Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth

: We consider the functional u f x Du x x ≔ ∫ , d, Ω ( ) ( ( )) where fxz ( ) , satisfies a (p q, )-growth condition with respect to z and can be approximated by means of a suitable sequence of functions. We consider BR ⋐ Ω and the spaces X = =∩ W B YW B W B , and , , . p R N p R N q R 1, 1, N loc 1, () () () We prove that the lower semicontinuous envelope of ∣Y coincides with or, in other words, that the Lavrentiev term is equal to zero for any admissible function u ∈ W B , p R 1, N ( ). We perform the approximations by means of functions preserving the values on the boundary of BR. regularity, minimizer, variational, integral, Lavrentiev, phenomenon