A New Unified Method for Boundary Hölder Continuity of Parabolic Equations

In a recent work due to Lian et al., the authors proposed new geometric conditions to study the pointwise boundary Hölder regularity for kinds of elliptic equations. Here, we aim at developing a unified method to discuss the relation between the boundary geometric properties and the boundary regularity for parabolic equations. The method built in the present paper is applicable for many types of parabolic equations, we only give detailed proofs for the linear equations, the p-Laplace equations, and the fractional Laplace equations. The geometric conditions required in this paper are weak. When we consider the boundary Hölder continuity for the equation \(u_t-\Delta u=0\) $u_t-\mathrm{\Delta }u=0$ , the boundary condition in our theorem is different from the positive geometric density condition required in the previous literature, and it exhibits that the measure of the complement of the domain near the boundary point concerned can be zero. The key idea in proving our results is to sufficiently investigate the information provided by the geometric conditions and transfer them into the desired oscillation estimates. Meanwhile, we also need to construct proper upper solutions and choose cylinders according to the regularity operators.