Editing Maximum principle (section)

==The classical strong maximum principle for linear elliptic PDE==
===Summary of proof===
Let {{mvar|M}} be an open subset of Euclidean space. Let <math>u:M\to\mathbb{R}</math> be a twice-differentiable function which attains its maximum value {{mvar|C}}. Suppose that
:<math>a_{ij}\frac{\partial^2u}{\partial x^i\,\partial x^j}+b_i\frac{\partial u}{\partial x^i}\geq 0.</math>

Suppose that one can find (or prove the existence of):
* a compact subset {{mvar|Ω}} of {{mvar|M}}, with nonempty interior, such that {{math|''u''(''x'') < ''C''}} for all {{mvar|x}} in the interior of {{mvar|Ω}}, and such that there exists {{math|''x''<sub>0</sub>}} on the boundary of {{mvar|Ω}} with {{math|''u''(''x''<sub>0</sub>) {{=}} ''C''}}.
* a continuous function <math>h:\Omega\to\mathbb{R}</math> which is twice-differentiable on the interior of {{mvar|Ω}} and with
::<math>a_{ij}\frac{\partial^2h}{\partial x^i\,\partial x^j}+b_i\frac{\partial h}{\partial x^i}\geq 0,</math>
: and such that one has {{math|''u'' + ''h'' ≤ ''C''}} on the boundary of {{mvar|Ω}} with {{math|''h''(''x''<sub>0</sub>) {{=}} 0}}
Then {{math|''L''(''u'' + ''h'' − ''C'') ≥ 0}} on {{mvar|Ω}} with {{math|''u'' + ''h'' − ''C'' ≤ 0}} on the boundary of {{mvar|Ω}}; according to the weak maximum principle, one has {{math|''u'' + ''h'' − ''C'' ≤ 0}} on {{mvar|Ω}}. This can be reorganized to say
:<math>-\frac{u(x)-u(x_0)}{|x-x_0|}\geq \frac{h(x)-h(x_0)}{|x-x_0|}</math>
for all {{mvar|x}} in {{mvar|Ω}}. If one can make the choice of {{mvar|h}} so that the right-hand side has a manifestly positive nature, then this will provide a contradiction to the fact that {{math|''x''<sub>0</sub>}} is a maximum point of {{mvar|u}} on {{mvar|M}}, so that its gradient must vanish.

===Proof===
The above "program" can be carried out. Choose {{mvar|Ω}} to be a spherical annulus; one selects its center {{math|''x''<sub>c</sub>}} to be a point closer to the closed set {{math|''u''<sup>−1</sup>(''C'')}} than to the closed set {{math|∂''M''}}, and the outer radius {{mvar|R}} is selected to be the distance from this center to {{math|''u''<sup>−1</sup>(''C'')}}; let {{math|''x''<sub>0</sub>}} be a point on this latter set which realizes the distance. The inner radius {{mvar|ρ}} is arbitrary. Define
:<math>h(x)=\varepsilon\Big(e^{-\alpha|x-x_{\text{c}}|^2}-e^{-\alpha R^2}\Big).</math>
Now the boundary of {{mvar|Ω}} consists of two spheres; on the outer sphere, one has {{math|''h'' {{=}} 0}}; due to the selection of {{mvar|R}}, one has {{math|''u'' ≤ ''C''}} on this sphere, and so {{math|''u'' + ''h'' − ''C'' ≤ 0}} holds on this part of the boundary, together with the requirement {{math|''h''(''x''<sub>0</sub>) {{=}} 0}}. On the inner sphere, one has {{math|''u'' < ''C''}}. Due to the continuity of {{mvar|u}} and the compactness of the inner sphere, one can select {{math|''δ'' > 0}} such that {{math|''u'' + ''δ'' < ''C''}}. Since {{mvar|h}} is constant on this inner sphere, one can select {{math|''ε'' > 0}} such that {{math|''u'' + ''h'' ≤ ''C''}} on the inner sphere, and hence on the entire boundary of {{mvar|Ω}}.

Direct calculation shows
:<math>\sum_{i=1}^n\sum_{j=1}^na_{ij}\frac{\partial^2h}{\partial x^i\,\partial x^j}+\sum_{i=1}^nb_i\frac{\partial h}{\partial x^i}=\varepsilon \alpha e^{-\alpha|x-x_{\text{c}}|^2}\left(4\alpha\sum_{i=1}^n\sum_{j=1}^n a_{ij}(x)\big(x^i-x_{\text{c}}^i\big)\big(x^j-x_{\text{c}}^j\big)-2\sum_{i=1}^n a_{ii}-2 \sum_{i=1}^n b_i\big(x^i-x_{\text{c}}^i\big)\right).</math>
There are various conditions under which the right-hand side can be guaranteed to be nonnegative; see the statement of the theorem below.

Lastly, note that the directional derivative of {{mvar|h}} at {{math|''x''<sub>0</sub>}} along the inward-pointing radial line of the annulus is strictly positive. As described in the above summary, this will ensure that a directional derivative of {{mvar|u}} at {{math|''x''<sub>0</sub>}} is nonzero, in contradiction to {{math|''x''<sub>0</sub>}} being a maximum point of {{mvar|u}} on the open set {{mvar|M}}.

===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.