On the condition of Λ-convexity in some problems of weak continuity and weak lower semicontinuity

Agnieszka Kałamajska

Colloquium Mathematicae, Tome 89 (2001), p. 43-59 / Harvested from The Polish Digital Mathematics Library

Résumé

We study the functional $I_{f} (u) = \int_{Ω} f (u (x)) d x$ , where u=(u₁, ..., uₘ) and each $u_{j}$ is constant along some subspace $W_{j}$ of ℝⁿ. 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 Λ-convex (see Definition 1.1 and Theorem 1.1). We also give a necessary and sufficient condition on ${W_{j}}_{j = 1, . . ., m}$ to have the equivalence: $I_{f}$ is weakly continuous if and only if f is Λ-affine.

Publié le : 2001-01-01

Zbl 1001.49021

EUDML-ID : urn:eudml:doc:286256

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     title = {On the condition of $\Lambda$-convexity in some problems of weak continuity and weak lower semicontinuity},
     journal = {Colloquium Mathematicae},
     volume = {89},
     year = {2001},
     pages = {43-59},
     zbl = {1001.49021},
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     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-1-3}
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Agnieszka Kałamajska. On the condition of Λ-convexity in some problems of weak continuity and weak lower semicontinuity. Colloquium Mathematicae, Tome 89 (2001) pp. 43-59. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-1-3/