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I've found many papers characterizing the weak lower semi-continuity of $$ \Phi(u)\triangleq \int_{x \in \Omega} f(x,u(x))\,dx, $$ on $L^2(\Omega)$ where $\Omega$ is a bounded subset of $\mathbb{R}^d$....

The heat kernel $u:\mathbb{R}^n\times (0,\infty)$ is defined as the solution to $$ u_t = \Delta u, $$ subject to certain boundary conditions and can alternatively be described, in variational form, as ...