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Consider a uniformly elliptic operator L of the form

$Lu=-\sum_{j,k=1}^d \frac{\partial}{\partial x_k}(a_{j,k} \frac{\partial u}{\partial x_j})+\sum_{j=1}^d b_j(x) \frac{\partial u}{\partial x_j} +c(x)u(x),$

then, there exists a constant $C$ only depending on the set $Ω$ and the uniform ellipticity constant $α$, such that:

$|u(x)| \leq \sup_{y \in \partial \Omega} |u(y)| + C \sup_{y \in \Omega} |Lu(y)| .$

I'm looking for something like this

$|u(x)| \leq \sup_{y \in \partial \Omega} |\frac{\partial u}{\partial n}(y)| + C \sup_{y \in \Omega} |Lu(y)| $,

or something similar.

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There is no such estimate. For example, let $B_1$ be the unit ball in $\mathbb R^n$. Let $u(x) = 1$ for all $x\in B_1$ and let $L = - \Delta$. Then $\frac{\partial u}{\partial u}$ vanishes identically on $\partial B_1$ and $L u = 0$ identically in $B_1$.

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