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.
Highlights
The Lavrentiev gap is a phenomenon that may occur in the study of variational problems
One of the goals of this paper is to provide new examples of variable exponents, such that the Lavrentiev gap occurs, but which do not need to cross the dimensional threshold, see Subsection 4.1
We show the non-density of smooth functions, i.e. H 1,p(·)( ) = W 1,p(·)( ) and the ambiguity of p(·)-harmonicity
Summary
The Lavrentiev gap is a phenomenon that may occur in the study of variational problems. Manià showed that there exists τ > 0 such that G(w) ≥ τ for all w ∈ C1([0, 1]) with w(0) = 0 and w(1) = 1. This gap between zero and τ is the so called Lavrentiev gap. In the example of Manià the integrand f (x, w, ξ ) := (x − w3)2|ξ |2 depends on x, w and ξ. If the integrand only depends on x and ξ , the Lavrentiev gap does not appear in the case of one-dimensional problems, see [14]. The corresponding question for two and higher dimensional problems with integrands of the form f (x, ∇w(x)) remained open for a very long time
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