Numerical approximations of the relative rearrangement : the piecewise linear case. Application to some nonlocal problems
ESAIM: Modélisation mathématique et analyse numérique, Tome 34 (2000) no. 2, pp. 477-499.
@article{M2AN_2000__34_2_477_0,
     author = {Rakotoson, Jean-Michel and Seoane, Maria Luisa},
     title = {Numerical approximations of the relative rearrangement : the piecewise linear case. {Application} to some nonlocal problems},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {477--499},
     publisher = {Dunod},
     address = {Paris},
     volume = {34},
     number = {2},
     year = {2000},
     mrnumber = {1765671},
     zbl = {0963.76052},
     language = {en},
     url = {http://archive.numdam.org/item/M2AN_2000__34_2_477_0/}
}
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: Modélisation mathématique et analyse numérique
PY  - 2000
SP  - 477
EP  - 499
VL  - 34
IS  - 2
PB  - Dunod
PP  - Paris
UR  - http://archive.numdam.org/item/M2AN_2000__34_2_477_0/
LA  - en
ID  - M2AN_2000__34_2_477_0
ER  - 
%0 Journal Article
%A Rakotoson, Jean-Michel
%A Seoane, Maria Luisa
%T Numerical approximations of the relative rearrangement : the piecewise linear case. Application to some nonlocal problems
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2000
%P 477-499
%V 34
%N 2
%I Dunod
%C Paris
%U http://archive.numdam.org/item/M2AN_2000__34_2_477_0/
%G en
%F M2AN_2000__34_2_477_0
Rakotoson, Jean-Michel; Seoane, Maria Luisa. Numerical approximations of the relative rearrangement : the piecewise linear case. Application to some nonlocal problems. ESAIM: Modélisation mathématique et analyse numérique, Tome 34 (2000) no. 2, pp. 477-499. http://archive.numdam.org/item/M2AN_2000__34_2_477_0/

[1] F. Almgren and E. Lieb, Symmetric rearrangement is sometimes continuous. J. Amer. Math. Soc. 2 (1989) 683-772. | MR | Zbl

[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. | MR | Zbl

[3] H. Berestycki and H. Brezis, On a free boundary problem arising in plasma physics. Nonlinear Anal. 4 (1980) 415-436. | MR | Zbl

[4] A. Bermúdez and C. Moreno, Duality methods for solving variational inequalities. Comp. and Math. Appl. 7 (1981) 43-58. | MR | Zbl

[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. | MR

[6] J. Blum, Numerical Simulation and Optimal Control in Plasma Physics, Wiley, Gauthier-Villars (1989). | MR | Zbl

[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. | MR | Zbl

[8] A. H. Boozer, Establishment of magnetic coordinates for given magnetic field. Phys. Fluids 25 (1982) 520-521. | Zbl

[9] H. Brezis, Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, North-Holland (1973). | Zbl

[10] G. Chiti, Rearrangements of functions and convergence in Orlicz spaces. Applicable Analysis 9 (1979). | MR | Zbl

[11] K. M. Chong and N. M. Rice, Equimesurable rearrangements of functions, Queen's University (1971). | MR | Zbl

[12] P. G. Ciarlet, Introduction to Numerical Linear Algebra and Optimization, Cambrigde University Press (1989). | MR | Zbl

[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. | EuDML | Numdam | MR | Zbl

[14] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol I., Interscience Pub. (1953). | MR | Zbl

[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). | MR

[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. | MR | Zbl

[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. | MR | Zbl

[20] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland (1976). | MR | Zbl

[21] E. Fernández-Cara and C. Moreno, Critical Point Approximation Through Exact Regularization. Math. Comp. 50 (1988) 139-153. | MR | Zbl

[22] J. P. Freidberg, Ideal Magnetohydrodynamics. Plenum Press (1987).

[23] A Friedman, Variational principles and free-boundary problems, John Wiley and Sons (1982). | MR | Zbl

[24] R. Glowinski, Numerical methods for non linear variational problems, Springer Verlag (1984). | MR

[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. | MR | Zbl

[27] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge University Press (1964). | JFM | MR | Zbl

[28] T. C. Hender and B. A. Carreras , Equilibrium calculation for helical axis Stellarators. Phys. Fluids 27 (1984) 2101-2120. | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[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. | Zbl

[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. | EuDML | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[36] J. Mossino, Inégalités isopérmétriques et applications en physique, Hermann (1984). | MR | Zbl

[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). | MR

[41] J. P. Puel, A nonlinear eigenvalue problem with free boundary, C.R. Acad. Sci. Paris A 284 (1977) 861-863. | Zbl

[42] J. M. Rakotoson, Some properties of the relative rearrangement. J. Math. Anal. Appl. 135 (1988) 488-500. | MR | Zbl

[43] J. M. Rakotoson, A differentiability result for the relative rearrangement. Diff. Int. Eq. 2 (1989) 363-377. | MR | Zbl

[44] J. M. Rakotoson, Relative rearrangement for highly nonlinear equations. Nonlinear Analysis. Theory, Meth. and Appl. 24 (1995) 493-507. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | EuDML | MR | Zbl

[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. | MR | Zbl

[50] R. T. Rockafellar, Convex Analysis, Princeton Unviversity Press (1970). | MR | Zbl

[51] V. D. Shafranov, On agneto-hydrodynamical equilibrium configurations. Soviet Physics JETP, 6 (1958) 5456-554. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[54] G. G. Talenti, Assembling a rearrangement. Arch. Rat. Mech. Anal. 98 (1987) 85-93. | MR | Zbl

[55] R. Temam, A nonlinear eigenvalue problem equilibrium shape of a confined plasma. Arch. Rat. Mech. Anal. 65 (1975) 51-73. | MR | Zbl

[56] R. Temam, Remarks on a free boundary problem arising in plasma physics. Comm. Par. Diff. Eq. 2 (1977) 563-585. | MR | Zbl

[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). | Zbl

[58] R. Temam, Analyse Numérique, Presses Universitaires de France (1971). | Zbl

[59] J. F. Toland, Duality in nonconvex optimization. J. Math. Appl. 66 (1978) 399-415. | MR | Zbl

[60] J. F. Toland, A Duality Principle for Non-convex Optimisation and the Calculus the Variations. Arch. Rat. Mech. Anal. 71 (1979) 41-61. | MR | Zbl

[61] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall Inc. (1962). | MR | Zbl