Weak lower semicontinuity

Summary

A functional on a Banach space is weakly lower semicontinuous if it cannot jump up in a weak limit. That is, $J(u) \le \liminf_k J(u_k)$ whenever $u_k \to u$ weakly. It is semicontinuity with respect to the weak topology, usually in its sequential form. Together with coercivity it is the main hypothesis of the direct method. For integral functionals it is closely tied to convexity of the integrand in the gradient variable.