The Muskat problem with a large slope

In this paper, we establish local well-posedness results for the Muskat equation in any dimension using modulus of continuity techniques. By introducing a novel quantity $\beta_\sigma(f_0')$ which encapsulates local monotonicity and slope, we identify a new class of initial data within $W^{1,\infty}(\mathbb{R}^d)$. This includes scenarios where the product of the maximal and minimal slopes is large, thereby guaranteeing the local existence of a classical solution.