Isoperimetric inequalities for the first Neumann eigenvalue in Gauss space
Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 2, p. 199-216
We provide isoperimetric Szegö–Weinberger type inequalities for the first nontrivial Neumann eigenvalue μ 1 (Ω) in Gauss space, where Ω is a possibly unbounded domain of N . Our main result consists in showing that among all sets Ω of N symmetric about the origin, having prescribed Gaussian measure, μ 1 (Ω) is maximum if and only if Ω is the Euclidean ball centered at the origin.
DOI : https://doi.org/10.1016/j.anihpc.2011.10.002
Classification:  35B45,  35P15,  35J70
Keywords: Neumann eigenvalues, Symmetrization, Isoperimetric estimates
@article{AIHPC_2012__29_2_199_0,
     author = {Chiacchio, F. and Di Blasio, G.},
     title = {Isoperimetric inequalities for the first Neumann eigenvalue in Gauss space},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {29},
     number = {2},
     year = {2012},
     pages = {199-216},
     doi = {10.1016/j.anihpc.2011.10.002},
     zbl = {1238.35072},
     mrnumber = {2901194},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2012__29_2_199_0}
}
Chiacchio, F.; Di Blasio, G. Isoperimetric inequalities for the first Neumann eigenvalue in Gauss space. Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 2, pp. 199-216. doi : 10.1016/j.anihpc.2011.10.002. http://www.numdam.org/item/AIHPC_2012__29_2_199_0/

[1] R.A. Adams, General logarithmic Sobolev inequalities and Orlicz imbeddings, J. Funct. Anal. 34 (1979), 292-303 | MR 552707 | Zbl 0425.46020

[2] M.S. Ashbaugh, Isoperimetric and universal inequalities for eigenvalues, Spectral Theory and Geometry, Edinburgh, 1998, London Math. Soc. Lecture Note Ser. vol. 273, Cambridge Univ. Press, Cambridge (1999), 95-139 | MR 1736867 | Zbl 0937.35114

[3] M.S. Ashbaugh, R. Benguria, Sharp upper bound to the first nonzero Neumann eigenvalue for bounded domains in spaces of constant curvature, J. Lond. Math. Soc. (2) 52 no. 2 (1995), 402-416 | Zbl 0835.58037

[4] C. Bandle, Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics vol. 7, Pitman (Advanced Publishing Program), Boston, MA, London (1980) | MR 572958 | Zbl 0436.35063

[5] R.D. Benguria, H. Linde, A second eigenvalue bound for the Dirichlet Schrödinger operator, Comm. Math. Phys. 267 no. 3 (2006), 741-755 | MR 2249789 | Zbl 1126.35035

[6] M.F. Betta, F. Chiacchio, A. Ferone, Isoperimetric estimates for the first eigenfunction of a class of linear elliptic problems, Z. Angew. Math. Phys. 58 no. 1 (2007), 37-52 | MR 2293101 | Zbl 1161.35007

[7] B. Brandolini, F. Chiacchio, C. Trombetti, Hardy type inequalities and Gaussian measure, Commun. Pure Appl. Anal. 6 no. 2 (2007), 411-428 | MR 2289828 | Zbl 1154.46307

[8] V.I. Bogachev, Gaussian Measures, Mathematical Surveys and Monographs vol. 62, American Mathematical Society, Providence, RI (1998) | MR 1642391 | Zbl 0938.28010

[9] C. Borell, The Brunn–Minkowski inequality in Gauss space, Invent. Math. 30 (1975), 207-211 | MR 399402 | Zbl 0292.60004

[10] E. Carlen, C. Kerce, On the cases of equality in Bobkovʼs inequality and Gaussian rearrangement, Calc. Var. Partial Differential Equations 13 (2001), 1-18 | MR 1854254 | Zbl 1009.49029

[11] I. Chavel, Lowest-eigenvalue inequalities, Geometry of the Laplace Operator, Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979, Proc. Sympos. Pure Math. vol. XXXVI, American Mathematical Society, Providence, RI (1980), 79-89 | MR 573429 | Zbl 0467.58025

[12] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, New York (2001) | MR 768584

[13] D. Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl. 52 (1975), 189-289 | MR 385666 | Zbl 0317.49005

[14] K.M. Chong, N.M. Rice, Equimeasurable Rearrangements of Functions, Queenʼs Papers in Pure and Applied Mathematics vol. 28, Queenʼs University (1971) | MR 372140 | Zbl 0275.46024

[15] A. Cianchi, L. Pick, Optimal Gaussian Sobolev embeddings, J. Funct. Anal. 256 no. 11 (2009), 3588-3642 | MR 2514054 | Zbl 1203.46019

[16] R. Courant, D. Hilbert, Methods of Mathematical Physics, vols. I and II, Interscience Publishers, New York, London (1962) | MR 140802 | Zbl 0729.35001

[17] A. Ehrhard, Symmétrisation dans lʼspace de Gauss, Math. Scand. 53 (1983), 281-301 | MR 745081

[18] A. Ehrhard, Inégalités isopérimetriques et intégrales de Dirichlet gaussiennes, Ann. Sci. Ec. Norm. Super. 17 (1984), 317-332 | Numdam | MR 760680 | Zbl 0546.49020

[19] F. Feo, M.R. Posteraro, Logarithmic Sobolev Trace inequalities, preprint, No. 34, 2010, Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”. | MR 3119800

[20] S. Flügge, Practical Quantum Mechanics, I, II, Die Grundlehren der mathematischen Wissenschaften Bände 177 und 178, Springer-Verlag, Berlin, New York (1971) | MR 366247

[21] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1976), 1061-1083 | MR 420249 | Zbl 0318.46049

[22] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser, Basel (2006) | MR 2251558 | Zbl 1109.35081

[23] A. Henrot, M. Pierre, Variation et optimisation de formes. Une analyse géométrique, Mathématiques & Applications (Berlin) vol. 48, Springer-Verlag, Berlin (2005) | MR 2512810

[24] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics vol. 1150, Springer-Verlag, New York (1985) | MR 810619 | Zbl 0593.35002

[25] S. Kesavan, Symmetrization & Applications, Series in Analysis vol. 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2006) | MR 2238193 | Zbl 1110.35002

[26] E.T. Kornhauser, I. Stakgold, A variational theorem for 2 u+λu=0 and its applications, J. Math. Phys. 31 (1952), 45-54 | MR 47236 | Zbl 0046.32303

[27] R.S. Laugesen, B.A. Siudeja, Maximizing Neumann fundamental tones of triangles, J. Math. Phys. 50 no. 11 (2009), 112903 | MR 2567204 | Zbl 1304.35453

[28] C. Müller, Spherical Harmonics, Lecture Notes in Mathematics vol. 17, Springer-Verlag, Berlin, New York (1966) | MR 199449 | Zbl 0138.05101

[29] E. Pelliccia, G. Talenti, A proof of a logarithmic Sobolev inequality, Calc. Var. Partial Differential Equations 1 no. 3 (1993), 237-242 | MR 1261545 | Zbl 0796.49013

[30] H.F. Weinberger, An isoperimetric inequality for the N-dimensional free membrane problem, J. Rational Mech. Anal. 5 (1956), 633-636 | MR 79286 | Zbl 0071.09902

[31] J.M. Rakotoson, B. Simon, Relative rearrangement on a measure space application to the regularity of weighted monotone rearrangement, I, II, Appl. Math. Lett. 6 (1993), 75-78 | MR 1347759

[32] V.N. Sudakov, B.S. TsirelʼSon, Extremal properties of half-spaces for spherically invariant measures, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 (1974), 14-24 | MR 365680

[33] G. Szegö, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal. 3 (1954), 343-356 | MR 61749 | Zbl 0055.08802