Generalised second order vectorial $\mathrm{\infty }$-eigenvalue problems

We consider the problem of minimising the $L^{\mathrm{\infty }}$ norm of a function of the hessian over a class of maps, subject to a mass constraint involving the $L^{\mathrm{\infty }}$ 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 $L^{\mathrm{\infty }}$ minimiser, 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.