(Eigenfunctions of Laplacian) For a
bounded set
with
boundary
we introduce a nondecreasing sequence
such
that
for some
and the functions
form a basis in
.

Definition

(H-tilde spaces) For
we introduce the Hilbert spaces
according to the
relationships
The following proposition shows that
is independent of the choice of the basis
.

The proposition (
Parseval equality
)
illuminates the convergence of the series
.

Proposition

(Characterization of H-tilde
spaces) For
we have
where the boundary condition is understood in the sense of the proposition
(
Trace theorem
). The norms
and
are equivalent.

Proof

Let's introduce the convenience notation

First, we show that for a
we have
.
This would prove
because
is dense in
.
Indeed,
where we calculate for every term of the
sum
by the definition (
Eigenfunctions of
Laplacian
)
by the proposition (
Green formula
)-2 and
on
Hence, we continue
by the proposition (
Parseval
equality
)
and by the proposition (
Green
formula
)-1
Thus,
.
We also
extract
for reference below.

We now prove
using the same steps as
above:
by
with
The prove of
is similar.

We prove the inclusion
as follows. For
(forming dense set in
)
we have by the above
calculations
then by the proposition (
Boundary
elliptic
regularity
)
The boundary condition
at
is evident.