Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-10-30T13:48:31.515Z Has data issue: false hasContentIssue false
  • English
  • Français

A remark on comparison results via symmetrization

Published online by Cambridge University Press:  14 November 2011

A. Alvino ,
P.L. Lions  and
G. Trombetti
A. Alvino
Affiliation:
Istituto di Matematica, Universita di Napoli, Via Mezzocannone, 8, 80134 Naples, Italy
P.L. Lions
Affiliation:
CEREMADE, Université Paris IX-Dauphine, Pl. de Lattre de Tassigny, 75775 Paris Cedex 16, France
G. Trombetti
Affiliation:
Istituto di Matematica, Universita di Napoli, Via Mezzocannone, 8, 80134 Naples, Italy
Get access Rights & Permissions [Opens in a new window]

Synopsis

In this paper, we study the converse of comparison results for solutions to linear second-order elliptic equations. Namely, in the inequalities proved by G. Talenti and others, we study the case of equality and prove that “equalities are achieved only in the spherical situation”. We also present some applications of these results to semilinear elliptic equations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 102 , Issue 1-2 , 1986 , pp. 37 - 48
Copyright
Copyright © Royal Society of Edinburgh 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Alvino, A. and Trombetti, G.. Sulle miglioni costanti per una classe di equazioni ellitiche degeneri. Ricerche Mat. 27 (1978), 413–428.Google Scholar
2Alvino, A. and Trombetti, G.. Equazioni ellitiche contermini di ordine inferiore e riordinamenti. Rend. Accad. Naz. Lincei. 66 (1979), 1–5.Google Scholar
3Alvino, A. and Trombetti, G.. Sulle migliori constanti di maggiorazione per una classe de equazioni ellitiche degeneri e non. Ricerche Mat. 30 (1981), 15–33.Google Scholar
4Aronsonn, G. and Talenti, G.. Estimating the integral of a function in terms of a distribution function of its gradient. Boll. Un. Mat. Ital. 18 (1981), 885–894.Google Scholar
5Bandle, C.. Bounds for the solutions of boundary value problems. J. Math. Appl. 54 (1976), 706–716.Google Scholar
6Bandle, C.. Symetrisation et problemes paraboliques. C. R. Acad. Sci. Paris 280 (1975), 1113–1115.Google Scholar
7Bandle, C.. On symmetrizations in parabolic equations. J. Analyse Math. 30 (1976), 98–112.CrossRefGoogle Scholar
8Bandle, C.. Estimates for the Green's functions of elliptic operators. SIAM J. Math. Anal. 9 (1978), 1126–1136.CrossRefGoogle Scholar
9Bandle, C.. Isoperimetric inequalities and applications (London: Pitman, 1980).Google Scholar
10Berestycki, H. and Lions, P. L.. Nonlinear scalar fields equations. Part 1. Arch. Rational Mech. Anal. 82 (1983), 313–346.CrossRefGoogle Scholar
11Berestycki, H. and Lions, P. L.. Nonlinear scalar fields equations. Part 2. Arch. Rational Mech. Anal. 82 (1983), 347–376.CrossRefGoogle Scholar
12Brézis, H. and Lieb, E. H.. Work in preparation; see also “Some vector fields equations” by Lieb, E. H., in Proceedings Alabama Conference, 1983.Google Scholar
13Chiti, G.. Norme di Orlicz delle soluzioni di una classe di equazioni ellitche. Boll. Un. Mat. Ital. 16 (1979), 178–185.Google Scholar
14Coleman, S., Glazer, V. and Martin, A.. Action minima among solutions to a class of Euclidean scalar field equations. Comm. Math. Phys. 58 (1978), 211–221.CrossRefGoogle Scholar
15Gidas, B., Ni, W. M. and Nirenberg, L.. Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), 209–243.CrossRefGoogle Scholar
16Hardy, G. H., Littlewood, J. E. and Polya, G.. Inequalities (Cambridge Univ. Press, 1964).Google Scholar
17Lions, P. L.. Quelques remarques sur la symétrisation de Schwarz. In Nonlinear Partial Differential Equations and their application, Collège de France Seminar, vol. 1 (London: Pitman, 1981).Google Scholar
18Lions, P. L.. Two geometrical properties of solutions of semilinear problems. Applicable Anal. 12 (1981), 267–272.CrossRefGoogle Scholar
19Lions, P. L.. The concentration-compactness principle in the Calculus of Variations. The locally compact case, Part 2. Ann. Inst. H. Poincarée. Analyse non linéeaire (1984), 223–283.CrossRefGoogle Scholar
20Lions, P. L.. The concentration-compactness principle in the Calculus of Variations. The limit case, Part 1. Riv. Mat. Iberoamericana 1 (1985), 145–201.CrossRefGoogle Scholar
21Polya, G. and Szegö, G.. Isoperimetric inequalities in Mathematical Physics (Ann. of Math. Studies, no. 27) (Princeton, 1951).Google Scholar
22Talenti, G.. Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa 3 (1976), 697–718.Google Scholar
23Talenti, G.. Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces. Ann. Mat. Pura Appl. 120 (1979), 159–184.CrossRefGoogle Scholar
24Weinberger, H. F.. Symmetrization in uniformly elliptic problems. In Studies in Mathematical Analysis (Stanford Univ. Press, 1962).Google Scholar
25Oklawder, E. T.. Interpolation, Espacios de Lorentz y theorema de Marcinkiewicz. (Cursos y seminarios de matematica, Univ. de Buenos-Aires).Google Scholar