Lavrentiev gap for some classes of generalized Orlicz functions

TL;DR

This paper investigates the Lavrentiev gap phenomenon in generalized Orlicz spaces, establishing optimal conditions that distinguish between regular cases and those with gaps, especially for double phase potentials.

Contribution

It introduces new optimal conditions for the Lavrentiev gap in generalized Orlicz functions and provides results on the density of smooth functions in these spaces.

Findings

Identified optimal conditions separating regularity from Lavrentiev gap cases.

Established new results on the density of smooth functions in generalized Orlicz spaces.

Analyzed the borderline case of double phase potentials.

Abstract

In the present paper we find optimal conditions separating the regular case from the one with Lavrentiev gap for the borderline case of double phase potencial and related general classes of integrands. We present new results on density of smooth functions.