I found two interesting formulas related to co-area formula while reading some tricks done by Talenti regarding to the best constant of the Sobolev inequality. The first result is to derive a representation of

and the second result is to deal with differentiation of level sets. Having all these stuffs, I will derive a very short and beautiful proof concerning the lower bound of
where
, a positive solution to the following PDE

This proof I firstly learned from a paper published in Duke Math. J. in 1991 by W. Cheng and C. Li [here].
Co-area formula. Suppose that
is an open set in
, and
is a real-valued Lipschitz function on
. Then, for an integrable function 

where
is the
-dimensional Hausdorff measure.
The Sard theorem. Let
be
,
times continuously differentiable, where
. Let
be the critical set of
, the set of points
in
at which the Jacobian matrix of
has
. Then
has Lebesgue measure
in
.
The Sard theorem has some useful applications. For example, if
the space of test functions where
, then for almost every
in the range of
, we have that
on the level set
. Thus that level set will be an
-dimensional surface. Furthermore

and
.
Theorem. Let
be an open set and
. If
then for any
, we have

where
over
.
(more…)