Debye sources, Beltrami fields, and a complex structure on Maxwell fields. (English) Zbl 1342.35368
Authors’ abstract: The Debye source representation for solutions to the time-harmonic Maxwell equations is extended to bounded domains with finitely many smooth boundary components. A strong uniqueness result is proved for this representation. Natural complex structures are identified on the vector spaces of time-harmonic Maxwell fields. It is shown that these complex structures are uniformized by the Debye source representation, that is, represented by a fixed linear map on a fixed vector space, independent of the frequency. This complex structure relates time-harmonic Maxwell fields to constant-\(k\) Beltrami fields, i.e., solutions of the equation
\[
\nabla\times\mathbf E=k\mathbf E
\]
A family of self-adjoint boundary conditions are defined for the Beltrami operator. This leads to a proof of the existence of zero-flux, constant-\(k\), force-free Beltrami fields for any bounded region in \(\mathbb R^3\), as well as a constructive method to find them. The family of self-adjoint boundary value problems defines a new spectral invariant for bounded domains in \(\mathbb R^3\).
Reviewer: Aleksander Pankov (Baltimore)
MSC:
| 35Q61 | Maxwell equations |
References:
| [1] | Alpert, B. K.Hybrid Gauss‐trapezoidal quadrature rules. SIAM J. Sci. Comput.20 (1999), no. 5, 1551-1584 (electronic). doi: 10.1137/S1064827597325141 · Zbl 0933.41019 |
| [2] | Bauer, F.; Betancourt, O.; Garabedian, P.Magnetohydrodynamic equilibrium and stability of stellarators. Springer, New York, 1984. · Zbl 0545.76135 |
| [3] | Beyn, W.‐J.An integral method for solving nonlinear eigenvalue problems. Linear Algebra Appl.436 (2012), no. 10, 3839-3863. doi: 10.1016/j.laa.2011.03.030 · Zbl 1237.65035 |
| [4] | Cantarella, J.Topological structure of stable plasma flows. Thesis, University of Pennsylvania, 1999. |
| [5] | Cantarella, J.; DeTurck, D.; Gluck, H.; Teytel, M.The spectrum of the curl operator on spherically symmetric domains. Phys. Plasmas7 (2000), no. 7, 2766-2775. doi: 10.1063/1.874127 |
| [6] | Colton, D., and Kress, R.Integral equation methods in scattering theory. Krieger Publishing, Malabar, Fla., 1992. |
| [7] | Epstein, C. L.; Greengard, L.Debye sources and the numerical solution of the time harmonic Maxwell equations. Comm. Pure Appl. Math.63 (2010), no. 4, 413-463. doi: 10.1002/cpa.20313 · Zbl 1190.35215 |
| [8] | Epstein, C. L.; Greengard, L.; O’Neil, M.Debye sources and the numerical solution of the time harmonic Maxwell equations II. Comm. Pure Appl. Math.66 (2013), no. 5, 753-789. doi: 10.1002/cpa.21420 · Zbl 1291.35383 |
| [9] | Griffiths, P.; Harris, J.Principles of algebraic geometry. Pure and Applied Mathematics. Wiley, New York, 1978. · Zbl 0408.14001 |
| [10] | Hiptmair, R.; Kotiuga, P. R.; Tordeux, S.Self‐adjoint curl operators. Ann. Mat. Pura Appl. (4)191 (2012), no. 3, 431-457. doi: 10.1007/s10231-011-0189-y · Zbl 1275.47100 |
| [11] | Hudson, S. R.; Dewar, R. L.; Dennis, G.; Hole, M. J.; McGann, M.Non‐axisymmetric, multi‐region relaxed magnetohydrodynamic equilibrium solutions. Plasma Phys. Control. Fusion54 (2012), 014005. doi: 10.1088/0741-3335/54/1/014005 |
| [12] | Hudson, S. R.; Dewar, R. L.; Dennis, G.; Hole, M. J.; McGann, M.; vonNessi, G.; Lazer‐son, S.Computation of multi‐region relaxed magnetohydrodynamic equilibria. Phys. Plasmas19 (2012), 112502. doi: 10.1063/1.4765691 |
| [13] | Jackson, J. D.Classical electrodynamics. Third edition. Wiley, Hoboken, N.J., 1998. · Zbl 0913.00013 |
| [14] | Kress, R.A boundary integral equation method for a Neumann boundary problem for force‐free fields. J. Engrg. Math.15 (1981), no. 1, 29-48. doi: 10.1007/BF00039842 · Zbl 0477.73077 |
| [15] | Kress, R.On constant‐alpha force‐free fields in a torus. J. Engrg. Math.20 (1986), no. 4, 323-344. doi: 10.1007/BF00044609 · Zbl 0637.76121 |
| [16] | Marsh, G. E.Force‐free magnetic fields: solutions, topology and applications. World Scientific, Singapore, 1996. |
| [17] | Mü;ller, C.Foundations of the mathematical theory of electromagnetic waves. Die Grundlehren der mathematischen Wissenschaften, 155. Springer, New York‐Heidelberg1969. · Zbl 0181.57203 |
| [18] | Nédélec, J.‐C.Acoustic and electromagnetic equations. Integral representations for harmonic problems. Applied Mathematical Sciences, 144. Springer, New York, 2001. doi: 10.1007/978-1-4757-4393-7 · Zbl 0981.35002 |
| [19] | Papas, C. H.Theory of electromagnetic wave propagation. Dover, New York, 1988. |
| [20] | Picard, R.On a selfadjoint realization of curl and some of its applications. Ricerche Mat.47 (1998), no. 1, 153-180. · Zbl 0926.35104 |
| [21] | Picard, R.On a selfadjoint realization of curl in exterior domains. Math. Z.229 (1998), no. 2, 319-338. doi: 10.1007/PL00004656 · Zbl 0935.35029 |
| [22] | Rodríguez, R.; Venegas, P.Numerical approximation of the spectrum of the curl operator. Math. Comp.83 (2014), no. 286, 553-577. doi: 10.1090/S0025-5718-2013-02745-7 · Zbl 1288.65162 |
| [23] | Taylor, M. E.Partial differential equations I: basic theory. Applied Mathematical Sciences, 115. Springer, New York, 1996. · Zbl 0869.35002 |
| [24] | Young, P.; Hao, S.; Martinsson, P. G.A high‐order Nyström discretization scheme for boundary integral equations defined on rotationally symmetric surfaces. J. Comput. Phys.231 (2012), no. 11, 4142-4159. doi: 10.1016/j.jcp.2012.02.008 · Zbl 1250.65146 |
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