報(bào)告人：劉祖漢 教授

報(bào)告題目：Dimension estimates of the singular set for a fractional MEMS problem

報(bào)告時(shí)間：2026年4月13日（周一）上午9：00

報(bào)告地點(diǎn)：云龍校區(qū)6號(hào)樓304報(bào)告廳

主辦單位：數(shù)學(xué)與統(tǒng)計(jì)學(xué)院、數(shù)學(xué)研究院、科學(xué)技術(shù)研究院

報(bào)告人簡(jiǎn)介：

劉祖漢，揚(yáng)州大學(xué)教授，博士生導(dǎo)師。歷任揚(yáng)州大學(xué)數(shù)學(xué)科學(xué)學(xué)院院長(zhǎng)，江蘇師范大學(xué)黨委常委、副校長(zhǎng)，揚(yáng)州大學(xué)黨委常委、副校長(zhǎng)、副書(shū)記；2018年10月至2022年8月任鹽城工學(xué)院黨委書(shū)記。長(zhǎng)期從事偏微分方程的研究，多次參加國(guó)家自然科學(xué)基金項(xiàng)目會(huì)議評(píng)審，在SIAM J. Math. Anal.，J. Funct. Anal.，SIAM J. Appl. Math., JDE，CVPDE，European J. Applied. Math.等重要國(guó)際數(shù)學(xué)期刊上發(fā)表研究論文100余篇。

報(bào)告摘要：

We consider the following semilinear elliptic equation involving the fractional Laplacian

\begin{eqnarray*}(-\triangle)^su=-u^{-p} \hbox{~~in~} B_1,\end{eqnarray*}

where $p>1$, $s\in(0,1)$, $(-\triangle)^s$ is the $s$-Laplacian and $B_1=B_1(0)$ is the unit ball in $\mathbb{R}^N$. We first establish an optimal H\{o}lder regularity estimate for solutions by using blow-up analysis and Liouville-type theorems. Subsequently, we give a convergence result for sequences of solutions with uniform H\{o}lder continuity. These results are also used to show that the Hausdorff dimension of the rupture set $\{u=0\}$ satisfies:

$\dim_{\mathcal{H}} \{u=0\} \leq N-2 \hbox{~if~} \frac{p+1}{2p}<s<1;$< p="">

$\dim_{\mathcal{H}} \{u=0\} \leq N-1 \hbox{~if~} 0<s\leq\frac{p+1}{2p}$.< p="">

In particular, the latter one is a new phenomenon arising from the fractional Laplacian.