Editing Maximum principle (section)

===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}}.