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On Maxwell’s and Poincaré’s constants. (English) Zbl 1321.35226

The author studies lower and upper estimates of the Maxwell constant, i.e. the constant \(c_m\) such that (\( \operatorname{rot} = \operatorname{curl}\)) \[ \int_\Omega| v|^2 dx \leq c_m^2 \int_\Omega(|\operatorname{rot}v|^2 + \operatorname{div}v|^2) dx \] for any \(v\) with square integrable \(\operatorname{rot}v\) and \(\operatorname{div}v\) with either normal and tangential component of the velocity zero such that it is orthonormal to any function with zero \(\operatorname{rot}\) and \(\operatorname{curl}\) and zero the corresponding velocity component.
He proves that for any bounded convex \(\Omega \subset \mathbb R^3\) with Lipschitz boundary \[ c_{p,0} \leq c_m \leq c_p \leq \operatorname{diam} \Omega/\pi \] where \[ \int_\Omega| u|^2 dx \leq c_{p,0}^2 \int_\Omega|\nabla u|^2 dx \] for any \(u\) with square integrable gradient and zero trace on the boundary, and \[ \int_\Omega | u|^2 dx \leq c_p^2 \int_\Omega| \nabla u|^2 dx \] for any \(u\) with square integrable gradient and zero mean value over \(\Omega\).
The proof is based on the Maxwell estimates. It is closely connected with an estimate of the second Maxwell eigenvalue. The proof of the corresponding result in two space dimensions, which has been known before, is presented in the Appendix.

MSC:

35Q61 Maxwell equations
35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
35E10 Convexity properties of solutions to PDEs with constant coefficients
35F15 Boundary value problems for linear first-order PDEs
46E40 Spaces of vector- and operator-valued functions
53A45 Differential geometric aspects in vector and tensor analysis
78A30 Electro- and magnetostatics
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References:

[1] C. Amrouche, Vector potentials in three-dimensional non-smooth domains,, Math. Methods Appl. Sci., 21, 823 (1998) · Zbl 0914.35094
[2] C. Amrouche, Weak vector and scalar potentials. Applications to Poincaré’s theorem and Korn’s inequality in Sobolev spaces with negative exponents,, Anal. Appl. (Singap.), 8, 1 (2010) · Zbl 1190.49004
[3] M. Bebendorf, A note on the Poincaré inequality for convex domains,, Z. Anal. Anwendungen, 22, 751 (2003) · Zbl 1057.26011
[4] M. Costabel, A coercive bilinear form for Maxwell’s equations,, J. Math. Anal. Appl., 157, 527 (1991) · Zbl 0738.35095
[5] N. Filonov, On an inequality for the eigenvalues of the Dirichlet and Neumann problems for the Laplace operator,, St. Petersburg Math. J., 16, 413 (2005) · Zbl 1078.35081
[6] V. Girault, <em>Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms</em>,, Springer (Series in Computational Mathematics) (1986) · Zbl 0585.65077
[7] V. Gol’dshtein, Hodge decompositions with mixed boundary conditions and applications to partial differential equations on Lipschitz manifolds,, J. Math. Sci. (N.Y.), 172, 347 (2011) · Zbl 1230.58018
[8] P. Grisvard, <em>Elliptic Problems in Nonsmooth Domains</em>,, Pitman (Advanced Publishing Program) (1985) · Zbl 0695.35060
[9] T. Jakab, On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains,, Indiana Univ. Math. J., 58, 2043 (2009) · Zbl 1190.35056
[10] F. Jochmann, A compactness result for vector fields with divergence and curl in \(L^q(\Omega)\) involving mixed boundary conditions,, Appl. Anal., 66, 189 (1997) · Zbl 0886.35042
[11] P. Kuhn, Regularity results for generalized electro-magnetic problems,, Analysis (Munich), 30, 225 (2010) · Zbl 1225.35229
[12] R. Leis, Zur Theorie elektromagnetischer Schwingungen in anisotropen inhomogenen Medien,, Math. Z., 106, 213 (1968)
[13] R. Leis, <em>Initial Boundary Value Problems in Mathematical Physics</em>,, Teubner (1986) · Zbl 0599.35001
[14] D. Pauly, Low frequency asymptotics for time-harmonic generalized Maxwell equations in nonsmooth exterior domains,, Adv. Math. Sci. Appl., 16, 591 (2006) · Zbl 1119.35098
[15] D. Pauly, Generalized electro-magneto statics in nonsmooth exterior domains,, Analysis (Munich), 27, 425 (2007) · Zbl 1132.35487
[16] D. Pauly, Complete low frequency asymptotics for time-harmonic generalized Maxwell equations in nonsmooth exterior domains,, Asymptot. Anal., 60, 125 (2008) · Zbl 1179.35322
[17] D. Pauly, Hodge-Helmholtz decompositions of weighted Sobolev spaces in irregular exterior domains with inhomogeneous and anisotropic media,, Math. Methods Appl. Sci., 31, 1509 (2008) · Zbl 1159.58002
[18] D. Pauly, On constants in Maxwell inequalities for bounded and convex domains,, Zapiski POMI, 435, 46 (2014)
[19] L. Payne, An optimal Poincaré inequality for convex domains,, Arch. Rational Mech. Anal., 5, 286 (1960) · Zbl 0099.08402
[20] R. Picard, Randwertaufgaben der verallgemeinerten Potentialtheorie,, Math. Methods Appl. Sci., 3, 218 (1981) · Zbl 0466.31016
[21] R. Picard, On the boundary value problems of electro- and magnetostatics,, Proc. Roy. Soc. Edinburgh Sect. A, 92, 165 (1982) · Zbl 0516.35023
[22] R. Picard, An elementary proof for a compact imbedding result in generalized electromagnetic theory,, Math. Z., 187, 151 (1984) · Zbl 0527.58038
[23] R. Picard, Some decomposition theorems and their applications to non-linear potential theory and Hodge theory,, Math. Methods Appl. Sci., 12, 35 (1990) · Zbl 0702.35212
[24] R. Picard, Time-harmonic Maxwell equations in the exterior of perfectly conducting, irregular obstacles,, Analysis (Munich), 21, 231 (2001) · Zbl 1075.35085
[25] J. Saranen, On an inequality of Friedrichs,, Math. Scand., 51, 310 (1982) · Zbl 0524.35100
[26] C. Weber, A local compactness theorem for Maxwell’s equations,, Math. Methods Appl. Sci., 2, 12 (1980) · Zbl 0432.35032
[27] N. Weck, Maxwell’s boundary value problems on Riemannian manifolds with nonsmooth boundaries,, J. Math. Anal. Appl., 46, 410 (1974) · Zbl 0281.35022
[28] K.-J. Witsch, A remark on a compactness result in electromagnetic theory,, Math. Methods Appl. Sci., 16, 123 (1993) · Zbl 0778.35105
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