Minimization with integrands composed of minimum of convex functions (Q5946001)

Let \(\Omega\) be a smooth bounded domain of \({\mathbb R}^n\). Then the paper is concerned with minimization problems of the form NEWLINE\[NEWLINE\alpha=\inf\left\{\int_\Omega\min\{f(v,Dv),g(v,Dv)\}dx : v\in H^1(\Omega;{\mathbb R}^m)\right\},NEWLINE\]NEWLINE under the assumption that the two integral functionals NEWLINE\[NEWLINEv\to\int_\Omega f(v,Dv)dx\;\text{ and } v\to\int_\Omega g(v,Dv)dxNEWLINE\]NEWLINE are sequentially weakly lower semicontinuous and coercive in \(H^1(\Omega;{\mathbb R}^m)\).NEWLINENEWLINENEWLINEThe proposed approach is to search not only for the minimizer \(u\) but also for functions \(\chi_f\) and \(\chi_g\colon\Omega\to[0,1]\) with \(\chi_f+\chi_g\equiv 1\) together with \({\mathcal R}={\mathcal R}(u,\chi_f,\chi_g)\) such that NEWLINE\[NEWLINE\int_\Omega[\chi_ff(u,Du)+\chi_gg(u,Du)]dx-{\mathcal R}=\alpha.NEWLINE\]NEWLINE It is clear that in such approach explicit representation formulas for \({\mathcal R}\) play a central role.NEWLINENEWLINENEWLINEIn this setting, an approximation result of \((u,\chi_f,\chi_g)\) is proposed, together with some explicit formulas for \(\mathcal R\).NEWLINENEWLINENEWLINEExemplifications to one- and three-dimensional problems related to well potentials are also given.