Parabolic problem with relaxed boundary approximation.

roblem

(Partially
inverted semi discrete parabolic problem) Assuming existence of the operator
as in the condition (
Properties of
solution operator
) we pose the problem of finding
such
that

Proposition

(Partial inversion lemma) Let
for
where the operator
is
nonnegative
for some scalar product
.
Then for the corresponding norm
we
have

Proof

We apply the operation
to the relationship
and
obtain
We also have
and
.
Hence, using nonnegativeness of
We
integrate:
Let
,
,
then
It remains to note
that