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

### Fernando Farroni

Università Telematica Pegaso, Napoli, Italy### Raffaella Giova

Università degli Studi di Napoli Parthenope, Italy### François Murat

Université Pierre et Marie Curie, Paris, France

## 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.

## Cite this article

Fernando Farroni, Raffaella Giova, François Murat, An Example of a Functional which is Weakly Lower Semicontinuous on $W_0^{1,p}$ for every $p>2$ but not on $H_0^1$. Z. Anal. Anwend. 30 (2011), no. 1, pp. 59–69

DOI 10.4171/ZAA/1423