具有平流的MEMS模型极值解的正则性
Regularity of the extremal solution in a MEMS model with advection
摘要：
本文考虑非线性特征值问题（S）_λ \qquad {rcr} -Δu + c（x）\cdot \nabla u &= \frac λ{（1-u）^2} \qquad {in $ Ω$ }，u &= 0 \qquad {on $ \pOm$ }极值解的正则性，其中$ Ω$是$ \IR^N$中的光滑有界区域，$ c(x)$是$\bar Ω$上的光滑有界向量场.我们证明了，就像在对流自由模型（$c\equiv 0$），所有的半稳定的解决方案是光滑的，当（且仅当）的尺寸$N\leq 7$。这里的新颖性来自于缺乏一个合适的半稳定性假设变分表征。我们克服了这个困难，通过使用一个一般版本的哈代不等式。在即将发表的论文\cite{CG2}中，我们指出这种方法如何适用于许多其他非线性特征值问题，涉及平流（包括Gelfand问题），表明他们都基本上有相同的临界尺寸为他们的对流免费同行。
英文摘要：
We consider the regularity of the extremal solution of the nonlinear eigenvalue problem (S)_λ\qquad {rcr} -Δu + c(x) \cdot \nabla u &=& \fracλ{(1-u)^2} \qquad {in $ Ω$}, u &=& 0 \qquad {on $ \pOm$}, where $ Ω$ is a smooth bounded domain in $ \IR^N$ and $ c(x)$ is a smooth bounded vector field on $\bar Ω$. We show that, just like in the advection-free model ($c\equiv 0$), all semi-stable solutions are smooth if (and only if) the dimension $N\leq 7$. The novelty here comes from the lack of a suitable variational characterization for the semi-stability assumption. We overcome this difficulty by using a general version of Hardy's inequality. In a forthcoming paper \cite{CG2}, we indicate how this method applies to many other nonlinear eigenvalue problems involving advection (including the Gelfand problem), showing that they all essentially have the same critical dimension as their advection-free counterparts.
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