Existence of weak solutions for elliptic Dirichlet problem.

roposition

(Existence of weak solution for elliptic Dirichlet problem 1). Let
be a bounded open subset of
with a
-boundary
and
satisfy the definition (
Elliptic
differential operator
). There exists a number
dependent only on
and
such that for any
and any function
there exists a weak solution of the
problem

We introduce the following "adjoint" operator
and a bilinear form
:

Definition

The function
is a weak solution of the
problem
if

Proposition

(Existence of weak solution for elliptic Dirichlet problem 2). Let
be a bounded open subset of
with a
-boundary
and
satisfy the definition (
Elliptic
differential operator
).

(Existence of weak solution for elliptic Dirichlet problem 3)

1. There exists at most countable set
such that the boundary value problem
has a unique weak solution for
iff
.

2. If
is infinite then
where the values
are nondecreasing and
.

Proof

We have established during the proof of the previous proposition that the
operator
is compact in
if defined. The rest follows from the proposition
(
Spectrum of compact operator
).

Proposition

(Elliptic boundedness of inverse)
Let
be a weak solution of
for
then
for a constant
depending only on
and
.
The constant
blows up if
approaches
.