Une inégalité de Cheeger pour le spectre de Steklov
Jammes, Pierre
Annales de l'Institut Fourier, Tome 65 (2015), p. 1381-1385 / Harvested from Numdam

On montre une inégalité de Cheeger pour la première valeur propre de Steklov. Elle fait intervenir deux constantes isopérimétriques.

We prove a Cheeger inequality for the first positive Steklov eigenvalue. It involves two isoperimetric constants.

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/aif.2960
Classification:  35P15,  58J50
Mots clés: inégalité de Cheeger, spectre de Steklov
@article{AIF_2015__65_3_1381_0,
     author = {Jammes, Pierre},
     title = {Une in\'egalit\'e de Cheeger pour le spectre de Steklov},
     journal = {Annales de l'Institut Fourier},
     volume = {65},
     year = {2015},
     pages = {1381-1385},
     doi = {10.5802/aif.2960},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_2015__65_3_1381_0}
}
Jammes, Pierre. Une inégalité de Cheeger pour le spectre de Steklov. Annales de l'Institut Fourier, Tome 65 (2015) pp. 1381-1385. doi : 10.5802/aif.2960. http://gdmltest.u-ga.fr/item/AIF_2015__65_3_1381_0/

[1] Agranovich, M. S. On a mixed Poincaré-Steklov Type Spectral Problem in a Lipschitz Domain, Russ. J. Math. Phys., Tome 13 (2005) no. 3, pp. 239-244 | Article | MR 2262827 | Zbl 1162.35351

[2] Bandle, C. Isoperimetric inequalities and applications, Pitman, Monographs and Studies in Mathematics, Tome 7 (1980) | MR 572958 | Zbl 0519.53037

[3] Brooks, R. The bottom of the spectrum of a Riemannian covering, J. Reine Angew. Math., Tome 357 (1985), pp. 101-114 | MR 783536 | Zbl 0553.53027

[4] Buser, P. On Cheeger inequality λ 1 h 2 /4, Geometry of the Laplace operator, Amer. Math. Soc. (Proc. Sympos. Pure Math., XXXVI) (1980), pp. 29-77 | MR 573428 | Zbl 0432.58024

[5] Cheeger, J. A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969), Princeton Univ. Press (1970) | MR 402831 | Zbl 0212.44903

[6] Cheng, S.-Y.; Oden, K. Isoperimetric inequalities and the gap between the first and second eigenvalues of an Euclidean domain, J. Geom. Anal., Tome 7 (1997) no. 2, pp. 217-239 | Article | MR 1646764 | Zbl 0920.58054

[7] Colbois, B.; El Soufi, A.; Girouard, A. Isoperimetric control of the Steklov spectrum, J. Funct. Anal., Tome 261 (2011) no. 5, pp. 1384-1399 | Article | MR 2807105 | Zbl 1235.58020

[8] Escobar, J. F. The geometry of the first non-zero Stekloff eigenvalue, J. Funct. Anal., Tome 150 (1997) no. 2, pp. 544-556 | Article | MR 1479552 | Zbl 0888.58066

[9] Escobar, J. F. An isoperimetric Inequality and the first Steklov Eigenvalue, J. Funct. Anal., Tome 165 (1999) no. 1, pp. 101-116 | Article | MR 1696453 | Zbl 0935.58015

[10] Girouard, A.; Polterovich, I. On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues, Functional Analysis and its Applications, Tome 44 (2010) no. 2, pp. 106-117 | Article | MR 2681956 | Zbl 1217.35125

[11] Guérini, P. Prescription du spectre du laplacien de Hodge-de Rham, Ann. scient. Éc. norm. sup. (4), Tome 37 (2004) no. 2, pp. 270-303 | Numdam | MR 2061782 | Zbl 1068.58016

[12] Krantz, S. T.; Parks, H. R. Geometric Integration Theory, Birkäuser (2008) | MR 2427002 | Zbl 1149.28001

[13] Sylvester, J.; Uhlmann, G. The Dirichlet to Neumann map and applications, Inverse problems in partial differential equations (Arcata, CA, 1989), SIAM (1990), pp. 101-139 | MR 1046433 | Zbl 0713.35100

[14] Uhlmann, G. Electrical impedance tomography and Calderón’s problem, Inverse Problems, Tome 25 (2009) no. 12, pp. 123011, 39 | Article | Zbl 1181.35339