differential equation then the same analysis would show that cannot take on a minimum value. The possibility of such analysis is not even limited to partial differential equations. For instance, if is a function such that which is a sort of "non-local" differential equation, then the automatic strict positivity of the right-hand side shows, by the same analysis as above, that cannot attain a maximum value. There are many methods to extend the applicability of this kind of analysis in various ways. For instance, if is a harmonic function, then the above sort of contradiction does not directly occur, since the existence of a point where is not in contradiction to the requirement everywhere. However, one could consider, for an arbitrary real number , the function defined by It is straightforward to see that By the above analysis, if then cannot attain a maximum value. One might wish to consider the limit as to 0 in order to conclude that also cannot attain a maximum value. However, it is possible for the pointwise limit of a sequence of functions without maxima to have a maxima. Nonetheless, if has a boundary such that together with its boundary is compact, then supposing that can be continuously extended to the boundary, it follows immediately that both and attain a maximum value on Since we have shown that , as a function on , does not have a maximum, it follows that the maximum point of , for any , is on By the sequential compactness of it follows that the maximum of is attained on This is the weak maximum principle for harmonic functions. This does not, by itself, rule out the possibility that the maximum of is also attained somewhere on . That is the content of the "strong maximum principle," which requires further analysis. The use of the specific function above was very inessential. All that mattered was to have a function which extends continuously to the boundary and whose Laplacian is strictly positive. So we could have used, for instance, with the same effect. The classical strong maximum principle for linear elliptic PDE Summary of proof Let be an open subset of Euclidean space. Let be a twice-differentiable function which attains its maximum value . Suppose that Suppose that one can find (or prove the existence of): a compact subset of , with nonempty interior, such that for all in