Existence and regularity for scalar minimizers of affine nonconvex simple integrals.
The author studies the existence of minimizers for functionals of the form \(\int_a^b \{ \rho(x)h(x') + \varphi(x)\}dt\), where \(\rho~:~{\mathbf R}\mapsto [1,\infty)\), \(h~:~{\mathbf R}\mapsto [1,\infty]\), \(\varphi~:~{\mathbf R}\mapsto {\mathbf R} \). It is supposed that \(\varphi\) is lower semicontinuous and bounded from below, and that \(h\) is not necessarily convex, but it satisfies the condition \(h^{**}(0)=h(0)\). It is proved that the above functional admits absolutely continuous minimizers satisfying prescribed values at \(a\) and \(b\). Under some additional conditions the author proves that the minimizers are Lipschitz continuous.
scientific article; zbMATH DE number 1900942
scientific article