Smoothness Properties of the Lower Semicontinuous Quasiconvex Envelope

Abstract

Assume that $K\subset R^{nm}$ is a convex body with $o\in \mathrm{int}(K)$ and $\tilde{f}:R^{nm}\to R$ is a Lipschitz resp. $C^1$-function. Defining the unbounded function $f:R^{nm}\to R\cup \{(+\infty )\}$ through

we provide sufficient conditions in order to guarantee that its lower semicontinuous quasiconvex envelope

is globally Lipschitz continuous on $K$ or differentiable in $v\in \mathrm{int}(K)$, respectively. An example shows that the partial derivatives of $f^{(qc)}$ do not necessarily admit a representation with a “supporting measure” for $f^{qc}$ in $v_0$.