Approximation in nonconvex problems. (English) Zbl 0793.41027

Bandle, C. (ed.) et al., Progress in partial differential equations: calculus of variations, applications. 1st European conference on Elliptic and parabolic problems, Pont-à-Mousson, France, June 1991. Harlow, Essex: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 267, 150-157 (1993).
Let \(\oint:\mathbb{R}^ n \to \mathbb{R}\) be a continuous function, \(\oint^{**}\) be the convex envelope of \(\oint\) i.e. the function defined by \(\oint^{**} (\alpha)=\sup \{g(\alpha) : g\) convex, \(g \leq \oint\}\), \(\Omega\) be some bounded, polygonal domain of \(\mathbb{R}^ n\), \(n \geq 1\), with a boundary \(\Gamma\), \(\tau\) denotes a triangulation of \(\Omega\) into simplifices \(K\) of mesh size \(h=\text{Max}_{K \in \tau} \text{diam} K\), \(P_ 1(K)\) denotes the set of polynomials of degree 1 on \(K\) and \(V^ 0_ h=\{v:\Omega \to \mathbb{R}\): \(v\) is continuous, \(v |_ K \in P_ 1(K)\) \(\forall K \in \tau\), \(v=0\) on \(\Gamma\}\) (\(v |_ K\) denotes the restriction of \(v\) on \(K)\). Define the approximated convex envelope of \(\oint\) by \[ \oint^{**}_ h(\alpha)=\inf_{v \in V^ 0_ h} \left\{ {1 \over \Omega} \int_ \Omega \oint (\alpha+\nabla v(x)) dx \right\}. \] clearly \(\oint^{**}_ h(\alpha) \geq \oint^{**} (\alpha)\). In this note, estimates for the difference \(\oint^{**}_ h- \oint^{**}\) are derived in terms of the mesh size \(h\). Restricting to the one dimensional case, the following theorem is proved: Assume that \(\lim_{| \xi | \to \infty} {\oint (\xi) \over | \xi |}=+\infty\), \(\Omega=(w_ -,w_ +)\). Denote by \(x_ 0,x_ 1,\dots,x_{n+1}\) a subdivision of \(\Omega\) such that \(w_ -=x_ 0, x_ 1, \dots,x_ n\), \(x_{n+1}=w_ +\) and \(h=\max_{i=0,\dots,n} x_{i+1}-x_ i\). Then there exists a constant \(c=c(\alpha,\oint)\) such that \[ 0 \leq \oint_ h^{**} (\alpha)-\oint^{**} (\alpha) \leq {c \over | \Omega |} \cdot h \] where \(| \Omega |=w_ =-w_ -\).
For the entire collection see [Zbl 0780.00014].


41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)