Necessary and sufficient conditions in semicontinuity and convergence theorems with a functional
Mikhail A. Sychev · Sbornik Mathematics · 1995
For the functional ( satisfies the Caratheodory condition, and , , ) it is proved that: 1) is lower semicontinuous on a fixed pair of function with respect to convergence of to in and weak convergence of to in if an only if for a.e. the function is convex at the point ; 2) strong convergence of to in , weak convergence of to in , and convergence of the values of the functional to imply strong convergence of to if and only if for a.e. the function is strictly convex at the point . Analogous results are obtained for problems with restrictions on the ranges of the functions and in the gradient scalar case: , , .