We now continue this topic. Today we discuss a very powerful technique known as the Nash-Moser iteration technique. Let us consider a sub-solution of the following variational inequality

where . We assume is of class and do satisfy an ellipticity condition. For the sake of simplicity we assume . Then by using the test function

with a cut-off function defined as in this topic we arrive at

which yields

.

Thus by the ellipticity condition

and thus

.

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