Some properties of monotone rearrangement with applications to elliptic equations in cylinders

Mathematische Nachrichten

Volumn 266, Issue , 2004, Pages 3-19

  Berestycki, H ^a   Lachand Robert, T ^b  

^a ÉCOLE DES HAUTES ÉTUDES EN SCIENCES SOCIALES   (France)

^b UNIVERSITÉ DE SAVOIE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 4644290043     PISSN: 0025584X     EISSN: None     Source Type: Journal    
DOI: 10.1002/mana.200310139     Document Type: Article

References (30)

1. G. Alberti, Some remarks about a notion of rearrangement, Ann, Scuola Norm. Sup. Pisa (4) 29, 457-472 (2000).
2. F. J. Almgren Jr and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc. 2, 683-773 (1989).
3. F. J. Almgren Jr and E. H. Lieb, The (non)continuity of symmetric decreasing rearrangement, Symposia Mathematica, Cortona 30, 89-102 (1988) (see also: Differential Equations and Malhematical Physics (Birmingham, 1990), pp. 183200).
4. F. J. Almgren Jr and E. H. Lieb, The (non)continuity of symmetric decreasing rearrangement, Symposia Mathematica, Cortona 30, 89-102 (1988) (see also: Differential Equations and Malhematical Physics (Birmingham, 1990), pp. 183200).
5. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14, 349-381 (1973).
6. T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Diff. Geom. 11, 573-598 (1976).
7. C. Bandle, Isoperimetric Inequalities and Applications (Pitman, Boston, 1980).
8. H. Berestycki and L. Nirenberg, Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, J. Geometry and Physics 5, 237-275 (1988). (Volume in honor of I. M. Gelfand.)
9. H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil, de Mat. Nova Ser. 22, 1-37(1991).
10. F. Brock, One-dimensionality of solutions of semilinear elliptic problems in cylindrical domains, Z. Angew. Math. Mech. 80, No. 5, 345-350 (2000).
11. J. Brothers and W. P. Ziemer, Minimal rearrangements of Sobolev functions, J. Reine. Angew. Math. 384, 153-179 (1988).
12. 1/p, Geom. Funct. Anal. 7, No. 5, 823-860 (1997).
13. G. Carbou, Unicité et mmimalité des solutions d'une équation de Ginzburg-Landau, Ann. Inst. Henri Poincaré 12, No. 3, 305-318 (1995).
14. P. G. Ciarlet and P.-A. Raviart, Interpolation theory over curved elements, with applications to finite elements methods, Comp. Meth. Appl. Mech. Eng. 1, 217-249 (1972).
15. 1,p(ℝ), Ann. Scuola Norm. Sup. Pisa IV 11, 57-85 (1984).
16. I. Ekeland, Nonconvex minimisation problems, Bull. A. M. S. 1, 443-474 (1979).
17. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed. (Springer-Verlag, 1983).
18. G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities (Cambridge Univ. Press, Cambridge, 1958).
19. B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lectures Notes in Math. 1150 (Springer-Verlag, New York, 1985).
20. E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math. 57, 93-105 (1977).
21. P. L. Lions, Quelques remarques sur la symétrisation de Schwartz, Pitman Res. Notes in Math. 53, 308-319 (1981).
22. P. L. Lions, Two geometrical properties of solutions of semi-linear problems, Appl. Anal. 12, 267-272 (1981).
23. T. Lachand-Robert, Bifurcation and symmetry breaking in a non-linear elliptic equation in a cylinder, Nonlinear Anal. 22-23, 327-346 (1994).
24. T. Lachand-Robert, Bifurcation et rupture de symétrie pour un problème elliptique sur-linéaire dans un cylindre, Compte-Rendus Acad. Sci. Paris Sér. I 314, 1009-1014 (1992).
25. P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Expository lect. CMBS Reg. Conf. Univ. Miami (1984).
26. G. Talenti, Best constants in Sobolev inequality, Ann. Pura Appl. 110, 353-372 (1976).
27. G. Talenti, Elliptic equations and rearrangements, Ann. Sc. Norm. Sup. Pisa, s IV 3, 697-718 (1976).
28. M. Taylor, Estimate on the fundamental frequency of a drum, Duke Math. J. 46, 447-453 (1979).
29. R. E. L. Turner, Internal waves in fluids with rapidly varying density, Ann. Scuola Norm. Sup. Pisa, s. IV 8, 513-573 (1981).
30. R. E. L. Turner, private communication.