On the condition of ${\mit \Lambda }$-convexity in some problems of weak continuity and weak lower semicontinuity

Tom 89 / 2001

Agnieszka Kałamajska Colloquium Mathematicum 89 (2001), 43-59 MSC: 49J45, 35E10. DOI: 10.4064/cm89-1-3

Streszczenie

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.

Autorzy

  • Agnieszka KałamajskaInstitute of Mathematics
    Warsaw University
    Banacha 2
    02-097 Warszawa, Poland
    e-mail

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