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Self-adjoint curl operators. (English) Zbl 1275.47100

This paper deals with the study of the exterior derivative as a symmetric unbounded operator on square integrable 1-forms on bounded domains in the three dimensional Euclidean space. In the first part, the authors summarize the properties of self-adjoint extensions through complete Lagrangian subspaces of certain factor spaces. Next, the authors apply these abstract results to trace spaces for 1-forms and the corresponding curl operator. They also discuss the properties of Lagrangian subspaces spawned by the Hodge decomposition of 1-forms on surfaces. In relationship with these properties, there are established concrete boundary conditions for self-adjoint curl operators induced by the complete Lagrangian subspaces. The final sections of this paper deal with the spectral properties of the classes of self-adjoint curls examined before and explore their relationships with curl-curl operators.

MSC:

47F05 General theory of partial differential operators
46N20 Applications of functional analysis to differential and integral equations
53D12 Lagrangian submanifolds; Maslov index
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[1] Amrouche, C.; Bernardi, C.; Dauge, M.; Girault, V., Vector potentials in three-dimensional nonsmooth domains, Math. Meth. Appl. Sci, 21, 823-864 (1998) · Zbl 0914.35094 · doi:10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B
[2] Arnold, D.; Falk, R.; Winther, R., Finite element exterior calculus, homological techniques, and applications, Acta Numerica, 15, 1-155 (2006) · Zbl 1185.65204 · doi:10.1017/S0962492906210018
[3] Arnold, V.; Khesin, B., Topological Methods in Hydrodynamics, vol. 125 of Applied Mathematical Sciences (1998), New York: Springer, New York · Zbl 0902.76001
[4] Birman, M.; Solomyak, M., L_2-theory of the Maxwell operator in arbitrary domains, Russian Math. Surv., 42, 75-96 (1987) · Zbl 0653.35075 · doi:10.1070/RM1987v042n06ABEH001505
[5] Bott, R.; Tu, L., Differential Forms in Algebraic Topology (1982), New York: Springer, New York · Zbl 0496.55001
[6] Buffa, A., Hodge decompositions on the boundary of a polyhedron: the multiconnected case, Math. Mod. Meth. Appl. Sci., 11, 1491-1504 (2001) · Zbl 1014.58002 · doi:10.1142/S0218202501001434
[7] Buffa, A.; Carstensen, C.; Funken, S.; Hackbusch, W.; Hoppe, R.; Monk, P., Traces theorems on non-smooth boundaries for functional spaces related to Maxwell equations: An overwiew, Cmputational Electromagnetics, vol. 28 of Lecture Notes in Computational Science and Engineering., 23-34 (2003), Berlin: Springer, Berlin · Zbl 1061.53008
[8] Buffa, A.; Ciarlet, P., On traces for functional spaces related to Maxwell’s equations. Part I: an integration by parts formula in Lipschitz polyhedra, Math. Meth. Appl. Sci., 24, 9-30 (2001) · Zbl 0998.46012 · doi:10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2
[9] Buffa, A.; Ciarlet, P., On traces for functional spaces related to Maxwell’s equations. Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications, Math. Meth. Appl. Sci., 24, 31-48 (2001) · Zbl 0976.46023 · doi:10.1002/1099-1476(20010110)24:1<31::AID-MMA193>3.0.CO;2-X
[10] Buffa, A.; Costabel, M.; Sheen, D., On traces for H(curl, Ω) in Lipschitz domains, J. Math. Anal. Appl, 276, 845-867 (2002) · Zbl 1106.35304 · doi:10.1016/S0022-247X(02)00455-9
[11] Cartan, H., Differentialformen (1974), Zürich: Bibliographisches Institut, Zürich
[12] Chandrasekhar, S.; Kendall, P., On force-free magnetic fields, Astrophys. J., 126, 457-460 (1957) · doi:10.1086/146413
[13] Crager, J.; Kotiuga, P., Cuts for the magnetic scalar potential in knotted geometries and force-free magnetic fields, IEEE Trans. Magn., 38, 1309-1312 (2002) · doi:10.1109/TMAG.2002.996334
[14] Dłotko, P.; Specogna, R.; Trevisan, F., Automatic generation of cuts on large-sized meshes for the T-Ω geometric eddy-current formulation, Comput. Methods Appl. Mech. Eng., 198, 3765-3781 (2009) · Zbl 1230.78038 · doi:10.1016/j.cma.2009.08.007
[15] Everitt, W.; Markus, L., Complex symplectic geometry with applications to ordinary differential equations, Trans. Am. Math. Soc., 351, 4905-4945 (1999) · Zbl 0936.34005 · doi:10.1090/S0002-9947-99-02418-6
[16] Everitt, W.; Markus, L., Elliptic Partial Differential Operators and Symplectic Algebra, no. 770 in Memoirs of the American Mathematical Society (2003), Providence: American Mathematical Society, Providence · Zbl 1021.35033
[17] Everitt, W.; Markus, L., Complex symplectic spaces and boundary value problems Bull, A. Math. Soc, 42, 461-500 (2005) · Zbl 1091.46002
[18] Flanders, H., Differential Forms with Applications to the Physical Sciences (1963), New York: Academic Press, New York · Zbl 0112.32003
[19] Gross, P.; Kotiuga, P., Electromagnetic Theory and Computation: A Topological Approach, vol. 48 of Mathematical Sciences Research Institute Publications (2004), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 1096.78001
[20] Hiptmair, R., Finite elements in computational electromagnetism, Acta Numerica, 11, 237-339 (2002) · Zbl 1123.78320 · doi:10.1017/S0962492902000041
[21] Jette, A., Force-free magnetic fields in resistive magnetohydrostatics, J. Math. Anal. Appl., 29, 109-122 (1970) · Zbl 0167.25904 · doi:10.1016/0022-247X(70)90104-6
[22] Kotiuga, P., On making cuts for magnetic scalar potentials in multiply connected regions, J. Appl. Phys., 61, 3916-3918 (1987) · doi:10.1063/1.338583
[23] Kotiuga, P., Topological duality in three-dimensional eddy-current problems and its role in computer-aided problem formulation, J. Appl. Phys., 9, 4717-4719 (1990) · doi:10.1063/1.344812
[24] Kotiuga, P., Topology-based inequalities and inverse problems for near force-free magnetic fields, IEEE. Trans. Magn., 40, 1108-1111 (2004) · doi:10.1109/TMAG.2004.824590
[25] Lundquist, S., Magneto-hydrostatic fields, Ark. Fysik, 2, 361-365 (1950) · Zbl 0041.59405
[26] Mcduff, D.; Salamon, D., Introduction to Symplectic Topology, Oxford Mathematical Monographs (1995), Oxford, UK: Oxford University Press, Oxford, UK · Zbl 0844.58029
[27] Mcleod, K.; Picard, R., A compact imbedding result on Lipschitz manifolds, Math. Annalen, 290, 491-508 (1991) · Zbl 0715.46012 · doi:10.1007/BF01459256
[28] Morrey, C., Multiple integrals in the calculus of variations. vol. 130 of Grundlehren der mathematischen Wissenschaften (1966), New York: Springer, New York · Zbl 0142.38701
[29] Paquet, L., Problemes mixtes pour le systeme de Maxwell, Ann. Fac. Sci. Toulouse, 4, 103-141 (1982) · Zbl 0529.58038 · doi:10.5802/afst.576
[30] Picard, R., Ein Randwertproblem in der Theorie kraftfreier Magnetfelder, Z. Angew. Math. Phys., 27, 169-180 (1976) · Zbl 0337.35060 · doi:10.1007/BF01590803
[31] Picard, R., An elementary proof for a compact imbedding result in generalized electromagnetic theory, Math. Z., 187, 151-161 (1984) · Zbl 0527.58038 · doi:10.1007/BF01161700
[32] Picard, R., Über kraftfreie Magnetfelder, Wissenschaftliche Zeitschrift der technischen Universität Dresden, 45, 14-17 (1996)
[33] Picard, R., On a selfadjoint realization of curl and some of its applications, Riceche di Matematica, XLVII, 153-180 (1998) · Zbl 0926.35104
[34] Picard, R., On a selfadjoint realization of curl in exterior domains, Mathematische Zeitschrift, 229, 319-338 (1998) · Zbl 0935.35029 · doi:10.1007/PL00004656
[35] Rudin, W., Functional Analysis (1973), NY: McGraw-Hill, NY · Zbl 0253.46001
[36] Salamon, D.; Wehrheim, K., Instanton Floer homology with Lagrangian boundary conditions, Geom. Topol., 12, 747-918 (2008) · Zbl 1166.57018 · doi:10.2140/gt.2008.12.747
[37] Weber, C., A local compactness theorem for Maxwell’s equations, Math. Meth. Appl. Sci., 2, 12-25 (1980) · Zbl 0432.35032 · doi:10.1002/mma.1670020103
[38] Weck, N., Traces of differential forms on Lipschitz boundaries, Analysis, Int. Math. J. Anal. Appl., 24, 147-169 (2004) · Zbl 1187.35023
[39] Weidmann, J., Linear Operators in Hilbert spaces, vol. 68 of Graduate Texts in Mathematics (1980), New York: Springer, New York · Zbl 0434.47001
[40] Yoshida, Z.; Giga, Y., Remarks on spectra of operator rot, Math. Z., 204, 235-245 (1990) · Zbl 0676.47012 · doi:10.1007/BF02570870
[41] Yosida, K., Functional Analysis, Classics in Mathematics (1980), Berlin: Springer, Berlin · Zbl 0152.32102
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