Isoperimetric inequalities for the first Neumann eigenvalue in Gauss space

Chiacchio, F.; Di Blasio, G.

Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 2, pp. 199-216.

We provide isoperimetric Szegö–Weinberger type inequalities for the first nontrivial Neumann eigenvalue ${\mu }_1\left(\Omega \right)$ in Gauss space, where Ω is a possibly unbounded domain of ${ℝ}^N$. Our main result consists in showing that among all sets Ω of ${ℝ}^N$ symmetric about the origin, having prescribed Gaussian measure, ${\mu }_1\left(\Omega \right)$ is maximum if and only if Ω is the Euclidean ball centered at the origin.

Classification: 35B45, 35P15, 35J70
Keywords: Neumann eigenvalues, Symmetrization, Isoperimetric estimates

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

[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 | Zbl

[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

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

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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

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

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

[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

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

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

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

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

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

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

[26] E.T. Kornhauser, I. Stakgold, A variational theorem for ${\nabla }^{2}u+\lambda u=0$ and its applications, J. Math. Phys. 31 (1952), 45-54 | MR | Zbl

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

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

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

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

[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

[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 | EuDML | MR

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