论文标题
对于第一个卷曲特征值的轴对称最佳结构域的不存在
Non-existence of axisymmetric optimal domains with smooth boundary for the first curl eigenvalue
论文作者
论文摘要
我们说，如果$μ_1（ω）\ leqμ_1（ω'）$对于任何域$ω'$，则有限的域$ω$对于第一个正卷曲特征值$μ_1（ω）$是最佳的。尽管$μ_1（ω）$在体积方面是统一的下限，但在本文中，我们证明没有$ c^{2，α} $边界的轴对称最佳（甚至本地最小化）域，可以满足一个温和的技术假设。作为一种特殊情况，这种情况排除了$ c^{2，α} $带有凸面部分的最佳轴对称域的存在。在第一个负卷曲特征值的情况下，类似结果。
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.
