Editing Maximum principle (section)

===A partial formulation of the strong maximum principle===
Here we consider the simplest case, although the same thinking can be extended to more general scenarios. Let {{mvar|M}} be an open subset of Euclidean space and let {{mvar|u}} be a {{math|''C''<sup>2</sup>}} function on {{mvar|M}} such that
:<math>\sum_{i=1}^n\sum_{j=1}^n a_{ij}\frac{\partial^2u}{\partial x^i\,\partial x^j}=0</math>
where for each {{mvar|i}} and {{mvar|j}} between 1 and {{mvar|n}}, {{math|''a''<sub>''ij''</sub>}} is a function on {{mvar|M}} with {{math|''a''<sub>''ij''</sub> {{=}} ''a''<sub>''ji''</sub>}}.

Fix some choice of {{mvar|x}} in {{mvar|M}}. According to the [[spectral theorem]] of linear algebra, all eigenvalues of the matrix {{math|[''a''<sub>''ij''</sub>(''x'')]}} are real, and there is an orthonormal basis of {{math|ℝ<sup>''n''</sup>}} consisting of eigenvectors. Denote the eigenvalues by {{math|''λ''<sub>''i''</sub>}} and the corresponding eigenvectors by {{math|''v''<sub>''i''</sub>}}, for {{mvar|i}} from 1 to {{mvar|n}}. Then the differential equation, at the point {{mvar|x}}, can be rephrased as
:<math>\sum_{i=1}^n \lambda_i \left. \frac{d^2}{dt^2}\right|_{t=0}\big(u(x+tv_i)\big)=0.</math>
The essence of the maximum principle is the simple observation that if each eigenvalue is positive (which amounts to a certain formulation of "ellipticity" of the differential equation) then the above equation imposes a certain balancing of the directional second derivatives of the solution. In particular, if one of the directional second derivatives is negative, then another must be positive. At a hypothetical point where {{mvar|u}} is maximized, all directional second derivatives are automatically nonpositive, and the "balancing" represented by the above equation then requires 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 {{mvar|a}}), that {{mvar|u}} must be constant if there is a point of {{mvar|M}} where {{mvar|u}} is maximized.

Note that the above reasoning is unaffected if one considers the more general partial differential equation
:<math>\sum_{i=1}^n\sum_{j=1}^n a_{ij}\frac{\partial^2u}{\partial x^i \, \partial x^j}+\sum_{i=1}^n b_i\frac{\partial u}{\partial x^i}=0,</math>
since the added term is automatically zero at any hypothetical maximum point. The reasoning is also unaffected if one considers the more general condition
:<math>\sum_{i=1}^n\sum_{j=1}^n a_{ij}\frac{\partial^2u}{\partial x^i \, \partial x^j}+\sum_{i=1}^n b_i\frac{\partial u}{\partial x^i}\geq 0,</math>
in which one can even note the extra phenomena of having an outright contradiction if there is a strict inequality ({{mvar|>}} rather than {{mvar|≥}}) in this condition at the hypothetical maximum point. This phenomenon is important in the formal proof of the classical weak maximum principle.