In the following we denote by

the set of
matrices with entries from
,

the set of all measurable functions
such that
,

the set of all the functions
that have generalized derivatives
for all
,

the closure of the set
in the Hilbert space
,

the set of all the functions
that have generalized derivatives
for all
.

We will use the following results.

Theorem 2.1 (cf. [3]).

Let us consider the following classical boundary-eigenvalue problem for the laplacien:

where
is a nonempty bounded open set in
and
.

This problem has a countable system of eigenvalues
and
as
.

(i) All the eigenvalues
have finite multiplicity
equal to the dimension of the corresponding eigenspace
.

(ii) Let
be a basis of the
for every
then the eigenvectors
form a complete orthonormal system in the space
Hence for all
we have
If we put
then we get
.

(iii) Also, the eigenfunctions
, where
is the space of infinitely continuously differentiable functions on
and compactly supported in
.

(iv) For all
we have
.

(v) The operator

generates an analytic semigroup

on

defined by

Definition 2.2.

Let

a real number, the operator

is defined by

In particular, we obtain
and
. Since
form a complete orthonormal system in the space
then it is dense in
, and hence
is dense in
.

Proposition 2.3 (cf. [4]).

Let
be a Hilbert separable space and
and
two families of bounded linear operators in
, with
a family of complete orthogonal projections such that

Define the following family of linear operators
Then

(a)
is a linear and bounded operator if
with
continiuous for

(b) under the above condition (a),

is a strongly continiuous semigroup in the Hilbert space

whose infinitesimal generator

is given by

Theorem 2.4 (cf. [5]).

Suppose
is connected,
is a real function in
, and
on a nonempty open subset of
. Then
in
.