×

Inequalities for the lowest magnetic Neumann eigenvalue. (English) Zbl 1415.35213

Summary: We study the ground-state energy of the Neumann magnetic Laplacian on planar domains. For a constant magnetic field, we consider the question whether the disc maximizes this eigenvalue for fixed area. More generally, we discuss old and new bounds obtained on this problem.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Ashbaugh, MS, Isoperimetric and universal inequalities for eigenvalues, Lond. Math. Soc. Lecture Note Ser., 273, 95-139, (2000) · Zbl 0937.35114
[2] Bauman, P.; Phillips, D.; Tang, Q., Stable nucleation for the Ginzburg-Landau system with an applied magnetic field, Arch. Ration. Mech. Anal., 142, 1-43, (1998) · Zbl 0922.35157
[3] Bernoff, A.; Sternberg, P., Onset of superconductivity in decreasing fields for general domains, J. Math. Phys., 39, 1272-1284, (1998) · Zbl 1056.82523
[4] Brasco, L.; Philippis, G.; Velichkov, B., Faber-Krahn inequality in sharp quantitative form, Duke Math. J., 104, 1777-1831, (2015) · Zbl 1334.49149
[5] Bucur, D.: Personal communication (2017, March)
[6] Bucur, D.; Giacomini, A., Faber-Krahn inequalities for the Robin-Laplacian: a free discontinuity approach, Arch. Ration. Mech. Anal., 218, 757-824, (2015) · Zbl 1458.35286
[7] Colbois, B., El Soufi, A., Ilias, S., Savo, A.: Eigenvalues upper bounds for the magnetic operator (2017). ArXiv:1709.09482v1. 27 Sep 2017
[8] Colbois, B., Savo, A.: Eigenvalue bounds for the magnetic Laplacian (2016). ArXiv:1611.01930v1
[9] Colbois, B.; Savo, A., Lower bounds for the first eigenvalue of the magnetic Laplacian, J. Funct. Anal., 274, 2818-2845, (2018) · Zbl 1386.58013
[10] Ekholm, T.; Kovařík, H.; Portmann, F., Estimates for the lowest eigenvalue of magnetic Laplacians, J. Math. Anal. Appl., 439, 330-346, (2016) · Zbl 1386.35295
[11] Erdös, L., Rayleigh-type isoperimetric inequality with a homogeneous magnetic field, Calc. Var. PDE, 4, 283-292, (1996) · Zbl 0846.35094
[12] Fournais, S.; Helffer, B., Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian, Ann. Inst. Fourier, 56, 1-67, (2006) · Zbl 1097.47020
[13] Fournais, S., Helffer, B.: Spectral Methods in Surface Superconductivity. Progress in Nonlinear Differential Equations and Their Applications, vol. 77. Birkhäuser, Basel (2010) · Zbl 1256.35001
[14] Fournais, S.; Persson Sundqvist, M., Lack of diamagnetism and the Little-Parks effect, Commun. Math. Phys., 337, 191-224, (2015) · Zbl 1315.82027
[15] Freitas, P., Laugesen, R.S.: From Neumann to Steklov and beyond, via Robin: the Weinberger way. arXiv:1810.07461
[16] Helffer, B.; Morame, A., Magnetic bottles in connection with superconductivity, Journal of Functional Analysis, 185, 604-680, (2001) · Zbl 1078.81023
[17] Helffer, B.; Persson Sundqvist, M., On the semi-classical analysis of the Dirichlet Pauli operator, J. Math. Anal. Appl., 449, 138-153, (2017) · Zbl 1356.81132
[18] Helffer, B.; Persson Sundqvist, M., On the semi-classical analysis of the Dirichlet Pauli operator-the non simply connected case, Probl. Math. Anal. J. Math. Sci., 226, 4, (2017) · Zbl 1414.35134
[19] Howard, R.; Treibergs, A., A reverse isoperimetric inequality, stability and extremal theorems for plane curves with bounded curvature, Rocky Mt. J. Math., 25, 635-684, (1995) · Zbl 0909.53002
[20] Kawohl, B., Overdetermined problems and the p-Laplacian, Acta Math. Univ. Comen., 76, 77-83, (2007) · Zbl 1138.35069
[21] Krejcirik, D., Lotoreichik, V.: Optimisation of the lowest Robin eigenvalue in the exterior of a compact set, II: non-convex domains and higher dimensions. arXiv:1707.02269 · Zbl 1401.35223
[22] Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence (2001)
[23] Lu, K.; Pan, X., Eigenvalue problems of Ginzburg-Landau operator in bounded domains, J. Math. Phys., 40, 2647-2670, (1999) · Zbl 0943.35058
[24] Pankrashkin, K., An inequality for the maximum curvature through a geometric flow, Arch. Math., 105, 297-300, (2015) · Zbl 1325.53008
[25] Pankrashkin, K.; Popoff, N., Mean curvature bounds and eigenvalues of Robin Laplacians, Calc. Var., 54, 1947-1961, (2015) · Zbl 1327.35273
[26] Pestov, G.; Ionin, V., On the largest possible circle embedded in a given closed curve, Dokl. Akad. Nauk SSSR, 127, 1170-1172, (1959) · Zbl 0086.36104
[27] Polya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics. Princeton University Press, Princeton (1951) · Zbl 0044.38301
[28] Raymond, N., Sharp asymptotics for the Neumann Laplacian with variable magnetic field in dimension 2, Ann. Henri Poincaré, 10, 95-122, (2009) · Zbl 1210.81034
[29] Sperb, R.: Maximum Principles and Their Applications. Academic Press, New York (1981) · Zbl 0454.35001
[30] Szegö, G., Inequalities for certain eigenvalues of a membrane of given area, J. Ration. Mech. Anal., 3, 343-356, (1954) · Zbl 0055.08802
[31] Talenti, G., Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3, 697-718, (1976) · Zbl 0341.35031
[32] Berg, M.; Ferone, V.; Nitsch, C.; Trombetti, C., On Polya’s inequality for torsional rigidity and first Dirichlet eigenvalue, Integr. Equ. Oper. Theory, 86, 579-600, (2016) · Zbl 1388.49012
[33] Weinberger, HF, An isoperimetric inequality for the N-dimensional free membrane problem, J. Ration. Mech. Anal., 5, 633-636, (1956) · Zbl 0071.09902
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.