Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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REMARKS ON THE INFINITY WAVE EQUATION

  • Received : 2020.04.18
  • Accepted : 2020.11.02
  • Published : 2021.03.31
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Abstract

We propose the infinity wave equation which can be derived from the exponential wave equation through the limit p → ∞. The solution of infinity Laplacian equation can be considered as a static solution of the infinity wave equation. We present basic observations and find some special solutions.

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References

  1. S. N. Armstrong and C. K. Smart, A finite difference approach to the infinity Laplace equation and tug-of-war games, Trans. Amer. Math. Soc. 364 (2012), no. 2, 595-636. https://doi.org/10.1090/S0002-9947-2011-05289-X
  2. G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967), 551-561 (1967). https://doi.org/10.1007/BF02591928
  3. G. Aronsson, On the partial differential equation ux2uxx+2uxuyuxy+uy2uyy = 0, Ark. Mat. 7 (1968), 395-425. https://doi.org/10.1007/BF02590989
  4. G. Aronsson, On certain singular solutions of the partial differential equation u2xuxx+2uxuyuxy+u2yuyy = 0, Manuscripta Math. 47 (1984), no. 1-3, 133-151. https://doi.org/10.1007/BF01174590
  5. G. Aronsson, M. G. Crandall, and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 4, 439-505. https://doi.org/10.1090/S0273-0979-04-01035-3
  6. D. P. Chi, G. Choi, and J. Chang, Asymptotic behavior of harmonic maps and exponentially harmonic functions, J. Korean Math. Soc. 39 (2002), no. 5, 731-743. https://doi.org/10.4134/JKMS.2002.39.5.731
  7. Y.-J. Chiang and Y.-H. Yang, Exponential wave maps, J. Geom. Phys. 57 (2007), no. 12, 2521-2532. https://doi.org/10.1016/j.geomphys.2007.09.003
  8. R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I, Interscience Publishers, Inc., New York, NY, 1953.
  9. J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), no. 5, 385-524. https://doi.org/10.1112/blms/20.5.385
  10. L. C. Evans, The 1-Laplacian, the ∞-Laplacian and differential games, in Perspectives in nonlinear partial differential equations, 245-254, Contemp. Math., 446, Amer. Math. Soc., Providence, RI, 2007. https://doi.org/10.1090/conm/446/08634
  11. L. C. Evans and O. Savin, C1,α regularity for infinity harmonic functions in two dimensions, Calc. Var. Partial Differential Equations 32 (2008), no. 3, 325-347. https://doi.org/10.1007/s00526-007-0143-4
  12. L. C. Evans and Y. Yu, Various properties of solutions of the infinity-Laplacian equation, Comm. Partial Differential Equations 30 (2005), no. 7-9, 1401-1428. https://doi.org/10.1080/03605300500258956
  13. M. C. Hong, Liouville theorems for exponentially harmonic functions on Riemannian manifolds, Manuscripta Math. 77 (1992), no. 1, 41-46. https://doi.org/10.1007/BF02567042
  14. H. Huh, Global solutions of the exponential wave equation with small initial data, Bull. Korean Math. Soc. 50 (2013), no. 3, 811-821. https://doi.org/10.4134/BKMS.2013.50.3.811
  15. A. D. Kanfon, A. Fuzfa, and D. Lambert, Some examples of exponentially harmonic maps, J. Phys. A 35 (2002), no. 35, 7629-7639. https://doi.org/10.1088/0305-4470/35/35/307
  16. H. Koch, Y. R.-Y. Zhang, and Y. Zhou, Some sharp Sobolev regularity for inhomogeneous infinity Laplace equation in plane, J. Math. Pures Appl. (9) 132 (2019), 483-521. https://doi.org/10.1016/j.matpur.2019.02.007
  17. P. Lewintan, The "wrong minimal surface equation" does not have the Bernstein property, Analysis (Munich) 31 (2011), no. 4, 299-303. https://doi.org/10.1524/anly.2011.1137
  18. E. Lindgren, On the regularity of solutions of the inhomogeneous infinity Laplace equation, Proc. Amer. Math. Soc. 142 (2014), no. 1, 277-288. https://doi.org/10.1090/S0002-9939-2013-12180-5
  19. J. M. Overduin and P. S. Wesson, Kaluza-Klein gravity, Phys. Rep. 283 (1997), no. 5-6, 303-378. https://doi.org/10.1016/S0370-1573(96)00046-4
  20. Y. Peres, O. Schramm, S. Sheffield, and D. B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (2009), no. 1, 167-210. https://doi.org/10.1090/S0894-0347-08-00606-1
  21. Y. Pinchover and J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press, Cambridge, 2005. https://doi.org/10.1017/CBO9780511801228
  22. C.-H. Wei and N.-A. Lai, Global existence of smooth solutions to exponential wave maps in FLRW spacetimes, Pacific J. Math. 289 (2017), no. 2, 489-509. https://doi.org/10.2140/pjm.2017.289.489