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Extending linear growth functionals to functions of bounded fractional variation

Part of: Functions of several variables Existence theories 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 (hidde.schoenberger@ku.de)
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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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References

Alibert, J. J. and Bouchitté, G.. Non-uniform integrability and generalized Young measures. J. Convex Anal. 4 (1997), 129–147.Google Scholar
Alicandro, R., Ansini, N., Braides, A., Piatnitski, A. and Tribuzio, A., A variational theory of convolution-type functionals. Preprint arXiv:2007.03993 (2020).Google Scholar
Ambrosio, L. and Dal Maso, G.. On the relaxation in ${\rm BV}(\Omega ;R^m)$ of quasi-convex integrals. J. Funct. Anal. 109 (1992), 76–97.CrossRefGoogle Scholar
Ambrosio, L., Fusco, N. and Pallara, D., Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. (The Clarendon Press, Oxford University Press, New York, 2000.CrossRefGoogle Scholar
Antil, H., Di, Z. W. and Khatri, R.. Bilevel optimization, deep learning and fractional Laplacian regularization with applications in tomography. Inverse. Probl. 36 (2020), 064001.CrossRefGoogle Scholar
Antil, H., Díaz, H., Jing, T. and Schikorra, A., Nonlocal bounded variations with applications. Preprint arXiv:2208.11746 (2022).Google Scholar
Arroyo-Rabasa, A., De Philippis, G. and Rindler, F.. Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints. Adv. Calc. Var. 13 (2020), 219–255.CrossRefGoogle Scholar
Aubert, G. and Kornprobst, P.. Can the nonlocal characterization of Sobolev spaces by Bourgain et al. be useful for solving variational problems?. SIAM J. Numer. Anal. 47 (2009), 844–860.CrossRefGoogle Scholar
Bellido, J. C., Cueto, J. and Mora-Corral, C.. Fractional Piola identity and polyconvexity in fractional spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), 955–981.CrossRefGoogle Scholar
Bellido, J. C., Cueto, J. and Mora-Corral, C.. $\Gamma$-convergence of polyconvex functionals involving $s$-fractional gradients to their local counterparts. Calc. Var. Partial Differ. Equ. 60 (2021), 7.CrossRefGoogle Scholar
Bellido, J. C., Cueto, J. and Mora-Corral, C., Nonlocal gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings. Preprint arXiv:2201.08793 (2022).CrossRefGoogle Scholar
Bellido, J. C. and Mora-Corral, C.. Lower semicontinuity and relaxation via Young measures for nonlocal variational problems and applications to peridynamics. SIAM J. Math. Anal. 50 (2018), 779–809.CrossRefGoogle Scholar
Bellido, J. C., Mora-Corral, C. and Pedregal, P.. Hyperelasticity as a $\Gamma$-limit of peridynamics when the horizon goes to zero. Calc. Var. Partial Differ. Equ. 54 (2015), 1643–1670.CrossRefGoogle Scholar
Bruè, E., Calzi, M., Comi, G. E. and Stefani, G.. A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II. C. R. Math. Acad. Sci. Paris 360 (2022), 589–626.Google Scholar
Comi, G. E., Spector, D. and Stefani, G.. The fractional variation and the precise representative of $BV^{\alpha,p}$ functions. Fract. Calc. Appl. Anal. 25 (2022), 520–558.CrossRefGoogle Scholar
Comi, G. E. and Stefani, G.. A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up. J. Funct. Anal. 277 (2019), 3373–3435.CrossRefGoogle Scholar
Comi, G. E. and Stefani, G.. A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I. Rev. Matemática Complut. (2022).Google ScholarPubMed
Comi, G. E. and Stefani, G., Failure of the local chain rule for the fractional variation. Preprint arXiv:2206.03197 (2022).Google Scholar
Comi, G. E. and Stefani, G.. Leibniz rules and Gauss-Green formulas in distributional fractional spaces. J. Math. Anal. Appl. 514 (2022), 126312.CrossRefGoogle Scholar
Dacorogna, B., Direct methods in the calculus of variations, volume 78 of Applied Mathematical Sciences, 2nd Ed. (Springer, New York, 2008).Google Scholar
Dal Maso, G.. Integral representation on ${\rm BV}(\Omega )$ of $\Gamma$-limits of variational integrals. Manuscripta Math. 30 (1979/80), 387–416.CrossRefGoogle Scholar
D'Elia, M., Gulian, M., Mengesha, T. and Scott, J. M.. Connections between nonlocal operators: from vector calculus identities to a fractional Helmholtz decomposition. Fract. Calc. Appl. Anal. 25 (2022), 2488–2531.CrossRefGoogle Scholar
D'Elia, M., Gulian, M., Olson, H. and Karniadakis, G. E.. Towards a unified theory of fractional and nonlocal vector calculus. Fract. Calc. Appl. Anal. 24 (2021), 1301–1355.CrossRefGoogle Scholar
DiPerna, R. J. and Majda, A. J.. Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Phys. 108 (1987), 667–689.CrossRefGoogle Scholar
Fonseca, I. and Müller, S.. Relaxation of quasiconvex functionals in ${\rm BV}(\Omega,\,R^p)$ for integrands $f(x,\,u,\,\nabla u)$. Arch. Rational Mech. Anal. 123 (1993), 1–49.CrossRefGoogle Scholar
Gilboa, G. and Osher, S.. Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7 (2008), 1005–1028.CrossRefGoogle Scholar
Giusti, E., Minimal surfaces and functions of bounded variation, volume 80 of Monographs in Mathematics. (Birkhüser Verlag, Basel, 1984).CrossRefGoogle Scholar
Goffman, C. and Serrin, J.. Sublinear functions of measures and variational integrals. Duke Math. J. 31 (1964), 159–178.CrossRefGoogle Scholar
Grafakos, L., Classical Fourier analysis, volume 249 of Graduate Texts in Mathematics, 3rd ed. (Springer, New York, 2014).CrossRefGoogle Scholar
Holler, G. and Kunisch, K.. Learning nonlocal regularization operators. Math. Control Relat. Fields 12 (2022), 81–114.CrossRefGoogle Scholar
Kreisbeck, C., Ritorto, A. and Zappale, E.. Cartesian convexity as the key notion in the variational existence theory for nonlocal supremal functionals. Nonlinear Anal. 225 (2022), 113111.CrossRefGoogle Scholar
Kreisbeck, C. and Schönberger, H.. Quasiconvexity in the fractional calculus of variations: characterization of lower semicontinuity and relaxation. Nonlinear Anal. 215 (2022), 112625.CrossRefGoogle Scholar
Kristensen, J. and Rindler, F.. Characterization of generalized gradient Young measures generated by sequences in $W^1,\,1$ and BV. Arch. Ration. Mech. Anal. 197 (2010), 539–598.CrossRefGoogle Scholar
Kristensen, J. and Rindler, F.. Relaxation of signed integral functionals in BV. Calc. Var. Partial Differ. Equ. 37 (2010), 29–62.CrossRefGoogle Scholar
Mengesha, T. and Du, Q.. On the variational limit of a class of nonlocal functionals related to peridynamics. Nonlinearity 28 (2015), 3999–4035.CrossRefGoogle Scholar
Mizuta, Y., Potential theory in Euclidean spaces, volume 6 of GAKUTO International Series. Mathematical Sciences and Applications (Gakkōtosho Co., Ltd., Tokyo, 1996).Google Scholar
Morrey, C. B. Jr. Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952), 25–53.CrossRefGoogle Scholar
Pedregal, P.. Weak lower semicontinuity and relaxation for a class of non-local functionals. Rev. Mat. Complut. 29 (2016), 485–495.CrossRefGoogle Scholar
Reshetnyak, Y. G.. Weak convergence of completely additive vector functions on a set. Sib. Math. J. 9 (1968), 1039–1045.CrossRefGoogle Scholar
Rindler, F., Calculus of variations. Universitext (Springer, Cham, 2018).CrossRefGoogle Scholar
Rindler, F. and Shaw, G.. Liftings, Young measures, and lower semicontinuity. Arch. Ration. Mech. Anal. 232 (2019), 1227–1328.CrossRefGoogle Scholar
Shieh, T.-T. and Spector, D. E.. On a new class of fractional partial differential equations. Adv. Calc. Var. 8 (2015), 321–336.CrossRefGoogle Scholar
Shieh, T.-T. and Spector, D. E.. On a new class of fractional partial differential equations II. Adv. Calc. Var. 11 (2018), 289–307.CrossRefGoogle Scholar
Silling, S. A.. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48 (2000), 175–209.CrossRefGoogle Scholar
Stein, E. M., Singular integrals and differentiability properties of functions. Princeton Mathematical Series, Vol. 30 (Princeton University Press, Princeton, N.J., 1970).Google Scholar
Šilhavý, M.. Fractional vector analysis based on invariance requirements (critique of coordinate approaches). Contin. Mech. Thermodyn. 32 (2020), 207–228.CrossRefGoogle Scholar
Ziemer, W. P., Weakly differentiable functions, volume 120 of Graduate Texts in Mathematics. (Springer-Verlag, New York, 1989). Sobolev spaces and functions of bounded variation.CrossRefGoogle Scholar