On lower semicontinuity of multiple integrals

We give a new short proof of the Morrey-Acerbi-Fusco-Marcellini Theorem on lower semicontinuity of the variational functional $\int_{Ω} F(x,u,∇u)dx$. The proofs are based on arguments from the theory of Young measures

Oscillation and concentration effects described by Young measures which control discontinuous functions

We study oscillation and concentration effects for sequences of compositions $\{ f(u^\nu)\}_{\nu\in\mathbb N}$ of $\mu$-measurable functions $u^\nu\colon \Omega\rightarrow{\mathbb R}^{m}$ where $\Omega$ is the compact subset of ${\mathbb R}^n$ and $f$ is the (possibly) discontinuous function. The limits are described in terms of Young measures which can control discontinuous functions recently introduced in [A. Kałamajska, On Young measures controlling discontinuous functions , J. Conv. Anal. 13 (2006), no. 1, 177–192]

Pointwise multiplicative inequalities and Nirenberg type estimates in weighted Sobolev spaces

We prove pointwise multiplicative inequalities on bounded domains with the cone property and on infinite cones, and derive a certain class of multiplicative inequalities in weighted Sobolev spaces with Muckenhoupt weights. We also find some new formulas to represent functions by their derivatives