Too many maximum and comparison principles Why are there so many maximum and comparison principles in the study of partial differential equations? It is scary to try and learn them because there are hundreds of them. Does every type of domain require a new principle?
 A: The most rough way I can say the "maximum principle" for elliptic PDE is as follows: Take a second order elliptic PDE $F(D^2u,Du,u,x) = 0$. (Ellipticity in this context means that if $M \geq N$ as matrices, then $F(M,p,z,x) \geq F(N,p,z,x)$). If I can find a family of strict subsolutions $v(t)$ that $v(0)$ is below $u$ everywhere and $v(t) \leq u$ on the boundary for all $t$, then $v(t)$ must always lie below $u$ on the interior. Indeed, if not then for some first time a $v(a)$ would touch $u$ by below at some point $x$ on the interior, giving the contradiction
$$0 < F(D^2v(x),Dv(x),v(x),x) \leq F(D^2u(x),Du(x),u(x),x) = 0.$$
Most of the maximum principles I know are special cases of this. If constants are solutions, then the minimum must occur on the boundary by sliding horizontal planes from below. (Actually, since planes aren't strict subsolutions we need to "convexify" planes a bit and assume something about the dependence of $F$ on $Du$ so that they become strict subsolutions and still touch by below on the interior, giving the desired contradiction).
Even the Hopf lemma boils down to this idea; you produce a radially symmetric subsolution that intersects horizontal planes at an angle. The assumption that $F$ decreases in the $u$ variable is usually imperative to the maximum principle because we want to create a family of subsolutions by taking the original one and sliding down.
I hope this helps!
