Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems
Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems
For a Hamiltonian $H \in C^2(R^{N \times n})$ and a map $u:\Omega \subseteq R^n \to R^N$, we consider the supremal functional\begin{eqnarray}E_\infty (u,\Omega) := \|H(Du)\|_{L^\infty(\Omega)} .\end{eqnarray}The ``Euler-Lagrange' PDE associated to (1) is the quasilinear system\begin{eqnarray}A_\infty u := (H_P \otimes H_P + H[H_P]^\bot H_{PP})(Du):D^2 u = 0. \end{eqnarray}(1) and (2) are the …