Abstract/Summary

n this paper we study nd-order-variational problems by seeking to minimise a supremal functional involving the Hessian of admissible functions as well as their lower-order terms, Specifically, we establish the existence of minimisers subject to (first-order) Dirichlet data on under natural assumptions, and, when , we also show the existence of absolute minimisers. We further derive a necessary fully nonlinear PDE of third-order which arises as the analogue of the Euler-Lagrange equation for absolute minimisers, We then rigorously derive this PDE from smooth absolute minimisers, and prove the existence of generalised (merely measurable) solutions to the (first-order) Dirichlet problem on bounded domains. This generalises the key results obtained in [26] which first studied problems of this type, providing at the same time some simpler streamlined proofs.