The classical Diaz-Saa inequality found in [1] can be derived from the standard Picone inequality [2]. Let us try to use the generalized Picone inequality found in [3] and [4] to get a corresponding version of the Diaz-Saa inequality.
The generalized Picone inequality has the form
\[|\nabla u|^{p-2} \nabla u \nabla \left(\frac{v^{q}}{u^{q-1}}\right) \leq |\nabla v|^{p-2} \nabla v \nabla \left(\frac{v^{q-p+1}}{u^{q-p}}\right),\]where $q$ and $p$ satisfy certain algebraic relations, see Theorem 1.8 in [4].
If we suppose that $u, v \in W_0^{1,p}(\Omega)$ are reasonably “good” functions so that one can integrate the inequality over $\Omega$ by parts, we arrive at
\[\int_\Omega (-\Delta_p u) \frac{v^{q}}{u^{q-1}} \,dx \leq \int_\Omega (-\Delta_p v) \frac{v^{q-p+1}}{u^{q-p}} \,dx,\]and, analogously,
\[\int_\Omega (-\Delta_p v) \frac{u^{q}}{v^{q-1}} \,dx \leq \int_\Omega (-\Delta_p u) \frac{u^{q-p+1}}{v^{q-p}} \,dx.\]Summing these inequalities and rearranging, we arrive at
\[\int_\Omega \left(\frac{-\Delta_p v}{v^{p-1}} - \frac{-\Delta_p u}{u^{p-1}}\right) (v^{2q-p} - u^{2q-p}) v^{p-q} u^{p-q} \,dx \geq 0.\]If $q=p$, then we end up with the classical Diaz-Saa inequality.
Last modified: 09-Aug-25
Bibliography
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