We give an example of an autonomous functional $F(u) = \int_\Omega f(u,Du) dx$ (where $\Omega$ is open subset of $R^2$ and $u:\Omega\to R^2$ belongs the Sobolev space $W^{1,1}$) which is sequentially weakly lower semicontinuous in $W^{1,p}$ for every $p \ge 1$ but does not agree with the relaxation of the same functional restricted to smooth functions when $p<2$. A Lavrentiev phenomenon occurs for a related boundary problem.
Autori interni: | |
Autori: | Alberti, Giovanni; Majer, Pietro |
Titolo: | Gap phenomenon for autonomous functionals |
Anno del prodotto: | 1994 |
Abstract: | We give an example of an autonomous functional $F(u) = \int_\Omega f(u,Du) dx$ (where $\Omega$ is open subset of $R^2$ and $u:\Omega\to R^2$ belongs the Sobolev space $W^{1,1}$) which is sequentially weakly lower semicontinuous in $W^{1,p}$ for every $p \ge 1$ but does not agree with the relaxation of the same functional restricted to smooth functions when $p<2$. A Lavrentiev phenomenon occurs for a related boundary problem. |
Appare nelle tipologie: | 1.1 Articolo in rivista |
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