We consider the problem of minimizing the $L^\infty$ norm of a function of the hessian over a class of maps, subject to a mass constraint involving the $L^\infty$ norm of a function of the gradient and the map itself. We assume zeroth and first order Dirichlet boundary data, corresponding to the “hinged” and the “clamped” cases. By employing the method of $L^p$ approximations, we establish the existence of a special $L^\infty$ minimizer, which solves a divergence PDE system with measure coefficients as parameters. This is a counterpart of the Aronsson-Euler system corresponding to this constrained variational problem. Furthermore, we establish upper and lower bounds for the eigenvalue.

We generalize to the p-LaplacianΔp a spectral inequality proved by M.-T.Kohler−Jobin. As a particular case of such a generalization, we obtain a sharp lower boundon the first Dirichlet eigenvalue of Δp of aset in terms of its p-torsional rigidity. The result is valid in everyspace dimension, for every1 < p < ∞ and for every openset with finite measure. Moreover, it holds by replacing the first eigenvalue with moregeneral optimal Poincaré-Sobolev constants. The method of proof is based on ageneralization of the rearrangement technique introduced by Kohler−Jobin.