Lower Semicontinuity of Integral Functionals with Nonconvex Integrands by Relaxation-Compactification

Abstract

A new approach to the lower semicontinuity of integral functionals is presented. By a topological embedding of the “control” and “state” spaces in the Hilbert cube and a simultaneous relaxation of the “control functions,” a powerful approach emerges whose main features include: (i) A generalized convexity condition is imposed upon the integrand of which the classical convexity condition is a special case. (ii) In the embedded setting the integrand can be supposed Lipschitz-continuous in “control” and “state” arguments without loss of generality. (iii) Convergence in measure of the “trajectories,” metamorphoses into $L_1 $-norm convergence in the embedded setting.