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The set V<sup>*</sup> is larger than it absolutely needs to be. It turns out that it suffices to use functionals of the form
:φ<sub>''v''</sub>(f)=∫<sub>0</sub><sup>1</sup>∫<sub>0</sub><sup>1</sup>''v''(x,y)''f''(x,y)dxdy,
where ''v'' satifies the specified (periodic) boundary conditions.
That is, we are now replacing the original problem with that of finding ''u'' in V so that
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