As mentioned before, today 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].
We assume
.
Denote by the following set
.
It follows from this topic that
.
Also, it follows from this topic that
.
Thus, by the Schwarz inequality and the isoperimetric inequality
.
Hence
.
So
.
It is worth noticing that
which yields
.
Integrating from to
gives
which implies
.