Optimization of Neumann Eigenvalues under convexity and geometric constraints

In this paper we study optimization problems for Neumann eigenvalues $\mu_k$ among convex domains with a constraint on the diameter or the perimeter. We work mainly in the plane, though some results are stated in higher dimension. We study the existence of an optimal domain in all considered cases. We also consider the case of the unit disk, giving values of the index $k$ for which it can be or cannot be extremal. We give some numerical examples for small values of $k$ that lead us to state some conjectures.