New examples of Lavrentiev gap for generalized Orlicz functions

We construct new examples on Lavrentiev phenomenon using fractal contact sets. Comparing to the well-known examples of Zhikov it is not important that at the saddle point the variable exponent crosses the threshold dimension. As a consequence we give the negative answer to the well-known conjecture that the dimension plays a critical role for the Lavrentiev gap to appear. We apply our method to the setting of variable exponents, the double phase potential and weighted p-energy. The talk is based on the joint project with Lars Diening from Bielefeld Unievrsity.