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

Colloquium Mathematicae (2001)

• Volume: 89, Issue: 1, page 43-59
• ISSN: 0010-1354

top

## Abstract

top
We study the functional ${I}_{f}\left(u\right)={\int }_{\Omega }f\left(u\left(x\right)\right)dx$, 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.

## How to cite

top

Agnieszka Kałamajska. "On the condition of Λ-convexity in some problems of weak continuity and weak lower semicontinuity." Colloquium Mathematicae 89.1 (2001): 43-59. <http://eudml.org/doc/286256>.

@article{AgnieszkaKałamajska2001,
abstract = {We study the functional $I_\{f\}(u)=∫_\{Ω\} f(u(x))dx$, 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.},
author = {Agnieszka Kałamajska},
journal = {Colloquium Mathematicae},
keywords = {integral functionals; weak continuity; weak lower semicontinuity},
language = {eng},
number = {1},
pages = {43-59},
title = {On the condition of Λ-convexity in some problems of weak continuity and weak lower semicontinuity},
url = {http://eudml.org/doc/286256},
volume = {89},
year = {2001},
}

TY - JOUR
AU - Agnieszka Kałamajska
TI - On the condition of Λ-convexity in some problems of weak continuity and weak lower semicontinuity
JO - Colloquium Mathematicae
PY - 2001
VL - 89
IS - 1
SP - 43
EP - 59
AB - We study the functional $I_{f}(u)=∫_{Ω} f(u(x))dx$, 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.
LA - eng
KW - integral functionals; weak continuity; weak lower semicontinuity
UR - http://eudml.org/doc/286256
ER -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.