New Examples on Lavrentiev Gap Using Fractals
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Calculus of Variations
New Examples on Lavrentiev Gap Using Fractals Anna Kh. Balci1
· Lars Diening1
· Mikhail Surnachev2
Received: 10 July 2019 / Accepted: 31 July 2020 / Published online: 24 September 2020 © The Author(s) 2020
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. Mathematics Subject Classification 35J60 · 46E35 · 35J20
1 Introduction The Lavrentiev gap is a phenomenon that may occur in the study of variational problems. In particular, the minimum of the integral functional G taken over smooth functions may differ from the one taken over the associated energy space.
Dedicated to Vasili˘ı Vasil’evich Zhikov. Communicated by T. Riviere. Anna Kh.Balci and Lars Diening thank the German Research Foundation (DFG) for the support through the CRC 1283. The research of Mikhail Surnachev was supported by the Russian Science Foundation under grant 19-71-30004. Mikhail Surnachev acknowledges warm hospitality of Bielefeld University.
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Anna Kh. Balci [email protected] Lars Diening [email protected] Mikhail Surnachev [email protected]
1
University Bielefeld, Universitätsstrasse 25, 33615 Bielefeld, Germany
2
Keldysh Institute of Applied Mathematics RAS, Miusskaya sq. 4, 125047 Moscow, Russia
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The first example for Lavrentiev gap was constructed by Lavrentiev in [14]. A simpler one was provided by Manià in [15], who considered the functional 1 2 2 x − (w(x))3 |w (x)| d x G(w) := 0
subject to the boundary condition w(0) = 0 and w(1) = 1. Now, Manià showed that there exists τ > 0 such that G(w) ≥ τ for all w ∈ C 1 ([0, 1]) with w(0) = 0 and w(1) = 1. 1 1 However, the function x 3 ∈ W 1,1 ((0, 1)) has strictly smaller energy, namely G(x 3 ) = 0. This gap between zero and τ is the so called Lavrentiev gap. In the example of Manià the integrand f (x, w, ξ ) := (x − w 3 )2 |ξ |2 depends on x, w and ξ . If the integrand only depends on x and ξ , then 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.
1.1 Zhikov’s Famous Checkerboard Example – Variable Exponents In 1986 Zhikov presented his famous two-dimensional checkerboard example with a Lavrentiev gap, see [19]. In par
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