Minimal partitions for anisotropic tori (Q402279)

Consider the two-dimensional torus \(T(a,b)=S^1(\frac{a}{2\pi})\times S^1(\frac{b}{2\pi})\) for \(a\geq b>0\). Write \({\mathcal D}_k\) for the \(k\)-partition \((D_1,\dots,D_k)\) of the torus, where \(D_1,\dots,D_k\) are disjoint open sets of \(T(a,b)\). Next define NEWLINE\[NEWLINE{\mathcal L}_k(T(a,b))=\inf_{\mathcal D}\max_j\lambda(D_j)NEWLINE\]NEWLINE where \(\lambda(D_j)\) is the ground state energy of the Dirichlet Laplacian in \(D_j\).NEWLINENEWLINEThe main result proved in the paper under review is stated as follows.NEWLINENEWLINETheorem (Theorem 1.1 in the text): There exists \(b_k>0\) such that, if \(b<b_k\) then \({\mathcal L}_k(T(a,b))=k^2\pi^2\) and the corresponding minimal \(k\)-partition \({\mathcal D}_k=(D_1,\dots,D_k)\) is represented in \([ 0;1]\times[ 0;b]\) by NEWLINE\[NEWLINED_i=\left]\frac{i-1}{k};\frac{i}{k}\right[\times\left[0;b\right[,\quad i=1,\dots,k.NEWLINE\]NEWLINE Moreover, one can take \(b_k=2/k\) when \(k\) is even, and \(b_k=1/k\) when \(k\) is odd.NEWLINENEWLINEToward the proof of the above theorem, the authors prove several interesting, and somewhat independent, results.