We show that the smoother the weight, the broader the range of exponents for which the Lavrentiev's gap is absent for the double phase functionals, i. e. , u _Ω (| u|ᵖ + a (x) | u|q) \, dx\, , 1 p q <, \, a () 0\,. In particular, if a C^, then no additional restrictions are required on p and q. For a C^k, α, we establish the optimal range of exponents, which reads q p + (k + α) (1, p/N). Thereby, we extend previously known results which consider Hölder continuous a (i. e. , q p + α (1, p/N) ), showing that the range of exponents extends naturally upon imposing more regularity on a.
Michał Borowski (Mon,) studied this question.