(Energy estimates for
the bilinear form B). Let
be the bilinear form given by the definition
(
Bilinear form B
) and
satisfy the definition (
Elliptic
differential operator
). Then for any
where the constants
,
,
are dependent only on the functions
and the set
.

Remark

We may add
to both sides and rewrite the second inequality
as

Proof

Based on the definition (
Bilinear form B
) of
we directly
estimate
Each of the integrals is dominated by
for some constant
dependent only on
.
Hence,

According to the definition
(
Elliptic differential
operator
) the matrix
is uniformly positive definite for all
.
Hence, there exists a constant
such that for any
We integrate the above and use the definition
(
Bilinear form B
) of
:

We use the formula (
Cauchy
inequality with epsilon
) to estimate the second
term:
and choose the
so
that
We continue the
estimation:
where the constant
depends only on the functions
and the set
.