Consider the functional I(u) = ∫Ω ‖Du|n − L det Du| dx, whose energy well consists of matrices satisfying |ξ|n = L det ξ. We show that the relaxations of this functional in various Sobolev spaces are significantly different. We also make several remarks concerning various p-growth semiconvex hulls of the energy-well set and prove an attainment result for a special Hamilton-Jacobi equation, |Du|n = L det Du, in the so-called grand Sobolev space W1,q)(Ω; Rn), with q = nL/(L + 1).