all directional second derivatives to be identically zero. This elementary reasoning could be argued to represent an infinitesimal formulation of the strong maximum principle, which states, under some extra assumptions (such as the continuity of ), that must be constant if there is a point of where is maximized. Note that the above reasoning is unaffected if one considers the more general partial differential equation since the added term is automatically zero at any hypothetical maximum point. The reasoning is also unaffected if one considers the more general condition in which one can even note the extra phenomena of having an outright contradiction if there is a strict inequality ( rather than ) in this condition at the hypothetical maximum point. This phenomenon is important in the formal proof of the classical weak maximum principle. Non-applicability of the strong maximum principle However, the above reasoning no longer applies if one considers the condition since now the "balancing" condition, as evaluated at a hypothetical maximum point of , only says that a weighted average of manifestly nonpositive quantities is nonpositive. This is trivially true, and so one cannot draw any nontrivial conclusion from it. This is reflected by any number of concrete examples, such as the fact that and on any open region containing the origin, the function certainly has a maximum. The classical weak maximum principle for linear elliptic PDE The essential idea Let denote an open subset of Euclidean space. If a smooth function is maximized at a point , then one automatically has: as a matrix inequality. One can view a partial differential equation as the imposition of an algebraic relation between the various derivatives of a function. So, if is the solution of a partial differential equation, then it is possible that the above conditions on the first and second derivatives of form a contradiction to this algebraic relation. This is the essence of the maximum principle. Clearly, the applicability of this idea depends strongly on the particular partial differential equation in question. For instance, if solves the differential equation then it is clearly impossible to have and at any point of the domain. So, following the above observation, it is impossible for to take on a maximum value. If, instead solved the differential equation then one would not have such a contradiction, and the analysis given so far does not imply anything interesting. If solved the