\(\mathcal{D}\)-solutions to the system of vectorial calculus of variations in \(L^\infty\) via the singular value problem (Q2402889)

The paper is devoted to the problem of the existence of appropriately defined solutions with Dirichlet boundary conditions to the second order system \[ A_{\infty}u := \left(\mathrm{H}_P \otimes \mathrm{H}_P + \mathrm{H}[\mathrm{H}_P]^{\perp}\mathrm{H}_{PP}\right) (D_u) : D^2u = 0. \] This system is an analogue of the Euler-Lagrange equation when one considers nonstandard vectorial variational problems in the space \(L_{\infty}\) for an appropriate supremum functional. An important feature of the paper is the fact that no convexity of the functional \(\mathrm{H}\) is assumed. It is proved that the \(\mathcal{D}\)-solutions have extra geometric properties being \(W^{1,\infty}\)-submersions and solving a certain system of vectorial Hamilton-Jacobi equations.