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; Raffaella Giova; François Murat

doi:10.4171/zaa/1423

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

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

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Abstract

In this note we give an example of a functional which is defined and coercive on $H_{0}^{1} (Ω)$ , which is sequentially weakly lower semicontinuous on $W_{0}^{1, p} (Ω)$ for every $p > 2$ , but which is not sequentially lower semicontinuous on $H_{0}^{1} (Ω)$ . This functional is non local.

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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