Isoperimetric profile and uniqueness for Neumann problems
Annales de l'I.H.P. Analyse non linéaire (2009)
- Volume: 26, Issue: 1, page 81-100
- ISSN: 0294-1449
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topLucia, Marcello. "Isoperimetric profile and uniqueness for Neumann problems." Annales de l'I.H.P. Analyse non linéaire 26.1 (2009): 81-100. <http://eudml.org/doc/78845>.
@article{Lucia2009,
author = {Lucia, Marcello},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {isoperimetric profile; Neumann eigenvalues; semilinear equations; mean field equations; uniqueness},
language = {eng},
number = {1},
pages = {81-100},
publisher = {Elsevier},
title = {Isoperimetric profile and uniqueness for Neumann problems},
url = {http://eudml.org/doc/78845},
volume = {26},
year = {2009},
}
TY - JOUR
AU - Lucia, Marcello
TI - Isoperimetric profile and uniqueness for Neumann problems
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2009
PB - Elsevier
VL - 26
IS - 1
SP - 81
EP - 100
LA - eng
KW - isoperimetric profile; Neumann eigenvalues; semilinear equations; mean field equations; uniqueness
UR - http://eudml.org/doc/78845
ER -
References
top- [1] Aubin T., Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. Zbl0896.53003MR1636569
- [2] Bandle C., Isoperimetric Inequalities and Applications, Pitman, London, 1980. Zbl0436.35063MR572958
- [3] V. Bayle, Propriétés de concavité du profil isopérimétrique et applications, Thèse de doctorat, Université Joseph-Fourier, 2003.
- [4] Brothers J.E., Ziemer W.P., Minimal rearrangements of Sobolev functions, J. Reine Angew. Math.384 (1988) 153-179. Zbl0633.46030MR929981
- [5] Burago Y.D., Zalgaller V.A., Geometric Inequalities, Grundlehren der Mathematischen Wissenschaften, vol. 285, Springer-Verlag, Berlin, 1988. Zbl0633.53002MR936419
- [6] Cabré X., Lucia M., Sanchón M., A mean field equation on a torus: one-dimensional symmetry of solutions, Comm. Partial Differential Equations30 (2005) 1315-1330. Zbl1115.35041MR2180306
- [7] Caglioti E., Lions P.L., Marchioro C., Pulvirenti M., A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, Commun. Math. Phys.143 (1992) 501-525. Zbl0745.76001MR1145596
- [8] Chang S.-Y.A., Non-linear Elliptic Equations in Conformal Geometry, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zurich, 2004. Zbl1064.53018MR2104700
- [9] Chang S.-Y.A., Yang P.C., Conformal deformation of metrics on , J. Differential Geom.27 (1988) 259-296. Zbl0649.53022MR925123
- [10] Chang S.-Y.A., Chen C.-C., Lin C.-S., Extremal functions for a mean field equation in two dimension, Lecture on Partial Differential Equations in honor of Louis Nirenberg's 75th birthday, International Press, 2003, (Chapter 4). Zbl1071.35040MR2055839
- [11] Chanillo S., Kiessling M., Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry, Commun. Math. Phys.160 (1994) 217-238. Zbl0821.35044MR1262195
- [12] Chavel I., Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Zbl0551.53001MR768584
- [13] Cheeger J., A lower bound for the smallest eigenvalue of the Laplacian, in: Problems in Analysis, Princeton Univ. Press, Princeton, NJ, 1970, pp. 195-199. Zbl0212.44903MR402831
- [14] Cianchi A., On relative isoperimetric inequalities in the plane, Boll. Un. Mat. Ital. B (7)3 (1989) 289-325. Zbl0674.49030MR997998
- [15] Cianchi A., Moser–Trudinger inequalities without boundary conditions and isoperimetric problems, Indiana Univ. Math. J.54 (2005) 669-705. Zbl1097.46016MR2151230
- [16] Evans L.C., Gariepy R.L., Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. Zbl0804.28001MR1158660
- [17] Fontana L., Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv.68 (1993) 415-454. Zbl0844.58082MR1236762
- [18] Gelbaum B., Problems in Analysis, Problem Books in Mathematics, Springer-Verlag, New York–Berlin, 1982. Zbl0494.00004MR676641
- [19] Giusti E., Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. Zbl0545.49018MR775682
- [20] Gromov M., Paul Lévy's Isoperimetric Inequality, Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics, vol. 152, Birkhäuser Boston, Inc., Boston, MA, 1999. Zbl0953.53002
- [21] Hong C.-W., A best constant and the Gaussian curvature, Proc. Amer. Math. Soc.97 (1986) 737-747. Zbl0603.58056MR845999
- [22] Horstmann D., From 1970 until present: the Keller–Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math.-Verein.105 (2003) 103-165. Zbl1071.35001MR2013508
- [23] Horstmann D., From 1970 until present: the Keller–Segel model in chemotaxis and its consequences II, Jahresber. Deutsch. Math.-Verein.106 (2004) 51-69. Zbl1072.35007MR2073515
- [24] D. Horstmann, M. Lucia, Symmetry and uniqueness for some chemotaxis systems, preprint. Zbl1219.92007
- [25] Howards H., Hutchings M., Morgan F., The isoperimetric problem on surfaces, Amer. Math. Monthly106 (1999) 430-439. Zbl1003.52011MR1699261
- [26] Jeanjean L., Toland J.F., Bounded Palais–Smale mountain-pass sequences, C. R. Acad. Sci. Paris Sér. I Math.327 (1998) 23-28. Zbl0996.47052MR1650239
- [27] Kazdan J.L., Warner F.W., Curvature functions for compact 2-manifolds, Ann. of Math. (2)99 (1974) 14-47. Zbl0273.53034MR343205
- [28] Kiessling M.K.-H., Statistical mechanics of classical particles with logarithmic interactions, Comm. Pure Appl. Math.46 (1993) 27-56. Zbl0811.76002MR1193342
- [29] Kiessling M.K.-H., Statistical mechanics approach to some problems in conformal geometry, Phys. A279 (2000) 353-368. MR1797146
- [30] Lichnerowicz A., Géometrie des groupes de transformations, Travaux et Recherches Mathematiques, vol. III, Dunod, Paris, 1958. Zbl0096.16001MR124009
- [31] Lin C.-S., Uniqueness of solutions to the mean field equations for the spherical Onsager vortex, Arch. Ration. Mech. Anal.153 (2000) 153-176. Zbl0968.35045MR1770683
- [32] Lin C.-S., Lucia M., Uniqueness of solutions for a mean field equation on torus, J. Differential Equations229 (2006) 172-185. Zbl1105.58005MR2265623
- [33] C.-S. Lin, M. Lucia, One-dimensional symmetry of periodic minimizers for a mean field equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., in press. Zbl1150.35036
- [34] Liouville J., Sur l’équation aux differences partielles , J. Math.18 (1853) 71-72.
- [35] Lojasiewicz S., An Introduction to the Theory of Real Functions, John Wiley & Sons, Ltd., Chichester, 1988. Zbl0653.26001MR952856
- [36] Lucia M., Zhang L., A priori estimates and uniqueness for some mean field equations, J. Differential Equations217 (2005) 154-178. Zbl1175.35053MR2170531
- [37] Lucia M., A deformation lemma with an application to a mean field equation, Topol. Methods Nonlinear Anal.30 (2007) 113-138. Zbl1135.58005MR2363657
- [38] Maz'ya V.M., Sobolev Spaces, Springer-Verlag, Berlin, 1985. Zbl0692.46023MR817985
- [39] Moser J., A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J.20 (1970/1971) 1077-1092. Zbl0213.13001MR301504
- [40] Obata M., Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan14 (1962) 333-340. Zbl0115.39302MR142086
- [41] Onofri E., On the positivity of the effective action in a theory of random surfaces, Commun. Math. Phys.86 (1982) 321-326. Zbl0506.47031MR677001
- [42] Osserman R., The isoperimetric inequality, Bull. Amer. Math. Soc.84 (1978) 1182-1238. Zbl0411.52006MR500557
- [43] Senba T., Suzuki T., Some structures of the solution set for a stationary system of chemotaxis, Adv. Math. Sci. Appl.10 (2000) 191-224. Zbl0999.35031MR1769174
- [44] Struwe M., Tarantello G., On multivortex solutions in Chern–Simons gauge theory, Boll. U.M.I. B (8)1 (1998) 109-121. Zbl0912.58046MR1619043
- [45] Suzuki T., Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire9 (1992) 367-397. Zbl0785.35045MR1186683
- [46] Weinberger H.F., An isoperimetric inequality for the N-dimensional free membrane problem, J. Ration. Mech. Anal.5 (1956) 633-636. Zbl0071.09902MR79286
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