Let \(\Omega \subset \mathbb{R}^N\) be a bounded open set; a functional \(F\) on \(\mathcal A \times W^{1,\infty} (\Omega),\) where \(\mathcal A\) is the class of open subsets of \(\Omega,\) is called a \(L^\infty\)-functional if it may be represented in the so-called supremal form: \[ F(u,A) = \underset {x \in A}{\text{ess\,sup}} f(x,u(x),Du(x)). \] Without the assumption on continuity of \(f(\cdot,u,\xi),\) it is demonstrated that \(F(\cdot,A)\) is weakly\(^*\) lower semicontinuous in \(W^{1,\infty} (\Omega)\) if and only if it may be represented through a level convex supremand. The properties of the lower semicontinuous envelope \(\overline{F}\) of \(F\) are described. In the case when \(f\) is continuous, a complete relaxation theorem is presented and in case when \(f\) is only the Carathéodory type function, it is demonstrated that \(\overline{F}\) coincides with the level convex envelope of \(F.\)