(Second order internal
elliptic regularity). Let
be a bounded open set,
,
,
and
be a weak solution of the elliptic PDE (see the definition
(
Elliptic differential
operator
))
Then
and for any
we have the
estimate
where the constant
depends only on
and
.

Proof

We follow the strategy described in the beginning of the section
(
Elliptic regularity section
). Let
and
is the corresponding cutoff function for
and
,
see the definition (
Cutoff function
). The weak
solution
is defined by the
equation
We
substitute
We
have
We aim to use uniform positiveness of
.
Hence, we transform the first term as
follows:
Due to the presence of the cutoff function, the proposition
(
Finite difference basics
)
applies:
Hence,
We reverse the sign of all terms and estimate the left hand side from below
using uniform positive definiteness of the matrix
:
We proceed to estimate the right hand side from above. Note that some of the
terms have derivatives of second order. Such terms have to be estimated using
the formula (
Cauchy inequality with
epsilon
) with the epsilon to be much smaller than
.
Then presence such terms would not violate the estimate from below. For
example,
Then the
to
be chosen to satisfy
.
We perform such estimates for all terms on the right hand side and use the
proposition (
Finite difference
in Sobolev space
) to estimate the first order finite differences. We
arrive to the
estimate
Also,
We use the proposition
(
Existence of
derivative via finite difference
) to
conclude
Finally, the
estimate
follows
from
with
and similar calculations.

Proposition

(High order internal
elliptic regularity). Let
be a bounded open set,
,
,
and
be a weak solution of the elliptic PDE (see the definition
(
Elliptic differential
operator
))
Then
and for any
we have the
estimate
where the constant
depends only on
and
.

Assuming that the statement is true for
we prove it for
as follows.

Let
.
Let
is a multi-index (see the section
(
Function spaces section
)) and
.
We set
in the
identity
for some
.
We integrate by parts and arrive to
where the
is an expression containing derivatives of
up to the order
and derivatives of
up to the order
.
Hence, by induction hypothesis,
and the we complete the induction using the proposition
(
Second order internal
elliptic regularity
).