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A Crank-Nicolson type minimization scheme for a hyperbolic free boundary problem
1. | National Institute of Technology, Gifu College, Motosu, Gifu, 501-0495, Japan |
2. | Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, Nakanoku, Tokyo, 164-8525, Japan |
3. | Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa, Ishikawa, 920-1192, Japan |
4. | Institute of Science and Engineering, Kanazawa University, Kanazawa, Ishikawa, 920-1192, Japan |
5. | Graduate School of Science, Kyoto University, Sakyoku, Kyoto, 606-8502, Japan |
We consider a hyperbolic free boundary problem by means of minimizing time discretized functionals of Crank-Nicolson type. The feature of this functional is that it enjoys energy conservation in the absence of free boundaries, which is an essential property for numerical calculations. The existence and regularity of minimizers is shown and an energy estimate is derived. These results are then used to show the existence of a weak solution to the free boundary problem in the 1-dimensional setting.
References:
[1] |
M. Bonafini, M. Novaga and G. Orlandi,
A variational scheme for hyperbolic obstacle problems, Nonlin. Anal., 188 (2019), 389-404.
doi: 10.1016/j.na.2019.06.008. |
[2] |
M. Bonafini, V.P.C. Le, M. Novaga and G. Orlandi, On the obstacle problem for fractional semilinear wave equations, Nonlinear Anal., 210 (2021), 112368.
doi: 10.1016/j.na.2021.112368. |
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X. Chen, S. Cui and A. Friedman,
A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior, Trans. Am. Math. Soc., 357 (2005), 4771-4804.
doi: 10.1090/S0002-9947-05-03784-0. |
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E. Ginder and K. Svadlenka, A variational approach to a constrained hyperbolic free boundary problem, Nonlinear Anal., 71 (2009), e1527–e1537.
doi: 10.1016/j.na.2009.01.228. |
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T. Iguchi and D. Lannes, Hyperbolic free boundary problems and applications to wave-structure interactions, Indiana Univ. Math. J., 70 (2021), 353–464, arXiv: 1806.07704. |
[7] |
K. Kikuchi,
Constructing a solution in time semidiscretization method to an equation of vibrating string with an obstacle, Nonlinear Anal., 71 (2009), 1227-1232.
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K. Kikuchi and S. Omata,
A free boundary problem for a one dimensional hyperbolic equation, Adv. Math. Sci. Appl., 9 (1999), 775-786.
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O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, 1$^st$ edition, Academic Press, 1968.
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S. Omata,
A numerical treatment of film motion with free boundary, Adv. Math. Sci. Appl., 14 (2004), 129-137.
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[11] |
S. Omata, A hyperbolic obstacle problem with an adhesion force, in Mathematics for Nonlinear Phenomena-Analysis and Computation (eds Y. Maekawa and S. Jimbo), Springer Proceedings in Mathematics and Statistics, 215 (2017), 261–269.
doi: 10.1007/978-3-319-66764-5_12. |
[12] |
K. Svadlenka and S. Omata,
Mathematical modeling of surface vibration with volume constraint and its analysis, Nonlinear Anal., 69 (2008), 3202-3212.
doi: 10.1016/j.na.2007.09.013. |
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H. Yoshiuchi, S. Omata, K. Svadlenka and K. Ohara,
Numerical solution of film vibration with obstacle, Adv. Math. Sci. Appl., 16 (2006), 33-43.
|
show all references
References:
[1] |
M. Bonafini, M. Novaga and G. Orlandi,
A variational scheme for hyperbolic obstacle problems, Nonlin. Anal., 188 (2019), 389-404.
doi: 10.1016/j.na.2019.06.008. |
[2] |
M. Bonafini, V.P.C. Le, M. Novaga and G. Orlandi, On the obstacle problem for fractional semilinear wave equations, Nonlinear Anal., 210 (2021), 112368.
doi: 10.1016/j.na.2021.112368. |
[3] |
X. Chen, S. Cui and A. Friedman,
A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior, Trans. Am. Math. Soc., 357 (2005), 4771-4804.
doi: 10.1090/S0002-9947-05-03784-0. |
[4] |
M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, 1983.
![]() ![]() |
[5] |
E. Ginder and K. Svadlenka, A variational approach to a constrained hyperbolic free boundary problem, Nonlinear Anal., 71 (2009), e1527–e1537.
doi: 10.1016/j.na.2009.01.228. |
[6] |
T. Iguchi and D. Lannes, Hyperbolic free boundary problems and applications to wave-structure interactions, Indiana Univ. Math. J., 70 (2021), 353–464, arXiv: 1806.07704. |
[7] |
K. Kikuchi,
Constructing a solution in time semidiscretization method to an equation of vibrating string with an obstacle, Nonlinear Anal., 71 (2009), 1227-1232.
|
[8] |
K. Kikuchi and S. Omata,
A free boundary problem for a one dimensional hyperbolic equation, Adv. Math. Sci. Appl., 9 (1999), 775-786.
|
[9] |
O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, 1$^st$ edition, Academic Press, 1968.
![]() ![]() |
[10] |
S. Omata,
A numerical treatment of film motion with free boundary, Adv. Math. Sci. Appl., 14 (2004), 129-137.
|
[11] |
S. Omata, A hyperbolic obstacle problem with an adhesion force, in Mathematics for Nonlinear Phenomena-Analysis and Computation (eds Y. Maekawa and S. Jimbo), Springer Proceedings in Mathematics and Statistics, 215 (2017), 261–269.
doi: 10.1007/978-3-319-66764-5_12. |
[12] |
K. Svadlenka and S. Omata,
Mathematical modeling of surface vibration with volume constraint and its analysis, Nonlinear Anal., 69 (2008), 3202-3212.
doi: 10.1016/j.na.2007.09.013. |
[13] |
H. Yoshiuchi, S. Omata, K. Svadlenka and K. Ohara,
Numerical solution of film vibration with obstacle, Adv. Math. Sci. Appl., 16 (2006), 33-43.
|





C-N | DMF | |
energy | conserved | decays |
free boundary condition | holds | holds |
high harmonic wave | preserved | decays |
including constraints | possible | possible |
phase shift | occurs | occurs |
C-N | DMF | |
energy | conserved | decays |
free boundary condition | holds | holds |
high harmonic wave | preserved | decays |
including constraints | possible | possible |
phase shift | occurs | occurs |
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