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A self-affine property of evolutional type appearing in a Hamilton-Jacobi flow starting from the Takagi function. (English) Zbl 1495.35087

Summary: In this paper, we study a Hamilton-Jacobi flow \(\{ H_t\tau\}_{t>0}\) starting from the Takagi function \(\tau\). The Takagi function is well known as a pathological function that is everywhere continuous and nowhere differentiable on \(\mathbb{R}\). As the first result of this paper, we derive an explicit representation of \(\{ H_t\tau\}\). It turns out that \(H_t\tau\) is a piecewise quadratic function at any time and that the points of intersection between the parabolas are given in terms of binary expansion of real numbers. Applying the representation formula, we next give the main result, which asserts that \(\{ H_t\tau\}\) has a self-affine property of evolutional type involving a time difference in the functional equality. Furthermore, we determine the optimal time until when the self-affine property is valid.

MSC:

35F21 Hamilton-Jacobi equations
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
35D40 Viscosity solutions to PDEs
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[1] P. Allaart and K. Kawamura, The Takagi function: a survey, Real Anal. Exchange 37 (2011/12), 1-54. · Zbl 1248.26007
[2] M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Birkhäuser, Boston, 1997. Reprint of the 1997 edition. · Zbl 0890.49011
[3] M. G. Crandall, H. Ishii, and P. L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992), 1-67. · Zbl 0755.35015
[4] L. C. Evans, Partial differential equations, second edition, Grad. Stud. Math., 19, Am. Math. Soc., Providence, 2010. · Zbl 1194.35001
[5] K. J. Falconer, Fractal geometry, third edition, Math. Found. Appl., Wiley, New York, 2014. · Zbl 1285.28011
[6] Y. Fujita, N. Hamamuki, A. Siconolfi, and N. Yamaguchi, A class of nowhere differentiable functions satisfying some concavity-type estimate, Acta Math. Hungar. 160 (2020), 343-359. · Zbl 1441.26003
[7] J. C. Lagarias, The Takagi function and its properties, RIMS Kôkyûroku Bessatsu 34 (2012), 153-189. · Zbl 1275.26011
[8] T. Takagi, A simple example of the continuous function without derivative, Proc. Phys.-Math. Soc. Jpn. 1 (1903), 176-177. The Collected Papers of Teiji Takagi, S. Kuroda, Ed., Iwanami (1973), 5-6. · JFM 34.0410.05
[9] J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 (1968), 21-25. · Zbl 0162.06303
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