Generalized second order vectorial ∞-eigenvalue problems

We consider the problem of minimizing the L^ norm of a function of the hessian over a class of maps, subject to a mass constraint involving the L^ 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 Lᵖ approximations, we establish the existence of a special L^ 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.