We prove that in dimension n ≥ 2, within the collection of unit-measure cuboids in R n (i.e. domains of the form n i=1 (0, an)), any sequence of minimising domains R D k for the Dirichlet eigenvalues λ k converges to the unit cube as k → ∞. Correspondingly we also prove that any sequence of maximising domains R N k for the Neumann eigenvalues µ k within the same collection of domains converges to the unit cube as k → ∞. For n = 2 this result was obtained by Antunes and Freitas in the case of Dirichlet eigenvalues and van den Berg, Bucur and Gittins for the Neumann eigenvalues. The Dirichlet case for n = 3 was recently treated by van den Berg and Gittins.In addition we obtain stability results for the optimal eigenvalues as k → ∞. We also obtain corresponding shape optimisation results for the Riesz means of eigenvalues in the same collection of cuboids. For the Dirichlet case this allows us to address the shape optimisation of the average of the first k eigenvalues.

We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with connected boundary: we prove that each eigenvalue is uniquely minimized by the ball. We also observe that each surface of revolution with connected boundary is isospectral to the disk.

We obtain results for the spectral optimisation of Neumann eigenvalues on rectangles in R 2 with a measure or perimeter constraint. We show that the rectangle with measure 1 which maximises the k'th Neumann eigenvalue converges to the unit square in the Hausdorff metric as k → ∞. Furthermore, we determine the unique maximiser of the k'th Neumann eigenvalue on a rectangle with given perimeter.

This paper is devoted to the determination of the cases where there is equality in Courant's nodal domain theorem in the case of a Robin boundary condition. For the square, we partially extend the results that were obtained by Pleijel, Bérard-Helffer, Helffer-Persson-Sundqvist for the Dirichlet and Neumann problems.After proving some general results that hold for any value of the Robin parameter h, we focus on the case when h is large. We hope to come back to the analysis when h is small in a second paper.We also obtain some semi-stability results for the number of nodal domains of a Robin eigenfunction of a domain with C 2,α boundary (α > 0) as h large varies.MSC classification (2010): 35P99, 58J50, 58J37.

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