# Weak lower semicontinuity by means of anisotropic parametrized measures

## TL;DR

This paper introduces a new analytical tool to study the interaction of oscillations and concentrations in bounded sequences, with applications to weak lower semicontinuity in Sobolev spaces.

## Contribution

It develops a novel approach using anisotropic parametrized measures to analyze mutual interference of oscillating and concentrating sequences.

## Key findings

- New method for handling oscillation and concentration interactions
- Explicit examples demonstrating diverse behaviors
- Applications to weak lower semicontinuity in Sobolev spaces

## Abstract

It is well known that besides oscillations, sequences bounded only in $L^1$ can also develop concentrations, and if the latter occurs, we can at most hope for weak$^*$ convergence in the sense of measures. Here we derive a new tool to handle mutual interferences of an oscillating and concentrating sequence with another weakly converging sequence. We introduce a couple of explicit examples showing a variety of possible kinds of behavior and outline some applications in Sobolev spaces.