Note on the Lower Semicontinuity with Respect to the Weak Topology on
Bollettino dell'Unione Matematica Italiana (2010)
- Volume: 3, Issue: 2, page 381-390
- ISSN: 0392-4041
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topČerný, Robert. "Note on the Lower Semicontinuity with Respect to the Weak Topology on $W^{1,p}(\Omega)$." Bollettino dell'Unione Matematica Italiana 3.2 (2010): 381-390. <http://eudml.org/doc/290647>.
@article{Černý2010,
abstract = {Let $\Omega \subset \mathbf\{R\}^\{N\}$ be an open bounded set with a Lipschitz boundary and let $g: \Omega \times \mathbf\{R\} \to \mathbf\{R\}$ be a Carathéodory function satisfying usual growth assumptions. Then the functional $$\Phi(u) = \int\_\{\Omega\} g(x,u(x)) \, dx$$ is lower semicontinuous with respect to the weak topology on $W^\{1,p\}(\Omega)$, $1 \le p \le \infty$, if and only if $g$ is convex in the second variable for almost every $x \in \Omega$.},
author = {Černý, Robert},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {381-390},
publisher = {Unione Matematica Italiana},
title = {Note on the Lower Semicontinuity with Respect to the Weak Topology on $W^\{1,p\}(\Omega)$},
url = {http://eudml.org/doc/290647},
volume = {3},
year = {2010},
}
TY - JOUR
AU - Černý, Robert
TI - Note on the Lower Semicontinuity with Respect to the Weak Topology on $W^{1,p}(\Omega)$
JO - Bollettino dell'Unione Matematica Italiana
DA - 2010/6//
PB - Unione Matematica Italiana
VL - 3
IS - 2
SP - 381
EP - 390
AB - Let $\Omega \subset \mathbf{R}^{N}$ be an open bounded set with a Lipschitz boundary and let $g: \Omega \times \mathbf{R} \to \mathbf{R}$ be a Carathéodory function satisfying usual growth assumptions. Then the functional $$\Phi(u) = \int_{\Omega} g(x,u(x)) \, dx$$ is lower semicontinuous with respect to the weak topology on $W^{1,p}(\Omega)$, $1 \le p \le \infty$, if and only if $g$ is convex in the second variable for almost every $x \in \Omega$.
LA - eng
UR - http://eudml.org/doc/290647
ER -
References
top- ČERNÝ, R. - HENCL, S. - KOLÁŘ, J., Integral functionals that are continuous with respect to the weak topology on , Nonlinear Anal., 71 (2009), 2753-2763. Zbl1166.49014MR2532801DOI10.1016/j.na.2009.01.117
- DACOROGNA, B., Direct Methods in the Calculus of Variations (Springer, 1989). Zbl0703.49001MR990890DOI10.1007/978-3-642-51440-1
- DACOROGNA, B., Weak Continuity and Weak Lower Semicontinuity of Non-Linear Functionals (Springer, 1982). Zbl0484.46041MR658130
- HENCL, S. - KOLÁŘ, J. - PANKRÁC, O., Integral functionals that are continuous with respect to the weak topology on , Nonlinear Anal., 63 (2005), 81-87. MR2167316DOI10.1016/j.na.2005.05.003
- ZIEMER, W. P., Weakly differentiable functions. Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics, 120 (Springer-Verlag, New York, 1989). Zbl0692.46022MR1014685DOI10.1007/978-1-4612-1015-3
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