Editing Maximum principle (section)

== The classical weak maximum principle for linear elliptic PDE ==
=== The essential idea ===
Let {{mvar|M}} denote an open subset of Euclidean space. If a smooth function <math>u:M\to\mathbb{R}</math> is maximized at a point {{mvar|p}}, then one automatically has:
* <math>(du)(p)=0</math>
* <math>(\nabla^2 u)(p)\leq 0,</math> 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 {{mvar|u}} is the solution of a partial differential equation, then it is possible that the above conditions on the first and second derivatives of {{mvar|u}} 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 {{mvar|u}} solves the differential equation
:<math>\Delta u=|du|^2+2,</math>
then it is clearly impossible to have <math>\Delta u\leq 0</math> and <math>du=0</math> at any point of the domain. So, following the above observation, it is impossible for {{mvar|u}} to take on a maximum value. If, instead {{mvar|u}} solved the differential equation <math>\Delta u=|du|^2</math> then one would not have such a contradiction, and the analysis given so far does not imply anything interesting. If {{mvar|u}} solved the differential equation <math>\Delta u=|du|^2-2,</math> then the same analysis would show that {{mvar|u}} cannot take on a minimum value.

The possibility of such analysis is not even limited to partial differential equations. For instance, if <math>u:M\to\mathbb{R}</math> is a function such that
:<math>\Delta u-|du|^4=\int_M e^{u(x)}\,dx,</math>
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 {{mvar|u}} cannot attain a maximum value.

There are many methods to extend the applicability of this kind of analysis in various ways. For instance, if {{mvar|u}} is a harmonic function, then the above sort of contradiction does not directly occur, since the existence of a point {{mvar|p}} where <math>\Delta u(p)\leq 0</math> is not in contradiction to the requirement <math>\Delta u=0</math> everywhere. However, one could consider, for an arbitrary real number {{mvar|s}}, the function {{math|''u''<sub>''s''</sub>}} defined by
:<math>u_s(x)=u(x)+se^{x_1}.</math>
It is straightforward to see that
:<math>\Delta u_s=se^{x_1}.</math>
By the above analysis, if <math>s>0</math> then {{math|''u''<sub>''s''</sub>}} cannot attain a maximum value. One might wish to consider the limit as {{mvar|s}} to 0 in order to conclude that {{mvar|u}} 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 {{mvar|M}} has a boundary such that {{mvar|M}} together with its boundary is compact, then supposing that {{mvar|u}} can be continuously extended to the boundary, it follows immediately that both {{mvar|u}} and {{math|''u''<sub>''s''</sub>}} attain a maximum value on <math>M\cup\partial M.</math> Since we have shown that {{math|''u''<sub>''s''</sub>}}, as a function on {{mvar|M}}, does not have a maximum, it follows that the maximum point of {{math|''u''<sub>''s''</sub>}}, for any {{mvar|s}}, is on <math>\partial M.</math> By the sequential compactness of <math>\partial M,</math> it follows that the maximum of {{mvar|u}} is attained on <math>\partial M.</math> This is the '''weak maximum principle''' for harmonic functions. This does not, by itself, rule out the possibility that the maximum of {{mvar|u}} is also attained somewhere on {{mvar|M}}. That is the content of the "strong maximum principle," which requires further analysis.

The use of the specific function <math>e^{x_1}</math> 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,
:<math>u_s(x)=u(x)+s|x|^2</math>
with the same effect.