β Scribed by M. Cahen; S. Gutt
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We determine all smooth solutions of Maxwell's equation in Segal's universe; furthermore we show that the group of diffeomorphisms stabilizing the space of solutions of these equations is the conformal group of Segal's model. 0. Maxwell's equations for free fields in Minkowski space have the following 'local' invariance property: the Lie algebra of smooth vector fields on Minkowski space which stabilize the space of free Maxwell fields is the Lie algebra of conformal vector fields. This algebra, which is isomorphic to so(2,4), contains vector fields which are not complete. Hence, if one calls a conformal group a Lie group with Lie algebra isomorphic to so(2,4), there is no natural global action of a conformal group on Minkowski space.
Segal's model is a compact manifold on which the group 0(2,4) acts naturally; it can be characterized, up to covering by the properties of being connected and homogeneous under the action of a conformal group with a 'reasonable' isotropy (cf. proposition 1, w 2).
The space of smooth solutions of Maxwell's equations for free fields in Segal's model can be described in great detail using a Fourier decomposition of these fields. Theorem 2, w 3 and Theorem 2, w 4, which are the main results of this paper, give this description.
The relation of the conformal group of Segal's model with the space of solutions of Maxwell's equations is quite nice. Indeed the conformal group is the group of diffeomorphisms of Segal's model which stabilize the space of free Maxwell fields (cf. Theorems 1, w 5 and 2, w 5).
There exists a conformal imbedding of Minkowski space in Segal's model, the image being an open dense submanifold; this gives the possibility to compare free Maxwell fields in Segal's model and certain free Maxwell fields in Minkowski space. This comparison, which seems to have both mathematical and physical interest, is left to be done.
We believe that similar results could be obtained for any other zero rest mass free fields. As we do not know how to treat all these fields simultaneously, we did consider only the example of the Maxwell fields.
Finally it is our belief that Theorem 2, w 3 and Theorem 2, w 4 could be used to develop quantum theory in the framework of Segal's model. *Aspirant du F.N.R.S.
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## Abstract This paper is concerned with the structure of the singular and regular parts of the solution of timeβharmonic Maxwell's equations in polygonal plane domains and their effective numerical treatment. The asymptotic behaviour of the solution near corner points of the domain is studied by m
E,H obeys Maxwell's equations 1.4 , 1.5 , and 1.6 . The unknown Ε½ . w . functions , , E, H depend on t, x g 0, Ο± , where t, x denote the time 1 2 and space variable resp. β ; β«ήβ¬ 3 is a bounded Lipschitz-domain with Ρ¨ β s β« j β« , where β« , β« are disjoint subsets of Ρ¨ β. β« represents the D N D N D pe