Lower semicontinuity of integral functionals

Abstract:

It is shown that the integral functional $I(y,z) = {\smallint _G}f(t,y(t),z(t))d\mu$ is lower semicontinuous on its domain with respect to the joint strong convergence of ${y_k} \to y$ in ${L_p}(G)$ and the weak convergence of ${z_k} \to z$ in ${L_p}(G)$, where $1 \leq p \leq \infty$ and $1 \leq q \leq \infty$, under the following conditions. The function $f:(t,x,w) \to f(t,x,w)$ is measurable in t for fixed (x, w), is continuous in (x, w) for a.e. t, and is convex in w for fixed (t, x).