I am studying the heat equation on a general bounded domain $\Omega \subset \mathbb{R}^+ \times \mathbb{R}^n$ with continuous Dirichlet data $\phi$ on the boundary,

$$ \left\{ \begin{array}{ccc} \partial_t u - \Delta u = 0 & \text{in } \Omega\\ u = \phi & \text{ on } \partial_n \Omega \end{array} \right. $$

Here $\partial_n \Omega$ denotes the normal boundary of $\Omega$, i.e. the points of the boundary on which one can solve the Dirichlet problem (with the Perron method for instance, see Watson's book Introduction to heat potential theory) for any $\phi \in \mathcal{C}(\partial_n \Omega)$. It essentially consists of points of the boundary $\partial \Omega$ which are not part of the "top" or "cap", more precisely any lower half ball centered at some point $X_0\in \partial_n \Omega$ meets the complementary of $\Omega$.

I want to derive a fine gradient bound on the solution $u$ on this normal boundary $\partial_n \Omega$. I don't seem to find better estimates than the one given in the book of Gerry Lieberman Second Order Parabolic Differential Equations. He finds that for domains $\Omega$ with particular structure conditions, one can obtain,

$$ \sup_{|X-X_0| \neq 0} \frac{u(X) - u(X_0)}{|X-X_0|} \leq C $$

His arguments mainly rely on finding sub and supersolutions in the form $w = \phi + f(d)$ where $d$ denotes the distance to some simple domain $D$ containing $\Omega$ and $f$ an increasing concave real-valued function. These sub/supersolutions agree with $u$ at $X_0$ so one can bound $u$ above and below through the maximum principle. But nothing more precise can be obtained with this method, and I have found nothing in the literature giving more precise gradient bounds.

I have the intuition that for $\mathcal{C}^2$ domains with bounded mean curvature, we can have $u \in \mathcal{C}^1(\overline{\Omega})$, meaning that the total gradient $Du$ has a well defined limit on the boundary of the domain.

I have tried studying the problem satisfied by $\partial_i u$ in $\Omega$ without much success : the boundary data is not well defined, not continuous a priori.

I would really appreciate any input on the issue. Ideas or papers/books related to this question. Thank you for your reading.