17 Gennaio 2009 alle 10:58
#30188
::
Ecco qua:
\begin{multline}\label{1est}
\int |v|^q\varphi\,dx\leq\left(\int |u|^p\varphi\,dx\right)^{\frac{1}{p}}\left(\int\frac{|\Delta\varphi|^{p’}}{\varphi^{p’-1}}\,dx\right)^{\frac{1}{p’}}\leq\:\text{(by \eqref{sol2})}\\
\leq \left(\int v(-\Delta\varphi)\,dx\right)^{\frac{1}{p}}\left(\int\frac{|\Delta\varphi|^{p’}}{\varphi^{p’-1}}\,dx\right)^{\frac{1}{p’}}\leq\:\text{(again by H\”{o}lder)}\\
\leq \left(\int |v|^q\varphi\,dx\right)^{\frac{1}{pq}}\left(\int\frac{|\Delta\varphi|^{q’}}{\varphi^{q’-1}}\,dx\right)^{\frac{1}{pq’}}\left(\int\frac{|\Delta\varphi|^{p’}}{\varphi^{p’-1}}\,dx\right)^{\frac{1}{p’}}
\end{multline}
Grazie; ciao,
davide