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.

It is a well-known fact that solutions to nonlinear parabolic partial differential equations of p-laplacian type are Hölder continuous. One of the main features of the proof, as originally given by DiBenedetto and DiBenedetto-Chen, consists in studying separately two cases, according to the size of the solution. Here we present a new proof of the Hölder continuity of solutions, which is based on the ideas used in the proof of the Harnack inequality for the same kind of equations recently given by E. DiBenedetto, U. Gianazza and V. Vespri. Our method does not rely on any sort of alternative, and has a strong geometric character.

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In this paper we generalize the recent result of DiBenedetto, Gianazza, Vespri on the Harnack inequality for degenerate parabolic equations to the case of a weighted p-Laplacian type operator in the spatial part. The weight is assumed to belong to the suitable Muckenhoupt class.

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