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 $a$.