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Asymptotic behaviour of minimising Dirichlet cuboids

GFM seminar
FCUL, C6, room 6.2.33
2017-03-14 14:00 .. 15:00
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by Katie Gittins (Université de Neuchâtel)

For \(m \ge 4 \) we consider the collection of all unit measure, \(m\)-dimensional cuboids in \(\mathbb{R}^m\), that is, sets of the form \(\prod_{i=1}^m(0,a_i)\). Our goal is to prove that any sequence of such cuboids \((R^*_k)_k\) that minimise the Dirichlet eigenvalues \(\lambda_k\) converges to the \(m\)-dimensional unit cube as \(k\to\infty\). The corresponding result for \(m=2\) was proved by Antunes and Freitas in 2012, while for \(m=3\) it was shown recently by van den Berg and Gittins using a similar approach. Some of the arguments used in these lower-dimensional cases cannot be used when \(m\ge 4\). To address the higher-dimensional cases, we interpret the Dirichlet counting function as a Riesz mean of order zero. By using the product structure of the cuboids, Pólya's inequality and properties of Riesz means, we show that any sequence of minimising cuboids is bounded independently of \(k\). This allows us to prove convergence to the \(m\)-dimensional unit cube. We also mention how this approach can be used to deal with similar shape optimisation problems. This is joint work with Simon Larson, KTH.