An Example of a Functional which is Weakly Lower Semicontinuous on $W_0^{1,p}$ for every $p>2$ but not on $H_0^1$

Abstract

In this note we give an example of a functional which is defined and coercive on $H^1_0(\Omega)$, which is sequentially weakly lower semicontinuous on $W^{1,p}_0(\Omega)$ for every $p>2$, but which is not sequentially lower semicontinuous on $H^{1}_0(\Omega)$. This functional is non local.