On a range of exponents for absence of Lavrentiev phenomenon for double phase functionals

TL;DR

This paper improves the understanding of conditions under which the Lavrentiev phenomenon is absent for double phase functionals with $(p,q)$-growth, using a novel regularization approach based on Musielak-Orlicz spaces.

Contribution

It introduces a new regularization method that only requires bounded functions and leverages $L^{ abla}$ bounds, expanding applicability to unbounded data and variable exponents.

Findings

Established an improved exponent range for absence of Lavrentiev phenomenon.

Proved the method works for unbounded boundary data and vectorial problems.

Showed the result is optimal for $p \,\leq\, d$.

Abstract

For a class of functionals having the $(p,q)$-growth, we establish an improved range of exponents $p$, $q$ for which the Lavrentiev phenomenon does not occur. The proof is based on a standard mollification argument and Young convolution inequality. Our contribution is two-fold. First, we observe that it is sufficient to regularise only bounded functions. Second, we exploit the $L^{\infty}$ bound on the function rather than the $L^p$ estimate on the gradient. Our proof does not rely on the properties of minimizers to variational problems but it is rather a consequence of the underlying Musielak-Orlicz function spaces. Moreover, our method works for unbounded boundary data, the variable exponent functionals and vectorial problems. In addition, the result seems to be optimal for $p\le d$.