Geodesic
On the condition of ${\mit \Lambda }$-convexity in some problems of weak continuity and weak lower semicontinuity
Agnieszka Kałamajska1
1 Institute of Mathematics Warsaw University Banacha 2 02-097 Warszawa, Poland
Colloquium Mathematicum, Tome 89 (2001) no. 1, pp. 43-59.

Voir la notice de l'article provenant de 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.
DOI : 10.4064/cm89-1-3
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

Agnieszka Kałamajska 1

1 Institute of Mathematics Warsaw University Banacha 2 02-097 Warszawa, Poland 
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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. http://geodesic.mathdoc.fr/articles/10.4064/cm89-1-3/

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