Generalised higher order vectorial $\infty$-eigenvalue problems

Abstract

We study the problem of minimising the $L^{\infty }$ norm of a function of the $k$-th derivative over a class of maps, subject to a constraint involving the $L^{\infty }$ norm of a function of the map and its lower-order derivatives, for any integer $k\ge 2$. We impose boundary conditions corresponding to the $k$-th order analogues of the classical ``clamped'' and ``hinged'' cases. By employing the method of $L^p$ approximations, we establish the existence of a special $L^{\infty }$ minimiser, which solves a divergence PDE system with measure coefficients as parameters. This system constitutes the counterpart of the Aronsson--Euler equations for the constrained variational problem under consideration. Furthermore, we establish a lower bound for the eigenvalue. The present work extends the second-order vectorial results of Clark and Katzourakis [Generalised second order vectorial $\infty$-eigenvalue problems, PRSE A, 1-21, 2024] to the general higher-order setting.

18 pages