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
.