and barriers to establish existence, uniqueness, and attainment of boundary data. A boundary point is called regular when the upper and lower Perron solutions converge to the prescribed boundary value at that point.

The paper constructs barriers for bottom and lateral boundary points of cylinders and for certain more general geometric configurations. The unresolved question asks whether regularity can also be proved under a temporal accessibility condition expressed through the set $\Omega_t^{x_0}$, and specifically what happens in the borderline case where the distance from $t_0$ to this set is zero. The authors note that the proposed barrier argument becomes problematic in this case and that suitable local conditions on the domain may be needed.