Smoothness of weight sharply discards Lavrentiev's gap for double phase functionals

Abstract: 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., $u \mapsto \int_{\Omega} \left(|\nabla u|^p + a(x)|\nabla u|^q\right)\,dx\,, \quad 1 \leq p \leq q < \infty,\, a(\cdot) \geq 0\,.$ In particular, if $a \in C^\infty$, then no additional restrictions are required on $p$ and $q$. For $a \in C^{k, \alpha}$, we establish the optimal range of exponents, which reads $q \leq p + (k + \alpha)\max(1, p/N)$. Thereby, we extend previously known results which consider H\"older continuous $a$ (i.e., $q \leq p + \alpha\max(1, p/N)$), showing that the range of exponents extends naturally upon imposing more regularity on $a$.