Numerical approximations of the relative rearrangement : the piecewise linear case. Application to some nonlocal problems
Jean-Michel Rakotoson; Maria Luisa Seoane
- Volume: 34, Issue: 2, page 477-499
- ISSN: 0764-583X
Access Full Article
topHow to cite
topRakotoson, Jean-Michel, and Seoane, Maria Luisa. "Numerical approximations of the relative rearrangement : the piecewise linear case. Application to some nonlocal problems." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 34.2 (2000): 477-499. <http://eudml.org/doc/193997>.
@article{Rakotoson2000,
author = {Rakotoson, Jean-Michel, Seoane, Maria Luisa},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {finite element approximation; nonlocal problems; fixed point method; plasma physics; relative rearrangement of functions},
language = {eng},
number = {2},
pages = {477-499},
publisher = {Dunod},
title = {Numerical approximations of the relative rearrangement : the piecewise linear case. Application to some nonlocal problems},
url = {http://eudml.org/doc/193997},
volume = {34},
year = {2000},
}
TY - JOUR
AU - Rakotoson, Jean-Michel
AU - Seoane, Maria Luisa
TI - Numerical approximations of the relative rearrangement : the piecewise linear case. Application to some nonlocal problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2000
PB - Dunod
VL - 34
IS - 2
SP - 477
EP - 499
LA - eng
KW - finite element approximation; nonlocal problems; fixed point method; plasma physics; relative rearrangement of functions
UR - http://eudml.org/doc/193997
ER -
References
top- [1] F. Almgren and E. Lieb, Symmetric rearrangement is sometimes continuous. J. Amer. Math. Soc. 2 (1989) 683-772. Zbl0688.46014MR1002633
- [2] E. Beretta and M. Vogelius, Symmetrie rearrangement is sometimes continuous, An inverse problem originating from Magnetohydrodynamics II. the case of the Grad-Shafranov equation. Indiana University Mathematics Journal 41 (1992) 1081-1117. Zbl0774.76096MR1206341
- [3] H. Berestycki and H. Brezis, On a free boundary problem arising in plasma physics. Nonlinear Anal. 4 (1980) 415-436. Zbl0437.35032MR574364
- [4] A. Bermúdez and C. Moreno, Duality methods for solving variational inequalities. Comp. and Math. Appl. 7 (1981) 43-58. Zbl0456.65036MR593554
- [5] A. Bermúdez and M. L. Seone, Numerical Solution of a Nonlocal Problem Arising in Plasma Physics. Mathematical and Computing Modelling. 27 (1998) 45-59. Zbl1185.76930MR1616788
- [6] J. Blum, Numerical Simulation and Optimal Control in Plasma Physics, Wiley, Gauthier-Villars (1989). Zbl0717.76009MR996236
- [7] J. Blum, T. Gallouet and J. Simon, Existence and Control of plasma equilibrium in a tokamak. SIAM J. Math. Anal. 17 (1986) 1158-1177. Zbl0614.35082MR853522
- [8] A. H. Boozer, Establishment of magnetic coordinates for given magnetic field. Phys. Fluids 25 (1982) 520-521. Zbl0501.76121
- [9] H. Brezis, Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, North-Holland (1973). Zbl0252.47055
- [10] G. Chiti, Rearrangements of functions and convergence in Orlicz spaces. Applicable Analysis 9 (1979). Zbl0424.46023MR536688
- [11] K. M. Chong and N. M. Rice, Equimesurable rearrangements of functions, Queen's University (1971). Zbl0275.46024MR372140
- [12] P. G. Ciarlet, Introduction to Numerical Linear Algebra and Optimization, Cambrigde University Press (1989). Zbl0672.65001MR1015713
- [13] J. M. Coron, The Continuity of the Rearrangement in W1,p (R). Annali della Scuola Normale Superiore di Pisa. Série IV 11 (1984) 57-85. Zbl0574.46021MR752580
- [14] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol I., Interscience Pub. (1953). Zbl0051.28802MR65391
- [15] J. I. Díaz, Modelos bidimensionales de equilibrio magnetohidrodinámico para Stellarators Formulación global de las ecuacion es diferenciales no lineales y de las condiciones de contorno, CIEMAT, Informe #1 (1991).
- [16] J. I. Díaz, Modelos bidimensionales de equilibrio magnetohidrodinámico para Stellarators. Resultados de existencia de soluciones, CIEMAT, Informe #2 (1992). MR1194210
- [17] J. I. Díaz, Modelos bidimensionales de equilibrio magnetohidrodinámico para Stellarators. Multiplicidad y dependencia de parámetros, CIEMAT, Informe #3 (1993).
- [18] J. I. Díaz and J. M. Rakotoson, On a two-dimensional stationary free boundary problem arising in the confinement of a plasma in a Stellarator. C.R. Acad. Sci. Paris Serie I 317 (1993) 353-358. Zbl0783.76106MR1235448
- [19] J. I. Díaz and J. M. Rakotoson, On a nonlocal stationary free boundary problem arising in the confinement of a plasma in a Stellarator geometry. Arch. Rat. Mech. Anal. 134 (1996) 53-95. Zbl0863.76092MR1392309
- [20] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland (1976). Zbl0322.90046MR463994
- [21] E. Fernández-Cara and C. Moreno, Critical Point Approximation Through Exact Regularization. Math. Comp. 50 (1988) 139-153. Zbl0636.65058MR917822
- [22] J. P. Freidberg, Ideal Magnetohydrodynamics. Plenum Press (1987).
- [23] A Friedman, Variational principles and free-boundary problems, John Wiley and Sons (1982). Zbl0564.49002MR679313
- [24] R. Glowinski, Numerical methods for non linear variational problems, Springer Verlag (1984). Zbl0536.65054MR737005
- [25] H. Grad, Mathematical problem arising in plasmas physics. Proc. Intern. Congr. Math. Nice (1970).
- [26] J. M. Greene and J. L. Johnson, Determination of Hydromagnetic Equilibria. Phys. Fluids 27 (1984) 2101-2120. Zbl0098.44106MR134607
- [27] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge University Press (1964). Zbl0047.05302JFM60.0169.01
- [28] T. C. Hender and B. A. Carreras, Equilibrium calculation for helical axis Stellarators. Phys. Fluids 27 (1984) 2101-2120. Zbl0559.76119
- [29] B. Heron and M. Sermange, Non convex methods for computing free boundary equilibria of axially symmetric plasmas, Rapport de Recherche, I.N.R.I.A. (1981). Zbl0495.65052MR666266
- [30] M. D. Kruskal and R. M. Kulsrud, Equilibrium of Magnetically Confined Plasma in a Toriod. Physics of Fluids 1, No. 4, (1958) 265-274. Zbl0098.22707MR112549
- [31] A. Marrocco and O. Pironneau, Optimum desing with lagrangian finite elements: desing of an electromagnet, Rapport de Recherche, I.N.R.I.A (1977).
- [32] F. Mignot and J. P. Puel, On a class of nonlinear problems with positive, increasmg, convex nonlinearity. Comm. Par. Diff. Eq. 5 (1980) 791-836. Zbl0456.35034
- [33] J. Mossin and J. M. Rakotoson, Isoperimetric inequalities in parabolic equations. Annali della Scuola Normale Superiore di Pisa. Séne IV 13, No. 1, (1986) 51-73. Zbl0652.35053MR863635
- [34] J. Mossino and R. Temam, Directional Derivative of the Increasing Rearrangement Mapping and Application to a Queer Differential Equation in Plasma Physics. Duke Mathematical Journal 48 (1981) 475-495. Zbl0476.35031MR630581
- [35] J. Mossino and R. Temam, Free boundary problems in plasma physics, review of results and new developments. Free Boundary Problems: theory and applications. Vol I-II. Proc. Montec atini Symposium (1981). A. Fasano and M. Primicerio Eds, Pitman (1983) 672-681. Zbl0512.76126
- [36] J. Mossino, Inégalités isopérmétriques et applications en physique, Hermann (1984). Zbl0537.35002MR733257
- [37] K. Miyamoto, Plasma Physics for Nuclear Fusion, The M.I.T. Press (1987).
- [38] J. F. Padial, EDPs no lineales originadas en plasmas de fusión y filtración en medios porosos, Thesis Doctoral, Universidad Complutense de Madrid (1995).
- [39] J. F. Padial, J. M. Rakotoson and L. Tello, Introduction to the monotone and relative rearrangements and applications, Rapport, Département de Mathématiques, Université de Poitiers (1993).
- [40] G. Pòlya and W. N. Szegö, Isopermetric inequalities in mathematical physics, Princenton Univ. Press (1951). Zbl0044.38301MR43486
- [41] J. P. Puel, A nonlinear eigenvalue problem with free boundary, C.R. Acad. Sci. Paris A 284 (1977) 861-863. Zbl0362.35024
- [42] J. M. Rakotoson, Some properties of the relative rearrangement. J. Math. Anal. Appl. 135 (1988) 488-500. Zbl0686.28003MR967224
- [43] J. M. Rakotoson, A differentiability result for the relative rearrangement. Diff. Int. Eq. 2 (1989) 363-377. Zbl0772.35018MR983687
- [44] J. M. Rakotoson, Relative rearrangement for highly nonlinear equations. Nonlinear Analysis. Theory, Meth. and Appl. 24 (1995) 493-507. Zbl0830.35036MR1315691
- [45] J. M. Rakotoson and M. L. Seoane, (in preparation).
- [46] J. M. Rakotoson, Galerkin approximations, strong continuity of the relative rearrangement map and application to plasma physics equations. Diff. Int. Eq. 12 (1999) 67-81. Zbl1005.76097MR1668537
- [47] J. M. Rakotoson and B. Simon, Relative rearrangement on a measure space. Application to the regularity of weighted monotone rearrangement. Part I-II. Appl. Math. Lett. 6 (1993) 75-78; 79-92. Zbl0781.49024MR1347759
- [48] J. M. Rakotoson and B. Simon, Relative rearrangement on a finite measure space. Application to weighted spaces and to P.D.E. Rev. R. Acad. Cienc. Exactas. Fís. Nat. (Esp ) 91 (1997) 33-45. Zbl0909.35028MR1646540
- [49] J. M. Rakotoson and R. Temam, A co-area formula with applications to monotone rearrangement and to regularity. Arch. Rational Mech. Anal. 109 (1991) 213-238. Zbl0735.49039MR1025171
- [50] R. T. Rockafellar, Convex Analysis, Princeton Unviversity Press (1970). Zbl0193.18401MR274683
- [51] V. D. Shafranov, On agneto-hydrodynamical equilibrium configurations. Soviet Physics JETP, 6 (1958) 5456-554. Zbl0081.21801MR93282
- [52] G. G. Talenti, Rearrangements of functions and and Partial Differential Equations. Nonlinear Diffusion Problems, A. Fasano and M. Primicerio Eds, Springer-Verlag (1986) 153-178. Zbl0607.65077MR877989
- [53] G. G. Talenti, Rearrangements and PDE. Inequalities, fifty years on from Hardy, Littlewood and Pòlya, W.N. Everitt Ed., Marcel Dekker Inc. (1991) 211-230. Zbl0733.35013MR1112579
- [54] G. G. Talenti, Assembling a rearrangement. Arch. Rat. Mech. Anal. 98 (1987) 85-93. Zbl0619.35113MR872748
- [55] R. Temam, A nonlinear eigenvalue problem equilibrium shape of a confined plasma. Arch. Rat. Mech. Anal. 65 (1975) 51-73. Zbl0328.35069MR412637
- [56] R. Temam, Remarks on a free boundary problem arising in plasma physics. Comm. Par. Diff. Eq. 2 (1977) 563-585. Zbl0355.35023MR602544
- [57] R. Temam, Monotone rearrangement of functions and the Grad-Mercier equation of plasma physics, Proc. Int. Conf. Recent Methods in Nonlinear Analysis and Applications, E. de Giogi and U. Mosco Eds (1978). Zbl0405.35066
- [58] R. Temam, Analyse Numérique, Presses Universitaires de France (1971). Zbl0187.11801
- [59] J. F. Toland, Duality in nonconvex optimization. J. Math. Appl. 66 (1978) 399-415. Zbl0403.90066MR515903
- [60] J. F. Toland, A Duality Principle for Non-convex Optimisation and the Calculus the Variations. Arch. Rat. Mech. Anal. 71 (1979) 41-61. Zbl0411.49012MR522706
- [61] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall Inc. (1962). Zbl0133.08602MR158502
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.