New Examples on Lavrentiev Gap Using Fractals

By Anna Kh. Balci, Lars Diening, Mikhail Surnachev. Calculus of Variations and Partial Differential Equations, 2019

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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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5005c44d4bcf15a9397ccb0307965658563dc683

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Citations in the corpus

Regularity for Double Phase Problems at Nearly Linear Growth (2023, background): “For more related results on the absence of Lavrentiev phenomenon we refer to the recent papers [1,3,4,16,50,51] and related references.”