ON SECOND-ORDER L^∞ VARIATIONAL PROBLEMS WITH LOWER-ORDER TERMS

In this paper we study 2nd-order L^∞-variational problems by seeking to minimise a supremal functional involving the Hessian of admissible functions as well as their lower-order terms, considering for fixed Ω ⊆ Rⁿ open, and H: Ω × (R × Rⁿ × Rⁿ_s^∅2) → R, the functional E∞(u, O):= ess sup H(·, u, Du, D²u), u ∈ W^2,∞(Ω), O ⊆ Ω measurable. O Specifically, we establish the existence of minimisers subject to (first-order) Dirichlet data on ∂Ω under natural assumptions, and, when n = 1, 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, and is given by H_X(·, u, Du, D²u): D(H(·, u, Du, D²u)) ∅ D(H(·, u, Du, D²u)) = 0 in Ω. 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 [27] which first studied problems of this type, providing at the same time some simpler streamlined proofs.