The shape of extremal functions for Poincaré-Sobolev-type inequalities in a ball

Journal of Functional Analysis

Volumn 237, Issue 1, 2006, Pages 194-223

  Girão, Pedro ^a   Weth, Tobias ^b  

^a INSTITUTO SUPERIOR TÉCNICO   (Portugal)

^b JUSTUS LIEBIG UNIVERSITY   (Germany)

Author keywords

Extremal functions; Inequalities; Poincar inequality; Sobolev inequality; Symmetry

Indexed keywords

EID: 33646508752     PISSN: 00221236     EISSN: 10960783     Source Type: Journal    
DOI: 10.1016/j.jfa.2006.01.001     Document Type: Article

References (39)

1. Adimurthi A., and Mancini G. The Neumann problem for elliptic equations with critical nonlinearity. A Tribute in Honour of G. Prodi. Sc. Norm. Super. di Pisa Quaderni (1991) 9-25
2. Aftalion A., and Pacella F. Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains. C. R. Math. Acad. Sci. Paris 339 (2004) 339-344
3. Bartsch T., and Weth T. A note on additional properties of sign changing solutions to superlinear elliptic equations. Topol. Methods Nonlinear Anal. 22 (2003) 1-14
4. Bartsch T., Weth T., and Willem M. Partial symmetry of least energy nodal solutions to some variational problems. J. Anal. Math. 96 (2005) 1-18
5. *-functions in n-space. Duke Math. J. 43 (1976) 245-268
6. Bañuelos R., and Burdzy K. On the "hot spots" conjecture of J. Rauch. J. Funct. Anal. 164 (1999) 1-33
7. Belloni M., and Kawohl B. A symmetry problem related to Wirtinger's and Poincaré's inequality. J. Differential Equations 156 (1999) 211-218
8. Brezis H., and Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88 (1983) 486-490
9. Brock A. Symmetry and monotonicity of solutions to some variational problems in cylinders and annuli. Electron. J. Differential Equations 2003 (2003) 1-20
10. Brock F., and Solynin P. An approach to symmetrization via polarization. Trans. Amer. Math. Soc. 352 (2000) 1759-1796
11. Buslaev A.P., Kondratiev V.A., and Nazarov A.I. On a family of extremum problems and the properties of an integral. Math. Notes 64 (1998) 719-725
12. Carlen E.A., and Loss M. On the minimization of symmetric functionals. Rev. Math. Phys. 6 (1994) 1011-1032
13. Casten R.G., and Holland C.J. Instability results for reaction diffusion equations with Neumann boundary conditions. J. Differential Equations 27 (1978) 266-273
14. Cherrier P. Meilleures constantes dans les inégalités relatives aux espaces de Sobolev. Bull. Sci. Math. 108 (1984) 227-262
15. Cordero-Erausquin D., Nazaret B., and Villani C. A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182 (2004) 307-332
16. Dacorogna B., Gangbo W., and Subía N. Sur une généralisation de l'inégalité de Wirtinger. Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992) 29-50
17. Dancer E.N. Real analyticity and non-degeneracy. Math. Ann. 325 (2003) 369-392
18. Egorov Y.V. On a Kondratiev problem. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 503-507
19. Gilbarg D., and Trudinger N.S. Elliptic Partial Differential Equations of Second Order. Grundlehren Math. Wiss. vol. 224 (1977), Springer, Berlin
20. Giusti E. Direct Methods in the Calculus of Variations (2003), World Sci. Publishing, River Edge, NJ
21. Kawohl B. Rearrangements and Convexity of Level Sets in PDE. Lecture Notes in Math. vol. 1150 (1985), Springer
22. Kawohl B. Symmetry results for functions yielding best constants in Sobolev-type inequalities. Discrete Contin. Dyn. Syst. 6 (2000) 683-690
23. Jerison D. Locating the first nodal line in the Neumann problem. Trans. Amer. Math. Soc. 352 (2000) 2301-2317
24. 2. Manuscr. Math. 84 (1994) 13-19
25. Maggi F., and Villani C. Balls have the worst best Sobolev inequalities. J. Geom. Anal. 15 1 (2005) 83-121
26. Maz'ya V.G. Sobolev Spaces (1985), Springer, Berlin
27. Müller-Pfeiffer E. On the number of nodal domains for elliptic differential operators. J. London Math. Soc. (2) 31 (1985) 91-100
28. Nazarov A.I. On an exact constant in the generalized Poincaré inequality. J. Math. Sci. (N.Y.) 112 (2002) 4029-4047
29. Nazarov A.I. On the "one-dimensionality" of the extremal in the Poincaré inequality in a square. J. Math. Sci. (N.Y.) 109 (2002) 1928-1939
30. Payne L.E., and Weinberger H.F. An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5 (1960) 286-292
31. Ren X.F., and Wei J.C. On a two-dimensional elliptic problem with large exponent in nonlinearity. Trans. Amer. Math. Soc. 343 (1994) 749-763
32. Simon B. Schrödinger semigroups. Bull. Amer. Math. Soc. (N.S.) 7 (1982) 447-526
33. Smets D., and Willem M. Partial symmetry and asymptotic behavior for some elliptic variational problems. Calc. Var. Partial Differential Equations 18 (2003) 37-75
34. Szegö G. Inequalities for certain eigenvalues of a membrane of given area. J. Ration. Mech. Anal. 3 (1954) 343-356
35. Wang X.J. Neumann problem of semilinear elliptic equations involving critical Sobolev exponents. J. Differential Equations 93 (1991) 283-310
36. Weinberger H.F. An isoperimetric inequality for the n-dimensional free membrane problem. J. Ration. Mech. Anal. 5 (1956) 633-636
37. Zhu M. Extremal functions of Sobolev-Poincaré inequality. Nonlinear Evolution Equations and Dynamical Systems (2003), World Sci. Publishing, River Edge, NJ 173-181
38. Zhu M. On the extremal functions of Sobolev-Poincaré inequality. Pacific J. Math. 214 (2004) 185-199
39. Zhu M. Sharp Poincaré-Sobolev inequalities and the shortest length of simple closed geodesics on a topological two-sphere. Commun. Contemp. Math. 6 (2004) 781-792