Speaker
Yanyan Guo, Central China Normal University
Abstract
We prove the following {\it Limiting Bliss inequalities}
\begin{equation}\nonumber
\sup\limits_{v(0) = 0, \int_0^1|v’|^Ndx=1 }\int_0^1 e^{\beta\left(\log\frac{e}{s}\right)\frac{v^N(s)}{s^{N-1}}}ds\leq C(N,\beta), \ \hbox{ for } \beta \le 1
\end{equation}
The inequalities are optimal with respect to $\beta \le 1$; there is compactness for $\beta<1$, and along the {\it infinitesimal Moser sequence} for $\beta = 1$. Moreover, we show that the improved inequalities
\begin{equation}\nonumber
\sup\limits_{v(0) = 0, \int_0^1|v’|^Ndx=1 }\int_0^1 e^{\left(\log\frac{e}{s}+\gamma\log\log\frac{e}{s}\right)\frac{v^N(s)}{s^{N-1}}}ds\leq C(N,\gamma)
\end{equation}
hold for \(\gamma\leq1\), and for \(\gamma=1\) the inequalities are critical with loss of compactness. The inequalities are optimal: no further improvement in the coefficient of the exponent is possible.
The second result extends the result in \cite{DRU} from \(N=2\) to general dimensions \(N\geq2\).