be a smooth function. We multiply the equation
and integrate over
We perform integration by
Hence, we require that
would satisfy the condition
Following tradition of the section (
) we use the
We arrive to the following formulation.
(Variational toy problem) For a given
find a function
We start from (
) and perform the integration by parts in opposite
Therefore the term
must vanish. By contradiction, if it does not vanish the we choose a sequence
to blow up in a shrinking neighborhood of
and remain constant elsewhere and thus obtain a contradiction.
by a similar argument.
The boundary condition
of the problem (
) is explicitly
incorporated in (
) as the requirement
Such condition is called "essential" boundary condition. The condition
is incorporated implicitly. It is called "natural" boundary condition.