# New Examples on Lavrentiev Gap Using Fractals

## TL;DR

This paper presents new fractal-based examples demonstrating the Lavrentiev gap in variational problems with variable exponents, challenging previous assumptions about the role of the dimensional threshold.

## Contribution

It introduces fractal constructions to generate Lavrentiev gaps, showing the threshold is not essential, and extends the analysis to various settings including variable exponents and weighted energies.

## Key findings

- Fractal methods produce new Lavrentiev gap examples.
- The dimensional threshold is not necessary for the gap.
- Smooth functions are not dense in these new examples.

## Abstract

Zhikov showed 1986 with his famous checkerboard example that functionals with variable exponents can have a Lavrentiev gap. For this example it was crucial that the exponent had a saddle point whose value was exactly the dimension. In 1997 he extended this example to the setting of the double phase potential. Again it was important that the exponents crosses the dimensional threshold. Therefore, it was conjectured that the dimensional threshold plays an important role for the Lavrentiev gap. We show that this is not the case. Using fractals we present new examples for the Lavrentiev gap and non-density of smooth functions. We apply our method to the setting of variable exponents, the double phase potential and weighted p-energy.