On Maxwell's and Poincaré's constants

Dirk Pauly

doi:10.3934/dcdss.2015.8.607

Article Contents

2015, Volume 8, Issue 3: 607-618. Doi: 10.3934/dcdss.2015.8.607

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On Maxwell's and Poincaré's constants

Dirk Pauly^1,

1.
Fakultät für Mathematik, Universität Duisburg-Essen, Campus Essen, Thea-Leymann-Str. 9, 45141 Essen

Received: November 2013

Revised: May 2014

Published: July 2015

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Abstract

We prove that for bounded and convex domains in three dimensions, the Maxwell constants are bounded from below and above by Friedrichs' and Poincaré's constants. In other words, the second Maxwell eigenvalues lie between the square roots of the second Neumann-Laplace and the first Dirichlet-Laplace eigenvalue.

Keywords:

Mathematics Subject Classification: 35A23, 35Q61, 35E10, 35F15, 35R45, 46E40, 53A45.

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References

[1]	C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci., 21 (1998), 823-864.doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B.
[2]	C. Amrouche, P. Ciarlet and P. J. Ciarlet, Weak vector and scalar potentials. Applications to Poincaré's theorem and Korn's inequality in Sobolev spaces with negative exponents, Anal. Appl. (Singap.), 8 (2010), 1-17.doi: 10.1142/S0219530510001497.
[3]	M. Bebendorf, A note on the Poincaré inequality for convex domains, Z. Anal. Anwendungen, 22 (2003), 751-756.doi: 10.4171/ZAA/1170.
[4]	M. Costabel, A coercive bilinear form for Maxwell's equations, J. Math. Anal. Appl., 157 (1991), 527-541.doi: 10.1016/0022-247X(91)90104-8.
[5]	N. Filonov, On an inequality for the eigenvalues of the Dirichlet and Neumann problems for the Laplace operator, St. Petersburg Math. J., 16 (2005), 413-416.doi: 10.1090/S1061-0022-05-00857-5.
[6]	V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer (Series in Computational Mathematics), Heidelberg, 1986.doi: 10.1007/978-3-642-61623-5.
[7]	V. Gol'dshtein, I. Mitrea and M. Mitrea, Hodge decompositions with mixed boundary conditions and applications to partial differential equations on Lipschitz manifolds, J. Math. Sci. (N.Y.), 172 (2011), 347-400.doi: 10.1007/s10958-010-0200-y.
[8]	P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman (Advanced Publishing Program), Boston, 1985.
[9]	T. Jakab, I. Mitrea and M. Mitrea, On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains, Indiana Univ. Math. J., 58 (2009), 2043-2071.doi: 10.1512/iumj.2009.58.3678.
[10]	F. Jochmann, A compactness result for vector fields with divergence and curl in $L^q(\Omega)$ involving mixed boundary conditions, Appl. Anal., 66 (1997), 189-203.doi: 10.1080/00036819708840581.
[11]	P. Kuhn and D. Pauly, Regularity results for generalized electro-magnetic problems, Analysis (Munich), 30 (2010), 225-252.doi: 10.1524/anly.2010.1024.
[12]	R. Leis, Zur Theorie elektromagnetischer Schwingungen in anisotropen inhomogenen Medien, Math. Z., 106 (1968), 213-224.doi: 10.1007/BF01110135.
[13]	R. Leis, Initial Boundary Value Problems in Mathematical Physics, Teubner, Stuttgart, 1986.doi: 10.1007/978-3-663-10649-4.
[14]	D. Pauly, Low frequency asymptotics for time-harmonic generalized Maxwell equations in nonsmooth exterior domains, Adv. Math. Sci. Appl., 16 (2006), 591-622.
[15]	D. Pauly, Generalized electro-magneto statics in nonsmooth exterior domains, Analysis (Munich), 27 (2007), 425-464.doi: 10.1524/anly.2007.27.4.425.
[16]	D. Pauly, Complete low frequency asymptotics for time-harmonic generalized Maxwell equations in nonsmooth exterior domains, Asymptot. Anal., 60 (2008), 125-184.
[17]	D. Pauly, Hodge-Helmholtz decompositions of weighted Sobolev spaces in irregular exterior domains with inhomogeneous and anisotropic media, Math. Methods Appl. Sci., 31 (2008), 1509-1543.doi: 10.1002/mma.982.
[18]	D. Pauly, On constants in Maxwell inequalities for bounded and convex domains, Zapiski POMI, 435 (2014), 46-54, & J. Math. Sci. (N.Y.).
[19]	L. Payne and H. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal., 5 (1960), 286-292.doi: 10.1007/BF00252910.
[20]	R. Picard, Randwertaufgaben der verallgemeinerten Potentialtheorie, Math. Methods Appl. Sci., 3 (1981), 218-228.doi: 10.1002/mma.1670030116.
[21]	R. Picard, On the boundary value problems of electro- and magnetostatics, Proc. Roy. Soc. Edinburgh Sect. A, 92 (1982), 165-174.doi: 10.1017/S0308210500020023.
[22]	R. Picard, An elementary proof for a compact imbedding result in generalized electromagnetic theory, Math. Z., 187 (1984), 151-164.doi: 10.1007/BF01161700.
[23]	R. Picard, Some decomposition theorems and their applications to non-linear potential theory and Hodge theory, Math. Methods Appl. Sci., 12 (1990), 35-53.doi: 10.1002/mma.1670120103.
[24]	R. Picard, N. Weck and K.-J. Witsch, Time-harmonic Maxwell equations in the exterior of perfectly conducting, irregular obstacles, Analysis (Munich), 21 (2001), 231-263.doi: 10.1524/anly.2001.21.3.231.
[25]	J. Saranen, On an inequality of Friedrichs, Math. Scand., 51 (1982), 310-322.
[26]	C. Weber, A local compactness theorem for Maxwell's equations, Math. Methods Appl. Sci., 2 (1980), 12-25.doi: 10.1002/mma.1670020103.
[27]	N. Weck, Maxwell's boundary value problems on Riemannian manifolds with nonsmooth boundaries, J. Math. Anal. Appl., 46 (1974), 410-437.doi: 10.1016/0022-247X(74)90250-9.
[28]	K.-J. Witsch, A remark on a compactness result in electromagnetic theory, Math. Methods Appl. Sci., 16 (1993), 123-129.doi: 10.1002/mma.1670160205.