Smoothness Properties of the Lower Semicontinuous Quasiconvex Envelope

Abstract

Assume that K⊂ℝ_nm_ is a convex body with o ∈ int (K) and f~: ℝ_nm_ → ℝ is a Lipschitz resp. C_1-function. Defining the unbounded function f: ℝ_nm → ℝ ∪ {(+∞)} through

f(v) = {f~ (ν), ν ∈ K

(+∞), v ∈ ℝ_nm_ K,

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

f(qc)(w) = sup {g(w) g: ℝ_nm_ → ℝ ∪ {(+∞)} quasiconvex and lower semicontinuous, g(ν)≤ f(ν) ∀ ν ∈ ℝ_nm_}

is globally Lipschitz continuous on K or differentiable in ν ∈ int (K), respectively. An example shows that the partial derivatives of f(qc) do not necessarily admit a representation with a “supporting measure” for fqc in ν0.