Smoothness Properties of the Lower Semicontinuous Quasiconvex Envelope
Marcus Wagner
Universität Graz, Austria

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.
Cite this article
Marcus Wagner, Smoothness Properties of the Lower Semicontinuous Quasiconvex Envelope. Z. Anal. Anwend. 29 (2010), no. 4, pp. 377–400
DOI 10.4171/ZAA/1413