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In Gilbarg-Trudinger's book 'Elliptic Partial Differential Equations of Second order', the maximum principle is used to derive the following gradient estimates for Poisson equations on Euclidean spaces. See Section 3.4; Eq. (3.15)

Let $u\in C^2(Q)\cap C^0(\bar Q)$ for the cube $Q=\{x \in \mathbb R^n \mid |x_i|<d\}$. Then we have the following gradient estimate: $$ |D_iu(0)| \le \frac{n}{d} \sup_{\partial Q} |u| + \frac{d}{2}\sup_Q |\Delta u| $$

However, in practice I am more interested in a possible analogous gradient estimate on some Riemannian manifold $(M,g)$.

Question: Let $\Omega\subset M$ be a bounded domain and $u\in C^2(\Omega)\cap C^0(\bar \Omega)$. Then can we find $c_1,c_2>0$ so that for $x\in M$ $$ \|\nabla^g u(x)\|_g \le c_1 \sup_{\partial \Omega} |u| + c_2 \mathrm{dist}(x,\partial \Omega) \cdot \sup_\Omega \|\Delta^g u\|_g $$

If this is true it should be some standard result. Do you know of any reference about this? Thanks!

In general if I found some estimate in PDE book (e.g. Gilbarg-Trudinger), then when can we apply it to a Riemannian manifold?

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  • $\begingroup$ What do you want the constants $c_1$ and $c_2$ to depend on? The scaling is off (and is clearly inconsistent with the first estimate) if you want them to be independent of the metric (which you seem to want to some extent based on the $dist(x,\partial \Omega)$ term. $\endgroup$ – RBega2 Dec 19 '18 at 21:43
  • $\begingroup$ I think it is ok to allow the constants depend on the metric. $\endgroup$ – Hang Dec 19 '18 at 21:45

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