Isoperimetric Profile and Uniqueness for Neumann Problems
Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 1, pp. 81-100.
@article{AIHPC_2009__26_1_81_0,
     author = {Lucia, Marcello},
     title = {Isoperimetric {Profile} and {Uniqueness} for {Neumann} {Problems}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {81--100},
     publisher = {Elsevier},
     volume = {26},
     number = {1},
     year = {2009},
     doi = {10.1016/j.anihpc.2007.07.002},
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     zbl = {1159.58013},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2007.07.002/}
}
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Lucia, Marcello. Isoperimetric Profile and Uniqueness for Neumann Problems. Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 1, pp. 81-100. doi : 10.1016/j.anihpc.2007.07.002. http://www.numdam.org/articles/10.1016/j.anihpc.2007.07.002/

[1] Aubin T., Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. | MR | Zbl

[2] Bandle C., Isoperimetric Inequalities and Applications, Pitman, London, 1980. | MR | Zbl

[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. | MR | Zbl

[5] Burago Y. D., Zalgaller V. A., Geometric Inequalities, Grundlehren der Mathematischen Wissenschaften, vol. 285, Springer-Verlag, Berlin, 1988. | MR | Zbl

[6] Cabré X., Lucia M., Sanchón M., A Mean Field Equation on a Torus: One-Dimensional Symmetry of Solutions, Comm. Partial Differential Equations 30 (2005) 1315-1330. | MR | Zbl

[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. | MR | Zbl

[8] Chang S.-Y. A., Non-Linear Elliptic Equations in Conformal Geometry, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zurich, 2004. | MR | Zbl

[9] Chang S.-Y. A., Yang P. C., Conformal Deformation of Metrics on S 2 , J. Differential Geom. 27 (1988) 259-296. | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[12] Chavel I., Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. | MR | Zbl

[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. | MR | Zbl

[14] Cianchi A., On Relative Isoperimetric Inequalities in the Plane, Boll. Un. Mat. Ital. B (7) 3 (1989) 289-325. | MR | Zbl

[15] Cianchi A., Moser-Trudinger Inequalities Without Boundary Conditions and Isoperimetric Problems, Indiana Univ. Math. J. 54 (2005) 669-705. | MR | Zbl

[16] Evans L. C., Gariepy R. L., Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. | MR | Zbl

[17] Fontana L., Sharp Borderline Sobolev Inequalities on Compact Riemannian Manifolds, Comment. Math. Helv. 68 (1993) 415-454. | MR | Zbl

[18] Gelbaum B., Problems in Analysis, Problem Books in Mathematics, Springer-Verlag, New York-Berlin, 1982. | MR | Zbl

[19] Giusti E., Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. | MR | Zbl

[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. | Zbl

[21] Hong C.-W., A Best Constant and the Gaussian Curvature, Proc. Amer. Math. Soc. 97 (1986) 737-747. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[24] D. Horstmann, M. Lucia, Symmetry and uniqueness for some chemotaxis systems, preprint.

[25] Howards H., Hutchings M., Morgan F., The Isoperimetric Problem on Surfaces, Amer. Math. Monthly 106 (1999) 430-439. | MR | Zbl

[26] Jeanjean L., Toland J. F., Bounded Palais-Smale Mountain-Pass Sequences, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998) 23-28. | MR | Zbl

[27] Kazdan J. L., Warner F. W., Curvature Functions for Compact 2-Manifolds, Ann. of Math. (2) 99 (1974) 14-47. | MR | Zbl

[28] Kiessling M. K.-H., Statistical Mechanics of Classical Particles With Logarithmic Interactions, Comm. Pure Appl. Math. 46 (1993) 27-56. | MR | Zbl

[29] Kiessling M. K.-H., Statistical Mechanics Approach to Some Problems in Conformal Geometry, Phys. A 279 (2000) 353-368. | MR

[30] Lichnerowicz A., Géometrie Des Groupes De Transformations, Travaux et Recherches Mathematiques, vol. III, Dunod, Paris, 1958. | MR | Zbl

[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. | MR | Zbl

[32] Lin C.-S., Lucia M., Uniqueness of Solutions for a Mean Field Equation on Torus, J. Differential Equations 229 (2006) 172-185. | MR | Zbl

[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. | Numdam | Zbl

[34] Liouville J., Sur L’équation Aux Differences Partielles d 2 log λ du0ex0exdv±λ 2a 2 =0, J. Math. 18 (1853) 71-72.

[35] Lojasiewicz S., An Introduction to the Theory of Real Functions, John Wiley & Sons, Ltd., Chichester, 1988. | MR | Zbl

[36] Lucia M., Zhang L., A Priori Estimates and Uniqueness for Some Mean Field Equations, J. Differential Equations 217 (2005) 154-178. | MR | Zbl

[37] Lucia M., A Deformation Lemma With an Application to a Mean Field Equation, Topol. Methods Nonlinear Anal. 30 (2007) 113-138. | MR | Zbl

[38] Maz'Ya V. M., Sobolev Spaces, Springer-Verlag, Berlin, 1985. | MR | Zbl

[39] Moser J., A Sharp Form of an Inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/1971) 1077-1092. | MR | Zbl

[40] Obata M., Certain Conditions for a Riemannian Manifold to Be Isometric With a Sphere, J. Math. Soc. Japan 14 (1962) 333-340. | MR | Zbl

[41] Onofri E., On the Positivity of the Effective Action in a Theory of Random Surfaces, Commun. Math. Phys. 86 (1982) 321-326. | MR | Zbl

[42] Osserman R., The Isoperimetric Inequality, Bull. Amer. Math. Soc. 84 (1978) 1182-1238. | MR | Zbl

[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. | MR | Zbl

[44] Struwe M., Tarantello G., On Multivortex Solutions in Chern-Simons Gauge Theory, Boll. U.M.I. B (8) 1 (1998) 109-121. | MR | Zbl

[45] Suzuki T., Global Analysis for a Two-Dimensional Elliptic Eigenvalue Problem With the Exponential Nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992) 367-397. | Numdam | MR | Zbl

[46] Weinberger H. F., An Isoperimetric Inequality for the N-Dimensional Free Membrane Problem, J. Ration. Mech. Anal. 5 (1956) 633-636. | MR | Zbl

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