I am studying the heat equation on a general bounded domain $\Omega \subset \mathbb{R}^+ \times \mathbb{R}^n$ with continuously differentiable 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 [2]) 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 Gary Lieberman [1]. 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 and 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.
References
[1] Gary M. Lieberman, Second order parabolic differential equations (English), Singapore: World Scientific Publishers, pp. xi+439 (1996), ISBN: 981-02-2883-X, MR1465184, Zbl 0884.35001.
[2] Neil A. Watson, Introduction to heat potential theory (English), Mathematical Surveys and Monographs 182. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4998-9/hbk), xiii, 266 p. (2012), MR2907452, Zbl 1251.31001.
