On the condition of ${\mit \Lambda }$-convexity in some problems of
weak continuity and weak lower semicontinuity
Colloquium Mathematicum, Tome 89 (2001) no. 1, pp. 43-59
Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences
We study the functional
$I_f(u)=\int_{{\mit\Omega}} f(u(x))\,dx$, where $u=(u_1, \ldots ,u_m)$
and each $u_j$ is constant along some subspace
$W_j$ of ${\mathbb R}^{n} $.
We show that if intersections of the $W_j$'s
satisfy a certain condition
then $I_f$ is weakly lower
semicontinuous if and only if $f$ is ${\mit\Lambda} $-convex (see
Definition 1.1 and Theorem 1.1).
We also give a necessary and sufficient condition
on $\{ W_j\}_{j=1, \ldots ,m}$ to have the equivalence:
$I_f$ is weakly continuous if and only if $f$ is ${\mit\Lambda} $-affine.
Keywords:
study functional int mit omega where ldots each constant along subspace mathbb intersections satisfy certain condition weakly lower semicontinuous only mit lambda convex see definition nbsp theorem nbsp necessary sufficient condition ldots have equivalence weakly continuous only mit lambda affine
Affiliations des auteurs :
Agnieszka Kałamajska 1
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title = {On the condition of ${\mit \Lambda }$-convexity in some problems of
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Agnieszka Kałamajska. On the condition of ${\mit \Lambda }$-convexity in some problems of
weak continuity and weak lower semicontinuity. Colloquium Mathematicum, Tome 89 (2001) no. 1, pp. 43-59. doi: 10.4064/cm89-1-3
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