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Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel-Kontorova models and homogenization of Hamilton-Jacobi equations. (English) Zbl 1037.65054
The paper is concerned with a min-plus integral eigenvalue problem. The author proves that the unique eigenvalue of this problem depends continuously on parameters involved in the kernel defining the problem. A convergence analysis is given for the numerical method introduced by W. Chou and R. B. Griffiths [Ground states of one-dimensional systems using effective potentials. Phys. Rev. B 34, 6219–6234 (1986)] to compute this eigenvalue. The author illustrates obtained results in two contexts: Frenkel-Kontorova models in solid-state physics, and homogenization of Hamilton-Jacobi equations.

MSC:
65J10 Numerical solutions to equations with linear operators
65Z05 Applications to the sciences
45C05 Eigenvalue problems for integral equations
65R20 Numerical methods for integral equations
49J20 Existence theories for optimal control problems involving partial differential equations
47A10 Spectrum, resolvent
65K10 Numerical optimization and variational techniques
Software:
Scilab
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References:
[1] S. Aubry , The new concept of transitions by breaking of analyticity in a crystallographic model , in Solitons and Condensed Matter Physics, A.R. Bishop and T. Schneider, Eds., Springer-Verlag, Berlin ( 1978 ) 264 - 277 .
[2] S. Aubry , The twist map, the extended Frenkel-Kontorova model and the devil’s staircase . Physica D 7 ( 1983 ) 240 - 258 . Zbl 0559.58013 · Zbl 0559.58013
[3] N. Bacaër , Min-plus spectral theory and travelling fronts in combustion , in Proceedings of the Workshop on Max-Plus Algebras, Prague, August ( 2001 ). Submitted to S. Gaubert, Ed., Elsevier Science, Amsterdam.
[4] N. Bacaër , Can one use Scilab’s max-plus toolbox to solve eikonal equations ?, in Proceedings of the Workshop on Max-Plus Algebras, Prague, August ( 2001 ). Submitted to S. Gaubert, Ed., Elsevier Science, Amsterdam.
[5] F. Baccelli , G.J. Olsder , J.P. Quadrat and G. Cohen , Synchronization and Linearity . Wiley, Chichester ( 1992 ). MR 1204266 | Zbl 0824.93003 · Zbl 0824.93003
[6] W. Chou and R.B. Griffiths , Ground states of one-dimensional systems using effective potentials . Phys. Rev. B 34 ( 1986 ) 6219 - 6234 .
[7] W. Chou and R.J. Duffin , An additive eigenvalue problem of physics related to linear programming . Adv. in Appl. Math. 8 ( 1987 ) 486 - 498 . Zbl 0639.65033 · Zbl 0639.65033
[8] J. Cochet-Terrasson , G. Cohen , S. Gaubert , M. Mc Gettrick and J.P. Quadrat , Numerical computation of spectral elements in max-plus algebra . http://amadeus.inria.fr/gaubert/HOWARD.html [9] M. Concordel , Periodic homogenization of Hamilton-Jacobi equations: additive eigenvalues and variational formula . Indiana Univ. Math. J. 45 ( 1996 ) 1095 - 1117 . Zbl 0871.49025 · Zbl 0871.49025
[9] P.I. Dudnikov and S.N. Samborskii , Endomorphisms of semimodules over semirings with idempotent operation . Math. USSR-Izv. 38 ( 1992 ) 91 - 105 . Zbl 0746.16034 · Zbl 0746.16034
[10] L.C. Evans and D. Gomes , Effective Hamiltonians and averaging for Hamiltonian dynamics I . Arch. Rational Mech. Anal. 157 ( 2001 ) 1 - 33 . Zbl 0986.37056 · Zbl 0986.37056
[11] J.S. Golan , The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science . Longman Scientific & Technical, Harlow ( 1992 ). MR 1163371 | Zbl 0780.16036 · Zbl 0780.16036
[12] R.B. Griffiths , Frenkel-Kontorova models of commensurate-incommensurate phase transitions , in Fundamental Problems in Statistical Mechanics. VII, H. van Beijeren, Ed., North-Holland, Amsterdam ( 1990 ) 69 - 110 .
[13] V.N. Kolokoltsov and V.P. Maslov , Idempotent Analysis and its Applications . Kluwer Academic Publishers, Dordrecht, The Netherlands ( 1997 ). MR 1447629 | Zbl 0941.93001 · Zbl 0941.93001
[14] G. Namah and J.M. Roquejoffre , The “hump” effect in solid propellant combustion . Interfaces Free Bound 2 ( 2000 ) 449 - 467 . Zbl 0967.35156 · Zbl 0967.35156
[15] S.J. Sheu and A.D. Wentzell , On the solutions of the equation arising from the singular limit of some eigen problems , in Stochastic Analysis, Control, Optimization and Applications, W.M. McEneaney et al., Eds., Birkhäuser, Boston ( 1999 ) 135 - 150 . Zbl 0920.49015 · Zbl 0920.49015
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