Criterion for weakly lower semicontinuity in problems of calculus of variations

Question

Let’s say I have to minimize a certain integral functional $F(u)= \int_{\Omega} L(x,u,\nabla u)$ with $\Omega$ regular and $u$ in a certain Sobolev space (don’t really care to be precise, I just want a general idea). To prove there exists a minimum, let’s say I want to use the direct method. No problems for compactness, but when I want the lower semicontinuity for weakly convergent sequences (the derivatives, typically) I am kinda lost. I know the norm is weakly lower semicontinuous, but is there a simple general criterion to have weakly lower semicontinuity, in general?