New Examples on Lavrentiev Gap Using Fractals

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.

Crouzeix-Raviart finite element method for non-autonomous variational problems with Lavrentiev gap

“…Further examples involve fractal contact sets [BDS20] or matrix-valued integrands [FHM03]. The latter example was already treated numerically in [Ort11].…”

Global higher integrability for minimisers of convex functionals with (p,q)-growth

“…For n ≥ 2 counterexamples to W 1,q regularity with 1 < p < n < n + α < q are due to [19], see also [24]. Recent work suggests that the condition p < n < q may be removed [3]. If q < (n+α) p n , it was proven in [19] for many standard examples that minimisers enjoy W 1,q loc regularity.…”

“…It is straightforward to adapt the arguments of Theorem 7 to to growth conditions that combine (H1.1) and (H1.2):+ |z i | 2 + |w i | 2 p i (x)−2 2 |z i − w i | 2 ≤ F(x, z) − F(z, w) − ∂ z F(x, w) • (z − w)where z i = (z i ) j 1≤ j≤n , w i = (w i ) j 1≤ j≤n and 1 < p ≤ p i (x) ≤ q. (H1 3).…”

Global higher integrability for minimisers of convex functionals with (p,q)-growth

“…For n ≥ 2 counterexamples to W 1,q regularity with 1 < p < n < n + α < q are due to [19], see also [24]. Recent work suggests that the condition p < n < q may be removed [3]. If q < (n+α)p n , it was proven in [19] for many standard examples that minimisers enjoy W 1,q loc regularity.…”