第一旋度为eigenvalue的光滑边界轴对称最优域的不存在性
Non-existence of axisymmetric optimal domains with smooth boundary for the first curl eigenvalue
摘要：
我们说一个有界域$Ω$对于第一个正旋度特征值$μ_1(Ω)$是最优的，如果对于任何具有相同体积的域$Ω'$有$μ_1(Ω)\leq μ_1(Ω')$。尽管$μ_1(Ω)$是一致下界的体积，在本文中，我们证明了没有轴对称的最佳（甚至局部最小化）域的$C^{2,α}$边界，满足一个温和的技术假设。作为一种特殊情况，这排除了具有凸截面的$C^{2,α}$最优轴对称域的存在性。一个类似的结果在第一个负旋度本征值的情况下成立。
英文摘要：
We say that a bounded domain $Ω$ is optimal for the first positive curl eigenvalue $μ_1(Ω)$ if $μ_1(Ω)\leq μ_1(Ω')$ for any domain $Ω'$ with the same volume. In spite of the fact that $μ_1(Ω)$ is uniformly lower bounded in terms of the volume, in this paper we prove that there are no axisymmetric optimal (and even locally minimizing) domains with $C^{2,α}$ boundary that satisfies a mild technical assumption. As a particular case, this rules out the existence of $C^{2,α}$ optimal axisymmetric domains with a convex section. An analogous result holds in the case of the first negative curl eigenvalue.
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