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
Theorem. Let be an open set and . If then for any , we have
where over .
Proof. Two cases are possible
If . Define
then for all we have
Here is the trick. Let . Set
Using this test function and observing that is supported on the set and that on this set , we get
Thus, differentiating with respect to , we get
Integrating this relation over the range of , i.e. on we get
For such that on the level set , the Green theorem implies
since on the level set the tangential derivatives of vanish and since inside this surface, we have
If . Due to the presence of singularity, we need to use an approximation technique. Let . Define
then for all we have
Choose the same test function as in the previous case we arrive at
We now pass the limit as on both sides. On the LHS, the integrand coverges pointwise to ; further, since , we have
which is integrable. Thus, by the Dominated Convergence Theorem, we get
On the RHS, one again observes that
which is integrable on the set . So again by the Dominated Convergence Theorem,
since . Thus another application of the Dominated Convergence Theorem yields
The proof follows.
Remark. Concerning the Schwarz symmetrization, the following
See also: Symmetrization And Applications (Series in Analysis) by S. Kesavan.