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Variational integrals on mappings of bounded variation and their lower semicontinuity. (English) Zbl 0737.49011

Functionals of the form \[ F(u)=\int_ \Omega f(x,u,Du)dx \] are considered, where \(\Omega\) is an open subset of \(\mathbb{R}^ n\) and \(u\) varies among all functions in \(C^ 1(\Omega;\mathbb{R}^ m)\). The integrand \(f: \Omega\times \mathbb{R}^ m\times \mathbb{R}^{mn}\to \mathbb{R}\) is assumed to be nonnegative, continuous, and such that \(f(x,s,\cdot)\) is convex on \(\mathbb{R}^{mn}\) for every \(x\in \Omega\) and \(s\in \mathbb{R}^ m\). Moreover, it is assumed that there exist two continuous nonnegative functions \(c(x,s)\) and \(C(x,s)\) such that
{(i)} \(c(x,s)| z| \leq f(x,s,z)\leq C(x,s)(1+| z|)\) for all \((x,s,z)\in \Omega\times\mathbb{R}^ m\times \mathbb{R}^{mn}\);
{(ii)} \(c(x,s)=0\) implies \(f(x,s,z)=0\) for all \(z\in \mathbb{R}^{mn}\).
Under this kind of assumptions the functional \(F\) turns out to be l.s.c. on \(C^ 1(\Omega;\mathbb{R}^ m)\) with respect to the \(L_{loc}^ 1(\Omega;\mathbb{R}^ m)\)-convergence. Denoting by \(\overline{F}\) the relaxation of \(F\) on the natural space \(BV_{loc}(\Omega;\mathbb{R}^ m)\) \[ \overline{F}(u)=\sup\{G(u): G\leq F\text{ on }C^ 1, GL_{loc}^ 1- \text{l.s.c. on }BV_{loc}\}, \] the natural question of representing \(\overline{F}\) in an explicit integral form arises. By using \(n\)-currents and “stitched graphs”, the authors construct a functional \({\mathfrak F}\) on \(BV_{loc}(\Omega;\mathbb{R}^ m)\) which is \(L_{loc}^ 1\) l.s.c. and equal to \(F\) on \(C^ 1(\Omega;\mathbb{R}^ m)\). Therefore \({\mathfrak F}\leq \overline{F}\) on \(BV_{loc}(\Omega;\mathbb{R}^ m)\). The conjecture that \({\mathfrak F}=\overline{F}\) is supported by the fact (proved in Theorem 8.3) that the equality \({\mathfrak F}(u)=\overline{F}(u)\) holds for every bounded continuous \(u\in BV_{loc}(\Omega;\mathbb{R}^ m)\), and for every \(u\in BV_{loc}(\Omega)\) if \(m=1\).
Reviewer: G.Buttazzo (Pisa)

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
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