June  2015, 8(3): 607-618. doi: 10.3934/dcdss.2015.8.607

On Maxwell's and Poincaré's constants

1. 

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

Received  November 2013 Revised  May 2014 Published  October 2014

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.
Citation: Dirk Pauly. On Maxwell's and Poincaré's constants. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 607-618. doi: 10.3934/dcdss.2015.8.607
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.  doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B.  Google Scholar

[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.  doi: 10.1142/S0219530510001497.  Google Scholar

[3]

M. Bebendorf, A note on the Poincaré inequality for convex domains,, Z. Anal. Anwendungen, 22 (2003), 751.  doi: 10.4171/ZAA/1170.  Google Scholar

[4]

M. Costabel, A coercive bilinear form for Maxwell's equations,, J. Math. Anal. Appl., 157 (1991), 527.  doi: 10.1016/0022-247X(91)90104-8.  Google Scholar

[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.  doi: 10.1090/S1061-0022-05-00857-5.  Google Scholar

[6]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms,, Springer (Series in Computational Mathematics), (1986).  doi: 10.1007/978-3-642-61623-5.  Google Scholar

[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.  doi: 10.1007/s10958-010-0200-y.  Google Scholar

[8]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman (Advanced Publishing Program), (1985).   Google Scholar

[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.  doi: 10.1512/iumj.2009.58.3678.  Google Scholar

[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.  doi: 10.1080/00036819708840581.  Google Scholar

[11]

P. Kuhn and D. Pauly, Regularity results for generalized electro-magnetic problems,, Analysis (Munich), 30 (2010), 225.  doi: 10.1524/anly.2010.1024.  Google Scholar

[12]

R. Leis, Zur Theorie elektromagnetischer Schwingungen in anisotropen inhomogenen Medien,, Math. Z., 106 (1968), 213.  doi: 10.1007/BF01110135.  Google Scholar

[13]

R. Leis, Initial Boundary Value Problems in Mathematical Physics,, Teubner, (1986).  doi: 10.1007/978-3-663-10649-4.  Google Scholar

[14]

D. Pauly, Low frequency asymptotics for time-harmonic generalized Maxwell equations in nonsmooth exterior domains,, Adv. Math. Sci. Appl., 16 (2006), 591.   Google Scholar

[15]

D. Pauly, Generalized electro-magneto statics in nonsmooth exterior domains,, Analysis (Munich), 27 (2007), 425.  doi: 10.1524/anly.2007.27.4.425.  Google Scholar

[16]

D. Pauly, Complete low frequency asymptotics for time-harmonic generalized Maxwell equations in nonsmooth exterior domains,, Asymptot. Anal., 60 (2008), 125.   Google Scholar

[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.  doi: 10.1002/mma.982.  Google Scholar

[18]

D. Pauly, On constants in Maxwell inequalities for bounded and convex domains,, Zapiski POMI, 435 (2014), 46.   Google Scholar

[19]

L. Payne and H. Weinberger, An optimal Poincaré inequality for convex domains,, Arch. Rational Mech. Anal., 5 (1960), 286.  doi: 10.1007/BF00252910.  Google Scholar

[20]

R. Picard, Randwertaufgaben der verallgemeinerten Potentialtheorie,, Math. Methods Appl. Sci., 3 (1981), 218.  doi: 10.1002/mma.1670030116.  Google Scholar

[21]

R. Picard, On the boundary value problems of electro- and magnetostatics,, Proc. Roy. Soc. Edinburgh Sect. A, 92 (1982), 165.  doi: 10.1017/S0308210500020023.  Google Scholar

[22]

R. Picard, An elementary proof for a compact imbedding result in generalized electromagnetic theory,, Math. Z., 187 (1984), 151.  doi: 10.1007/BF01161700.  Google Scholar

[23]

R. Picard, Some decomposition theorems and their applications to non-linear potential theory and Hodge theory,, Math. Methods Appl. Sci., 12 (1990), 35.  doi: 10.1002/mma.1670120103.  Google Scholar

[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.  doi: 10.1524/anly.2001.21.3.231.  Google Scholar

[25]

J. Saranen, On an inequality of Friedrichs,, Math. Scand., 51 (1982), 310.   Google Scholar

[26]

C. Weber, A local compactness theorem for Maxwell's equations,, Math. Methods Appl. Sci., 2 (1980), 12.  doi: 10.1002/mma.1670020103.  Google Scholar

[27]

N. Weck, Maxwell's boundary value problems on Riemannian manifolds with nonsmooth boundaries,, J. Math. Anal. Appl., 46 (1974), 410.  doi: 10.1016/0022-247X(74)90250-9.  Google Scholar

[28]

K.-J. Witsch, A remark on a compactness result in electromagnetic theory,, Math. Methods Appl. Sci., 16 (1993), 123.  doi: 10.1002/mma.1670160205.  Google Scholar

show all references

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.  doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B.  Google Scholar

[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.  doi: 10.1142/S0219530510001497.  Google Scholar

[3]

M. Bebendorf, A note on the Poincaré inequality for convex domains,, Z. Anal. Anwendungen, 22 (2003), 751.  doi: 10.4171/ZAA/1170.  Google Scholar

[4]

M. Costabel, A coercive bilinear form for Maxwell's equations,, J. Math. Anal. Appl., 157 (1991), 527.  doi: 10.1016/0022-247X(91)90104-8.  Google Scholar

[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.  doi: 10.1090/S1061-0022-05-00857-5.  Google Scholar

[6]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms,, Springer (Series in Computational Mathematics), (1986).  doi: 10.1007/978-3-642-61623-5.  Google Scholar

[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.  doi: 10.1007/s10958-010-0200-y.  Google Scholar

[8]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman (Advanced Publishing Program), (1985).   Google Scholar

[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.  doi: 10.1512/iumj.2009.58.3678.  Google Scholar

[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.  doi: 10.1080/00036819708840581.  Google Scholar

[11]

P. Kuhn and D. Pauly, Regularity results for generalized electro-magnetic problems,, Analysis (Munich), 30 (2010), 225.  doi: 10.1524/anly.2010.1024.  Google Scholar

[12]

R. Leis, Zur Theorie elektromagnetischer Schwingungen in anisotropen inhomogenen Medien,, Math. Z., 106 (1968), 213.  doi: 10.1007/BF01110135.  Google Scholar

[13]

R. Leis, Initial Boundary Value Problems in Mathematical Physics,, Teubner, (1986).  doi: 10.1007/978-3-663-10649-4.  Google Scholar

[14]

D. Pauly, Low frequency asymptotics for time-harmonic generalized Maxwell equations in nonsmooth exterior domains,, Adv. Math. Sci. Appl., 16 (2006), 591.   Google Scholar

[15]

D. Pauly, Generalized electro-magneto statics in nonsmooth exterior domains,, Analysis (Munich), 27 (2007), 425.  doi: 10.1524/anly.2007.27.4.425.  Google Scholar

[16]

D. Pauly, Complete low frequency asymptotics for time-harmonic generalized Maxwell equations in nonsmooth exterior domains,, Asymptot. Anal., 60 (2008), 125.   Google Scholar

[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.  doi: 10.1002/mma.982.  Google Scholar

[18]

D. Pauly, On constants in Maxwell inequalities for bounded and convex domains,, Zapiski POMI, 435 (2014), 46.   Google Scholar

[19]

L. Payne and H. Weinberger, An optimal Poincaré inequality for convex domains,, Arch. Rational Mech. Anal., 5 (1960), 286.  doi: 10.1007/BF00252910.  Google Scholar

[20]

R. Picard, Randwertaufgaben der verallgemeinerten Potentialtheorie,, Math. Methods Appl. Sci., 3 (1981), 218.  doi: 10.1002/mma.1670030116.  Google Scholar

[21]

R. Picard, On the boundary value problems of electro- and magnetostatics,, Proc. Roy. Soc. Edinburgh Sect. A, 92 (1982), 165.  doi: 10.1017/S0308210500020023.  Google Scholar

[22]

R. Picard, An elementary proof for a compact imbedding result in generalized electromagnetic theory,, Math. Z., 187 (1984), 151.  doi: 10.1007/BF01161700.  Google Scholar

[23]

R. Picard, Some decomposition theorems and their applications to non-linear potential theory and Hodge theory,, Math. Methods Appl. Sci., 12 (1990), 35.  doi: 10.1002/mma.1670120103.  Google Scholar

[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.  doi: 10.1524/anly.2001.21.3.231.  Google Scholar

[25]

J. Saranen, On an inequality of Friedrichs,, Math. Scand., 51 (1982), 310.   Google Scholar

[26]

C. Weber, A local compactness theorem for Maxwell's equations,, Math. Methods Appl. Sci., 2 (1980), 12.  doi: 10.1002/mma.1670020103.  Google Scholar

[27]

N. Weck, Maxwell's boundary value problems on Riemannian manifolds with nonsmooth boundaries,, J. Math. Anal. Appl., 46 (1974), 410.  doi: 10.1016/0022-247X(74)90250-9.  Google Scholar

[28]

K.-J. Witsch, A remark on a compactness result in electromagnetic theory,, Math. Methods Appl. Sci., 16 (1993), 123.  doi: 10.1002/mma.1670160205.  Google Scholar

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