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Zeitschrift für Analysis und ihre Anwendungen


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Volume 30, Issue 1, 2011, pp. 59–69
DOI: 10.4171/ZAA/1423

Published online: 2011-01-03

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[1], Raffaella Giova[2] and François Murat[3]

(1) Università Telematica Pegaso, Napoli, Italy
(2) Università degli Studi di Napoli Parthenope, Italy
(3) Université Pierre et Marie Curie, Paris, France

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.

Keywords: Lower semicontinuity, Hardy–Sobolev inequalities

Farroni Fernando, Giova Raffaella, Murat François: 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), 59-69. doi: 10.4171/ZAA/1423