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On Lebesgue points of entropy solutions to the eikonal equation

Part of: General first-order equations and systems Partial differential equations

Published online by Cambridge University Press:  23 May 2023

Xavier Lamy Open the ORCID record for Xavier Lamy [Opens in a new window]  and
Elio Marconi Open the ORCID record for Elio Marconi [Opens in a new window]
Xavier Lamy
Affiliation:
Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France (xlamy@math.univ-toulouse.fr)
Elio Marconi
Affiliation:
Dipartimento di Matematica ‘Tullio Levi Civita’, Università di Padova, via Trieste 63, 35121 Padova, PD, Italy (elio.marconi@unipd.it)
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Abstract

We consider entropy solutions to the eikonal equation $|\nabla u|=1$ in two-space dimensions. These solutions are motivated by a class of variational problems and fail in general to have bounded variation. Nevertheless, they share several of their fine properties with BV functions: we show in particular that the set of non-Lebesgue points has at least one co-dimension.

Keywords

Aviles–Gigaeikonal equationLagrangian representation

MSC classification

Primary: 35F20: Nonlinear first-order equations
Secondary: 35B65: Smoothness and regularity of solutions

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 4 , August 2024 , pp. 1047 - 1059
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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

Adams, D. R. and Hedberg, L. I.. Function spaces and potential theory. Grundlehren Math. Wiss., vol. 314 (Berlin: Springer-Verlag, 1995).10.1007/978-3-662-03282-4CrossRefGoogle Scholar
Alouges, F., Rivière, T. and Serfaty, S.. Néel and cross-tie wall energies for planar micromagnetic configurations. ESAIM Control Optim. Calc. Var. 8 (2002), 31–68. A tribute to J. L. Lions.10.1051/cocv:2002017CrossRefGoogle Scholar
Ambrosio, L., De Lellis, C. and Mantegazza, C.. Line energies for gradient vector fields in the plane. Calc. Var. Partial Differ. Equ. 9 (1999), 327–255.10.1007/s005260050144CrossRefGoogle Scholar
Ambrosio, L., Fusco, N. and Pallara, D.. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs (New York: The Clarendon Press, Oxford University Press, 2000).10.1093/oso/9780198502456.001.0001CrossRefGoogle Scholar
Aviles, P. and Giga, Y.. A mathematical problem related to the physical theory of liquid crystal configurations. In Miniconference on Geometry and Partial Differential Equations, 2 (Canberra, 1986), Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 12, pp. 1–16 (Canberra: Austral. Nat. Univ., 1987).Google Scholar
Contreras Hip, A. A., Lamy, X. and Marconi, E.. Generalized characteristics for finite entropy solutions of Burgers’ equation. Nonlinear Anal. 219 (2022), 112804.10.1016/j.na.2022.112804CrossRefGoogle Scholar
De Lellis, C. and Otto, F.. Structure of entropy solutions to the eikonal equation. J. Eur. Math. Soc. 5 (2003), 107–145.10.1007/s10097-002-0048-7CrossRefGoogle Scholar
DeSimone, A., Müller, S., Kohn, R. V. and Otto, F.. A compactness result in the gradient theory of phase transitions. Proc. R. Soc. Edinburgh Sect. A 131 (2001), 833–844.10.1017/S030821050000113XCrossRefGoogle Scholar
Ghiraldin, F. and Lamy, X.. Optimal Besov differentiability for entropy solutions of the eikonal equation. Commun. Pure Appl. Math. 73 (2020), 317–349.10.1002/cpa.21868CrossRefGoogle Scholar
Jabin, P.-E. and Perthame, B.. Compactness in Ginzburg–Landau energy by kinetic averaging. Commun. Pure Appl. Math. 54 (2001), 1096–1109.10.1002/cpa.3005CrossRefGoogle Scholar
Jin, W. and Kohn, R. V.. Singular perturbation and the energy of folds. J. Nonlinear Sci. 10 (2000), 355–390.10.1007/s003329910014CrossRefGoogle Scholar
Lamy, X. and Marconi, E.. Stability of the vortex in micromagnetics and related models. 2022, arXiv:2209.09662.10.2422/2036-2145.202209_012CrossRefGoogle Scholar
Lamy, X. and Otto, F.. On the regularity of weak solutions to Burgers’ equation with finite entropy production. Calc. Var. Partial Differ. Equ. 57 (2018), 4.10.1007/s00526-018-1380-4CrossRefGoogle Scholar
Marconi, E.. Characterization of minimizers of Aviles–Giga functionals in special domains. Arch. Ration. Mech. Anal. 242 (2021), 1289–1316.10.1007/s00205-021-01704-wCrossRefGoogle ScholarPubMed
Marconi, E.. Rectifiability of entropy defect measures in a micromagnetics model. Adv. Calc. Var. 16 (2021), 233–257.10.1515/acv-2021-0012CrossRefGoogle Scholar
Marconi, E.. On the structure of weak solutions to scalar conservation laws with finite entropy production. Calc. Var. Partial Differ. Equ. 61 (2022), 30. Id/No 32.10.1007/s00526-021-02137-9CrossRefGoogle Scholar
Rivière, T. and Serfaty, S.. Limiting domain wall energy for a problem related to micromagnetics. Commun. Pure Appl. Math. 54 (2001), 294–338.10.1002/1097-0312(200103)54:3<294::AID-CPA2>3.0.CO;2-S3.0.CO;2-S>CrossRefGoogle Scholar