Variational formulation, essential and natural
boundary conditions.

e present equivalent formulation of the problem (
Toy
problem
).

Let
be a smooth function. We multiply the equation
with
and integrate over
:
We perform integration by
parts
Hence, we require that
would satisfy the condition
.
Following tradition of the section (
Elliptic
PDE section
) we use the
notation
We arrive to the following formulation.

Problem

(Variational toy problem) For a given
find a function
such
that

We start from (
Variational toy
problem
) and perform the integration by parts in opposite
direction:
thus
Therefore the term
must vanish. By contradiction, if it does not vanish the we choose a sequence
of
to blow up in a shrinking neighborhood of
and remain constant elsewhere and thus obtain a contradiction.
Hence,
But
then
by a similar argument.

Remark

The boundary condition
of the problem (
Toy problem
) is explicitly
incorporated in (
Variational toy
problem
) as the requirement
.
Such condition is called "essential" boundary condition. The condition
is incorporated implicitly. It is called "natural" boundary condition.