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Extending linear growth functionals to functions of bounded fractional variation

Part of: Existence theories Functions of several variables Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  21 February 2023

Hidde Schönberger Open the ORCID record for Hidde Schönberger [Opens in a new window]
Hidde Schönberger*
Affiliation:
Mathematisch-Geographische Fakultät, Katholische Universität Eichstätt-Ingolstadt, Ostenstraße 28, 85072 Eichstätt, Germany ([email protected])
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Abstract

In this paper we consider the minimization of a novel class of fractional linear growth functionals involving the Riesz fractional gradient. These functionals lack the coercivity properties in the fractional Sobolev spaces needed to apply the direct method. We therefore utilize the recently introduced spaces of bounded fractional variation and study the extension of the linear growth functional to these spaces through relaxation with respect to the weak* convergence. Our main result establishes an explicit representation for this relaxation, which includes an integral term accounting for the singular part of the fractional variation and features the quasiconvex envelope of the integrand. The role of quasiconvexity in this fractional framework is explained by a technique to switch between the fractional and classical settings. We complement the relaxation result with an existence theory for minimizers of the extended functional.

Keywords

linear growth functionalsRiesz fractional gradientspaces of bounded variationrelaxationquasiconvexity

MSC classification

Primary: 49J45: Methods involving semicontinuity and convergence; relaxation
Secondary: 35R11: Fractional partial differential equations 26B30: Absolutely continuous functions, functions of bounded variation
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 1 , February 2024 , pp. 304 - 327
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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