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 .

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