Integral functionals that are continuous with respect to the weak topology on image Original Research Article

For continuous (or, locally bounded Carathéodory) functions g:[0,1]×R→Rg:[0,1]×R→R we prove that the functional View the MathML sourceΦ(u)=∫01g(x,u(x))dx is weakly continuous on View the MathML sourceW01,p(0,1), 1⩽p<∞1⩽p<∞, if and only if g is linear in the second variable.