Absence of the Lavrentiev phenomenon for a general class of parabolic double phase problems

Abstract

In this paper, we prove the absence of the Lavrentiev phenomenon for a general class of parabolic double phase functionals with Orlicz growth. The energy density is given by $$G(|Dw|)+a(x,t)H(|Dw|),$$ where $G$ and $H$ are Young functions satisfying the ${\Delta }_2$ and ${\nabla }_2$ conditions with $G\prec H$, and $a(\cdot )$ is a continuous nonnegative coefficient. Under suitable balance conditions between the growth gap of $G$ and $H$ and the modulus of continuity of $a(\cdot )$, we show that every finite-energy map can be approximated locally by smooth functions without loss of energy. The result extends the known parabolic double phase theory from power type growth to a broad Young function framework and identifies the natural space-time Orlicz energy class for the problem.