Min-Max Principle and Harnack's inequality I read a statement online that Harnack's inequality is a more accurate and quantitative version of the Maximum principle. But it seems that it's impossible to deduce the maximum principle from Harnack's inequality. The difficulties (among a few others) are:


*

*maximum principle has information on the boundary, while in Harnack's inequality, the estimates occur in a compact subset that is positively bounded away from the boundary. 

*even if 1. can be fixed somehow, Harnack's inequality gives you estimates up to a multiple of constant. But in maximum principle, there is no extra constant appearing in the estimate.
Any thought is appreciated.
 A: The Harnack inequality is a quantitative version of the following result:
$\textbf{Strong maximum principle:}$ If $u\in C(\bar{\Omega})$ is a non-constant solution of $-\Delta u=0$ in $\Omega$ then the maximum of $u$ can't be attained at an interior point of $\Omega$.
To see this assume that the Harnack inequality is known and pick an interior maximum $x_0\in \Omega$ with $u(x_0)=M=\max_{\Omega}u$. Consider, for $\varepsilon\geq0$, $v_{\varepsilon}(x)=(M+\varepsilon)-u(x)$ then $v_{\varepsilon}>0$ is a harmonic function and Harnack's inequality implies, for any compact set $x_0\in K \subset \Omega$,
$$
0\leq\sup_K v_0 \leq \sup_K v_{\varepsilon} \leq C(K,\Omega) \inf_K v_{\varepsilon}= C(K,\Omega)\varepsilon, \qquad 0<\varepsilon.
$$
Since $\varepsilon$ was arbitrary we get $v_0=0$, or $u=M$ a contradiction.
Two remarks.
$\textbf{1.}$ What I think you're thinking of as 'the' maximum principle is the following
$\textbf{Weak Maximum principle:}$ If $u\in C(\bar{\Omega})$ is a solution of $Lu=0$ in $\Omega$, then the maximum of $u$ is attained at the boundary $\partial \Omega$.
Notice that, as their names suggest, strong implies weak since by compactness the maximum is always attained at a point in $\bar{\Omega}$.
$\textbf{2.}$ Most of the difficulties you mention more or less dissapear as soon as you realize that both the maximum principle and Harnack's inequality are, in essence, pointwise estimates on $u$. The maximum principle gives then a qualitative estimate: a non-positive solution can have no zero; while Harnack's inequality gives that a non-positve non-constant solution must satisfy $-u\geq C >0$ on any compact set.
