Let \(\Omega\) be an open set. A nonnegative, convex function \(F=F(A)\) in \(C^2(\mathbb{R}^n\setminus {\mathcal K})\) with a closed set \({\mathcal K}\subsetneqq\mathbb{R}^n\) is considered. The author ascertains that local minimizers of \[ E(u,\Omega) := \int_{\Omega} F(Du), \quad \Omega \subseteq\mathbb{R}^n, \tag{1} \] in \((C^0\cap W_{\text{loc}}^{1,1})(\Omega) \) are “very weak” viscosity solutions on \(\Omega\) in the sense of [P. Juutinen and P. Lindqvist, Commun. Partial Differ. Equations 30, No. 3, 305–321 (2005; Zbl 1115.35029)] of the highly singular expanded Euler-Lagrange equation of (1): \[ F_{AA}(Du) : D^2u = 0.\tag{2} \] Moreover, if in addition \(F\) satisfies one of the three conditions given in the paper then feeble viscosity solutions of (2) on \(\Omega\), which are in \(W_{\text{loc}}^{1,1}(\Omega) \), are continuous weak solutions of the equation \[ \text{Div} (F_A(Du)) = 0. \]