By Jean-Claude Nedelec

This publication is dedicated to the research of the acoustic wave equation and of the Maxwell process, the 2 commonest wave equations encountered in physics or in engineering. the most target is to provide a close research in their mathematical and actual homes. Wave equations are time established. despite the fact that, use of the Fourier trans shape reduces their research to that of harmonic structures: the harmonic Helmholtz equation, on the subject of the acoustic equation, or the har monic Maxwell method. This e-book concentrates at the research of those harmonic difficulties, that are a primary step towards the examine of extra common time-dependent difficulties. In each one case, we provide a mathematical environment that enables us to end up lifestyles and distinctiveness theorems. we've systematically selected using variational formulations on the topic of concerns of actual strength. We learn the critical representations of the recommendations. those representa tions yield a number of quintessential equations. We research their crucial homes. We introduce variational formulations for those indispensable equations, that are the foundation of so much numerical approximations. assorted elements of this ebook have been taught for no less than ten years by means of the writer on the post-graduate point at Ecole Poly method and the collage of Paris 6, to scholars in utilized arithmetic. the particular presentation has been demonstrated on them. I desire to thank them for his or her energetic and positive participation, which has been tremendous necessary, and that i say sorry for forcing them to benefit a few geometry of surfaces.

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**Extra info for Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems**

**Example text**

Its dual space is S'(r) (which is also (D'(r)). 2 ). Thus, the above definition makes sense when s < O. The dual space of HS(r) is the space H-S(f). We now have all the necessary tools to prove the trace theorem, which specifies the links between a function on ni and its trace on f. We will use the space D(n i ) of functions indefinitely differentiable up to the boundary of the set ni . Let n be the exterior normal to the surface f. 56) can be extended as a continuous linear mapping of the Hilbert space Hm (n i ) into respectively Hm-l/2(r), Hm-3/2(f), ..

The proof is straightforward, using the properties of the Fourier transform, and ordering the partial derivatives. 1 that if H m - 1 (1R3 -). 58) Proof A direct integration yields (I { + 1~12)m-l/2Iu(C oW 1 0 a~ = 2~ _00(1 + 1~12)m-l/2ii(~, z) a~ (~, z)dz. 59) in 1R2 and we bound the right-hand side using the Cauchy-Schwartz inequality. 60) • The continuity for the first trace is now proved by localization and transport using the diffeomorphisms 1>i. The case of the following traces is more complex.

107) j=-l We have Is Yim Is Yim(f) Yf (Qf) Yf (Qf)da = (f)da = bj. 109) In other words, the matrix a is unitary. Consider now the sum 1 F( f , iJ) L = Yim( f)Y;" (iJ). 110) m= - l The function F is a spherical harmonic in the two variables f and iJ separately. 107) and the fact that the matrix a is unitary shows that F(Qf,QiJ) = F(f,iJ). 111) shows that F depends only on the angle of these two vectors and precisely on their scalar product t. It is also an harmonic function invariant by the rotation around the axis X3.