By Jean-Claude Nedelec

This e-book is dedicated to the learn of the acoustic wave equation and of the Maxwell approach, the 2 commonest wave equations encountered in physics or in engineering. the most target is to give an in depth research in their mathematical and actual houses. Wave equations are time established. even though, use of the Fourier trans shape reduces their research to that of harmonic platforms: the harmonic Helmholtz equation, with regards to the acoustic equation, or the har monic Maxwell method. This ebook concentrates at the examine of those harmonic difficulties, that are a primary step towards the examine of extra basic time-dependent difficulties. In every one case, we supply a mathematical environment that enables us to end up life and strong point theorems. we now have systematically selected using variational formulations concerning concerns of actual power. We research the essential representations of the suggestions. those representa tions yield numerous crucial equations. We research their crucial houses. We introduce variational formulations for those necessary equations, that are the foundation of such a lot numerical approximations. various elements of this booklet have been taught for no less than ten years via the writer on the post-graduate point at Ecole Poly procedure and the collage of Paris 6, to scholars in utilized arithmetic. the particular presentation has been proven on them. I desire to thank them for his or her energetic and confident participation, which has been super invaluable, and that i ask for forgiveness for forcing them to benefit a few geometry of surfaces.

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**Example text**

We need to prove that aU/aXl, aU/aX2, aU/aX3, are in Hk-l(B i ), and that raU/aXl, raU/aX2, raU/aX3 are in Wk-l(Be), when Ud is in Hk-l/2(S). 5. The Laplace Equation in IR? 118). 118) that when Ud is in the space Hk-l/2(S) , aU/aXi belongs to Hk-3/2(S) . This yields the result for the interior problem. 31 ) 1=0 m=-I and so r au ~ ~ m[ 1 aHi aXi (X) = ~ ~ ul r21 aXi (x) - (2l + 1) r 21Xi+2 HIm] (x) . 32 ) 1=0 m=-I It follows that on the unit sphere ::i f t = 1=0 m=-I [ura~l~ (x) - (2l + l)UrXiHi(X)] .

6 using localization and • transport. 4. Remark The compact injection result is wrong in the space HS(1R2) and also in the space Hm(n e)' Indeed, as these domains are non-bounded, we can exhibit simple counter-examples, based on sequences of functions with unit norm and support going to infinity, that do not converge. 79) Ulr = Ud· We seek a solution in the space HI (0,). In this space, this equation does not directly make sense.

Its component [3 along the vector ii does not vanish. u. ( --[1 3 n · ) ) an [3 az j=1 aXj az J 1 . 62) au/an is the trace on C of the function of the right-hand side. It follows from the above remark that (a/az)(u 0 ¢-1) is an element of H m - 1(JR3). It is also the case of the right-hand side. This gives the continuity of the second trace. For the next traces, we proceed by induction. For example, computing (0 2/ az 2)( u 0 ¢ -1), we can express (0 2/ an 2 )u as the trace of a function of H m - 2 , ....