Non-existence of axisymmetric optimal domains with smooth boundary for the first curl eigenvalue

2020/07/10 by Enciso, Alberto, Peralta-Salas, Daniel · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2007.05406

Abstract

We say that a bounded domain Ω is optimal for the first positive curl eigenvalue μ₁(Ω) if μ₁(Ω)≤ μ₁(Ω') for any domain Ω' with the same volume. In spite of the fact that μ₁(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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