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Chapitre 5 : Convergence de l’´equation desondes amorties `a l’int´erieur vers l’´equationdes ondes amorties sur le bord1 IntroductionThis article is devoted tothe comparison ofthe dynamics ofthe wave equation dampedin the interior of the domain Ω with the dynamics of the wave equation damped on theboundary of Ω, when the interior damping converges to a Dirac distribution supported bythe boundary.One of the physical motivation is the following. We consider a soundproof room, wherecarpetcoversallthewalls.Thissituationismodeledasfollows.LetΩbeasmoothboundedd ∞domain ofR (d =1,2 or 3) and letγ be a non-negative function inL (∂Ω) (the effectivedissipation of the carpet at a point of the wall). The propagation of waves in the room ismodeled by the wave equation damped in the boundaryu (x,t) = (Δ−Id)u(x,t)+f(x,u(x,t)) , (x,t)∈ Ω×R tt +∂u(x,t)+γ(x)u (x,t) = 0 , (x,t)∈∂Ω×R (1.1)t +∂ν 1 2(u,u ) =(u ,u )∈H (Ω)×L (Ω)t |t=0 0 1Notice that, in this model, the waves are not dissipated in the interior of the room butinstantaneously damped at each rebound on the walls. This corresponds to a ponctualdissipation of the form γ(x) ⊗ δ , where δ is the Dirac function supported byx∈∂Ω x∈∂Ωthe boundary. Of course, this is an approximation of the reality, as the carpet has somethickness. Thus, we can model more precisely the propagation of waves in the soundproofroom by the equationu (x,t)+γ (x)u (x,t) =(Δ−Id)u(x,t)+f(x,u(x,t)) , (x,t)∈ Ω×R tt n t +∂ u(x,t)= 0 , (x ...