Last time we discussed the Picone identity for general purpose [here]. Now we present a generalization of this for -Laplacian operator. This can be seen from the a paper by Walter Allegretto and Yin Xi Huang [here].
Let , be differentiable over a domain . Denote
Moreover, , and a.e. if and only if a.e. , i.e. for some constant in each component of the domain.
Proof. For each partial derivative, it holds
Thus . The rest of the proof follows from the Young inequality. Precisely, we first write
We then have
This together the fact that
prove the non-negativity of .
Lastly, if with , we must then have
On the other hand, if
then a.e. in and thus a.e. in . We conclude that a.e. in .