Editing Maximum principle (section)

===Statement of the theorem===
The following is the statement of the theorem in the books of Morrey and Smoller, following the original statement of Hopf (1927):
{{blockquote|Let {{mvar|M}} be an open subset of Euclidean space {{math|ℝ<sup>''n''</sup>}}. For each {{mvar|i}} and {{mvar|j}} between 1 and {{mvar|n}}, let {{math|''a''<sub>''ij''</sub>}} and {{math|''b''<sub>''i''</sub>}} be continuous functions on {{mvar|M}} with {{math|''a''<sub>''ij''</sub> {{=}} ''a''<sub>''ji''</sub>}}. Suppose that  for all {{mvar|x}} in {{mvar|M}}, the symmetric matrix {{math|[''a''<sub>''ij''</sub>]}} is positive-definite. If {{mvar|u}} is a nonconstant {{math|''C''<sup>2</sup>}} function on {{mvar|M}} such that
:<math>\sum_{i=1}^n\sum_{j=1}^na_{ij}\frac{\partial^2u}{\partial x^i\,\partial x^j}+\sum_{i=1}^nb_i\frac{\partial u}{\partial x^i}\geq 0</math>
on {{mvar|M}}, then {{mvar|u}} does not attain a maximum value on {{mvar|M}}.}}
The point of the continuity assumption is that continuous functions are bounded on compact sets, the relevant compact set here being the spherical annulus appearing in the proof. Furthermore, by the same principle, there is a number {{mvar|λ}} such that for all {{mvar|x}} in the annulus, the matrix {{math|[''a''<sub>''ij''</sub>(''x'')]}} has all eigenvalues greater than or equal to {{mvar|λ}}. One then takes {{mvar|α}}, as appearing in the proof, to be large relative to these bounds. Evans's book has a slightly weaker formulation, in which there is assumed to be a positive number {{mvar|λ}} which is a lower bound of the eigenvalues of {{math|[''a''<sub>''ij''</sub>]}} for all {{mvar|x}} in {{mvar|M}}.

These continuity assumptions are clearly not the most general possible in order for the proof to work. For instance, the following is Gilbarg and Trudinger's statement of the theorem, following the same proof:
{{blockquote|Let {{mvar|M}} be an open subset of Euclidean space {{math|ℝ<sup>''n''</sup>}}. For each {{mvar|i}} and {{mvar|j}} between 1 and {{mvar|n}}, let {{math|''a''<sub>''ij''</sub>}} and {{math|''b''<sub>''i''</sub>}} be functions on {{mvar|M}} with {{math|''a''<sub>''ij''</sub> {{=}} ''a''<sub>''ji''</sub>}}. Suppose that  for all {{mvar|x}} in {{mvar|M}}, the symmetric matrix {{math|[''a''<sub>''ij''</sub>]}} is positive-definite, and let {{math|λ(x)}} denote its smallest eigenvalue. Suppose that <math>\textstyle\frac{a_{ii}}{\lambda}</math> and <math>\textstyle\frac{|b_i|}{\lambda}</math> are bounded functions on {{mvar|M}} for each {{mvar|i}} between 1 and {{mvar|n}}. If {{mvar|u}} is a nonconstant {{math|''C''<sup>2</sup>}} function on {{mvar|M}} such that
:<math>\sum_{i=1}^n\sum_{j=1}^na_{ij}\frac{\partial^2u}{\partial x^i\,\partial x^j}+\sum_{i=1}^nb_i\frac{\partial u}{\partial x^i}\geq 0</math>
on {{mvar|M}}, then {{mvar|u}} does not attain a maximum value on {{mvar|M}}.}}

One cannot naively extend these statements to the general second-order linear elliptic equation, as already seen in the one-dimensional case. For instance, the ordinary differential equation {{math|''y''{{''}} + 2''y'' {{=}} 0}} has sinusoidal solutions, which certainly have interior maxima. This extends to the higher-dimensional case, where one often has solutions to "eigenfunction" equations {{math|Δ''u'' + ''cu'' {{=}} 0}} which have interior maxima. The sign of ''c'' is relevant, as also seen in the one-dimensional case; for instance the solutions to {{math|''y''{{''}} - 2''y'' {{=}} 0}} are exponentials, and the character of the maxima of such functions is quite different from that of sinusoidal functions.