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Splitting Type Variational Problems with Linear Growth Conditions

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Regularity properties of solutions to variational problems are established for a broad class of strictly convex splitting type energy densities of the principal form f : ℝ2 → ℝ, f(ξ1, ξ2) = f1(ξ1) + f2(ξ2), with linear growth. We show that, regardless of the

corresponding property of f2, the assumption (t ∈ ℝ)

$$ {c}_1{\left(1+\left|t\right|\right)}^{-{\upmu}_1}\le {f}_1^{\hbox{'}\hbox{'}}(t)\le {c}_2,\kern1em 1<{\upmu}_1<2, $$

is sufficient to obtain higher integrability of ∂1u for any finite exponent. We also include a series of variants of our main theorem. In the case f : ℝn → ℝ, similar results hold with the obvious changes in notation. Bibliography: 30 titles.

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Authors and Affiliations

  1. Universität des Saarlandes, Fachrichtung Mathematik, Postfach 15 11 50, D–66041, Saarbrücken, Germany

    M. Bildhauer & M. Fuchs

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  1. M. Bildhauer
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  2. M. Fuchs
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Correspondence to M. Bildhauer.

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Translated from Problemy Matematicheskogo Analiza 105, 2020, pp. 45-58.

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Bildhauer, M., Fuchs, M. Splitting Type Variational Problems with Linear Growth Conditions. J Math Sci 250, 232–249 (2020). https://doi.org/10.1007/s10958-020-05012-8

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  • DOI: https://doi.org/10.1007/s10958-020-05012-8

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