We establish the interior and boundary Hölder continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is${\partial }_t(|u{|}^{p-2}u)-{\mathrm{\Delta }}_{p}u=0,p>1.$ The proof relies on the property of expansion of positivity and the method of intrinsic scaling, all of which are realized by De Giorgi's iteration. Our approach, while emphasizing the distinct roles of sub(super)-solutions, is flexible enough to obtain the Hölder regularity of solutions to initial-boundary value problems of Dirichlet type or Neumann type in a cylindrical domain, up to the parabolic boundary. In addition, based on the expansion of positivity, we are able to give an alternative proof of Harnack's inequality for non-negative solutions. Moreover, as a consequence of the interior estimates, we also obtain a Liouville-type result.

我们建立了一类双非线性抛物方程的可能符号变化解的内部和边界 Hölder 连续性，其原型为${\partial }_{吨}(|你{|}^{磷-2}你)-{\mathrm{\Delta }}_{磷}你=0,磷>1.$证明依赖于正性扩展的性质和内在缩放的方法，所有这些都是通过德乔治的迭代实现的。我们的方法在强调子（超）解的不同作用的同时，足够灵活，可以在圆柱域中获得狄利克雷型或诺依曼型初边界值问题的 Hölder 正则性，直到抛物线边界。此外，基于正性的扩展，我们能够给出非负解的 Harnack 不等式的另一种证明。此外，作为内部估计的结果，我们还获得了 Liouville 类型的结果。