P Genevet

S Barland

M Giudici

J R Tredicce

Cavity Soliton Laser based on mutually coupled semiconductor microresonators

Keywords: numbers: 42.65.Tg, 42.55.Px, 42.65.Pc, 42.79.Ta

We report on experimental observation of localized structures in two mutually coupled broadarea semiconductor resonators. These structures coexist with a dark homogeneous background and they have the same properties as cavity solitons without requiring the presence of a driving beam into the system. They can be switched individually on and off by means of a local addressing beam.

Localized Structures (LS) form in large aspect-ratio media where two or several solutions coexist in the parameter space (see e.g. [START_REF]Dissipative solitons[END_REF] for a recent review). Cavity Solitons (CS) are LS generated in a cavity filled with a non linear medium driven by a coherent injected field (holding beam, HB) where they appear as single bistable bright intensity peaks coexisting with a homogeneous background. Their existence and mutual independence in semiconductors have been experimentally demonstrated in microcavities operated as optical amplifiers [2,3] since a local perturbation in form of a beam coherent with the HB can be used for switching CS on and off independently [2,4]. The possibility to control their location and their motion by introducing phase or amplitude gradients in the holding beam suggests their use as mobile pixels for all-optical processing units. Indeed, in the last decade, CS in semiconductor have attracted a growing interest since they combine the bistability and plasticity properties with the advantages of semiconductor media in terms of fast response and small size. The application potential of CS has been evidenced with some first-principle demonstration of new all-optical devices exploiting CS properties for optical memories [5] and delay lines [6]. Nevertheless, the tight conditions required for CS stability in present experimental schemes hinders their application to non prototypical devices. A radical simplification could be achieved implementing the concept of Cavity Soliton Laser, i.e. a device generating CS without an external injection beam. Some steps in this direction have been made recently with a scheme based on broad area VCSEL submitted to frequency selective feedback [7]. However, even if no HB is present it appears that the stability of localized structures in this case depends critically on feedback alignment and on the detuning between the resonator and the external frequency selecting element. An alternative approach is provided by a laser with a saturable absorber (SA). Indeed, this system is among the first ones theoretically shown to possess the necessary ingredients for the generation of localized structures in optics, called in this case dissipative autosolitons [8]. While this initial work was realized in the limit of fast materials [8,[START_REF] Rosanov | Spatial Hysteresis and Optical Patterns[END_REF], it was later extended to the case of finite relaxation times [START_REF] Rosanov | Spatial Hysteresis and Optical Patterns[END_REF][START_REF] Fedorov | [END_REF]. Finally, the case of slow absorber material (as would be the case for semiconductors) was examined in [11] and the case where the absorbing and gain media have equal response times has been studied in [12]. The authors of these last references show numerically that in a vertical cavity surface emitting laser (VCSEL) with saturable absorber, CS related to the existence of a modulational instability can be switched on and off by injecting a local optical perturbation. Despite this extensive theoretical and numerical research, we are not aware of any experimental observation of CS using a saturable absorber in semiconductor laser. In this letter we show the experimental realization of a CSL based on two mutually coupled micro resonators where one plays the role of an amplifier and the second of a saturable absorber. As we shall demonstrate below, this scheme allows a remarkably simple realization of a cavity soliton laser. We show that bistable solitary structures coexist with a dark homogeneous background and that they can be switched on and off independently by an incoherent beam.

We believe that this demonstration opens the way towards implementations in very compact monolithic devices including both the amplifying and saturable absorber sections [13]. The lasers we use are two nominally identical VCSELs provided by ULM Photonics.

They are oxydized bottom-emitter VCSELs emitting around 975nm [14]. Their transverse section is 200 µm. They are mounted in a mutually coupled configuration, where one laser is electrically biased above the transparency but below its standalone coherent emission threshold (L 1 ), while the other is biased below transparency (L 2 ). Both devices packages are temperature stabilized. In order to compensate the effects of diffraction during the propagation in the extended cavity (60 cm long) and to keep the system highly symmetric, we use identical collimators and identical lenses placed such that the two resonators are in a self imaging condition (see Fig. 1). This configuration allows to preserve the high Fresnel number required for the existence of LS [3]. A 20 % reflection beam splitter is inserted in the center of the cavity to extract two output beams from the system. Time-averaged nearfield profiles of both resonators are simultaneously imaged on two charge-coupled device (CCD) cameras. The use of two CCD cameras allows to check the output of each element of the coupled system independently. In the detection path of laser L 1 , an iris placed in an intermediate near field plane enables to select a small area of the profile for point-like temporally resolved detection. A photodetector Thorlabs PDA8GS (less than 100 ps rise time) coupled to a digital oscilloscope LeCroy Wavemaster 8600A (6 GHz analog bandwidth) monitors the output of this small portion of the L 1 profile. The solitary VCSELs light intensity output as function of the pumping current (

I L 1 ,L 2 )
indicate that the uncoupled lasers have very similar standalone coherent emission thresholds

I th L 1 ∼ I th L 2 ∼ 400mA.
The temperature control of each device is set such that the emission wavelength of laser L 1 is approximately 1 nm blue detuned with respect to laser L 2 when both devices are pumped by the same amount of current. In Fig. 2 we show the local intensity output of the compound system as a function of pumping current of L 1 , (I L 1 ), while I L 2 is kept fixed at a few mA, i.e. below the transparency pumping value. The monitored region has a diameter of about 20 µm and it is placed in the centre of the device. When I L 1 ≤ 100 mA (zone A in Fig. 2, the increase of emitted power is attributed to spontaneous emission, since the optical spectrum of the system does not show coherent emission. The first threshold (I th ∼ 100 mA, region B of Fig. 2) is reached when light linearly reflected on the output mirror of L 2 sufficiently reduces the losses of the compound system such that laser emission can be obtained. At that point, threshold reduction and coherent emission may therefore be attributed to losses reduction as is shown in conventional optical feedback experiments.

For, We note that while the curve shown in Fig. 2 was obtained for a particular setting of the temperature of each device's substrate, equivalent results may be obtained for different temperature and current settings provided a number of conditions are satisfied. First of all, if device L 2 is pumped at a too high value (that we interpret as above transparency??), we made no observation of the bistable response of the system. Second, if L 1 is initially red detuned with respect to L 2 , these results won't be observed since both devices won't be resonant for any current value of L 1 . The essential conditions are therefore an initial detuning allowing for current induced tuning of L 1 towards the resonance of L 2 when L 2 is kept in an absorbing regime. An additional condition is that if the suitable tuning condition is obtained for a too low value of pumping of device L 1 (pumping of device L 2 being kept constant) the total amount of field in the compound system does not seem to reach the saturation value.

I L 1 ≥ 200mA (region C of
In this case, although the effects of absorption in L 2 can clearly be identified, no bistable behavior was observed. We expect however that this condition, related to the saturation fluence of device L 2 , could also be satisfied by applying a demagnification factor in the imaging of device L 1 on device L 2 in our setup or decreasing the reflectivity of the central beam splitter.

A detailed study of required conditions has been performed and will be presented elsewhere, but for now we underline that provided that the previous conditions are fulfilled, the curve shown on Fig. 2 can be obtained for different settings of device temperatures and currents, with I L 2 allowing to tune the amount of absorption in the system. As an order of magnitude, the results presented here could be obtained for a broad range of temperature settings leading to the bistability cycle being observed for currents in L 2 ranging from 5 to 30 mA and currents in L 1 ranging from 120 to 380 mA. In Fig. 3 we show the near-field transverse profiles of L 1 and L 2 for different values of I L 1 , while I L 2 is kept constant at 15 mA. As discussed previously, in the self imaging scheme configuration, L 1 and L 2 are placed on self-conjugate planes with a magnification of one. In order to monitor the absorption, the devices are slightly shifted with respect to each other in the horizontal direction. This way, a small portion close to the border of L 1 will not interact with the corresponding portion of L 2 and it will be simply reflected back by the substrate of the device. When I L 1 is below the compound system's threshold both profiles (not shown) are homogeneous. When I L 1 is increased above the first threshold (corresponding to region B in Fig. 2, we observe the formation of complex and in general nonstationary patterns resulting from the linear feedback effect of L 2 's output mirror on L 1 (Fig. 3 a),b)). In Fig. 3,b) showing the near field profile of L 2 , the dark area in the rightmost part of the image corresponds to high intensity emitted by L 1 and reflected by the substrate of the device L 2 due to the lateral shift we introduced. The bright arc nearby, closer to the center, corresponds to the edge of L 2 .

Keeping increasing I L 1 the two resonators start to interact with L 2 absorbing the field emitted by L 1 (region C of Fig. 2) and the near field profile of both lasers is mostly dark and homogeneous except for the small portion close to the rightmost edge, due to reflection onto the substrate of L 2 . By increasing the current I L 1 above I c L 1 , bistable bright spots appear spontaneously (see Fig. 3 C,D).

Further increase of I L 1 leads to the formation of generally nonstationary filaments connecting the isolated spots together. For higher values of I L 1 (Fig. 3 E andF), corresponding to region D of Fig. 2), a complex pattern develops progressively through the whole transverse section. Temporally resolved detection reveals that the widely spread pattern is generally not stationary and exhibits complex dynamics.

When the parameters are set in region C of figure 2, the observed bright isolated spots (Fig. 3,C,D) are candidates for an interpretation in terms of CS, since they coexist with a homogeneous background as shown by the hysteretical behavior as a function of I L 1 (Fig. 2.

While they appear rather uncorrelated one to the other on the near field images, the full demonstration of their mutual independence can only be performed by switching them on and off with a local perturbation. Indeed, it has been shown numerically that localized states in a cavity soliton laser can be switched on or off by a coherent [11,12] or incoherent [15] local optical perturbation. In our experiment, we use a coherent beam whose optical frequency is close (here, within 0.1 nm) to the emission frequency of the coupled device and whose diameter is about 15 µm. This beam is obtained from an external-cavity tunable laser in Littman configuration and applied on device L 2 . By means of this local optical injection, we are able to demonstrate independent switching of localized structures as shown on Fig. 4.

The system is prepared in the low level emission state with the parameters set in the bistable region. Starting with no spot, applying the WB to a point in the transverse plane of L 2 , we generate a high intensity spot with a diameter of the order of 10µm. We then remove the WB and the bright spot remains on indefinitely. We then apply the WB at a different position and a second spot is generated without disturbing the first one, provided the distance between the two spots is sufficient (no distance smaller than 40 µm was observed). The two spots stay on even after the WB is removed. We note that, as in previous experiments in semiconductor devices [4,7], local device inhomogeneities seem to play a role in the stabilization of localized states since although it is possible to observe stable structures in different positions, not all positions appear to be stable. We make use of this to switch off the localized structures. In experiments involving an external forcing by a holding beam, the simplest procedure is to apply a coherent local perturbation with opposite phase with respect to the holding beam. In the present case, this approach is not possible. Therefore we take advantage of mobility properties of the structures to switch them off: by applying the same perturbation as before close to a CS, we can drag it to a region of space where it is unstable and therefore switches off (Fig. 4,F,H). It is also possible to switch off the structures without dragging them outside of their preferred location if the pumping current of each device is set such that the system is very close to the lower edge of the bistability region for the structure under consideration. In this case though, our optical perturbation did not appear to be sufficient to switch on a localized structure. Conversely, if the system parameter were set very close to the upper edge of the bistability region, it was not possible to switch localized structures off, except with the dragging procedure described above.

While the application of the local perturbation was quasi continuous in the measurements shown above, numerical simulations performed on a model for a monolithic semiconductor laser with saturable absorber [15] indicate the possibility of ns switching time. Preliminary experimental observations indicate that perturbations as short as 100 ns (limited by the modulator) can be sufficient to switch on or off a localized structure, although no optimization (e.g tuning and power of the writing beam) has been performed yet.

In conclusion, we have given evidence of single localised peaks that fulfill all the criteria required to be interpreted as cavity solitons in a compound semiconductor laser system with a saturable absorber. Since no injection beam is present in our experiment we believe it is a very promising realization of a semiconductor cavity soliton laser. One of the greatest strengths of this experiment is its possibly straightforward miniaturization to monolithic devices able to generate self localized, bistable and mobile laser beams. In the present version of the system, the observation of spatially localized periodically pulsed transient regimes suggests the suitability of the system to the generation of three dimensional localized structures.

FIG. 1 :

 1 FIG. 1: Schematic drawing of the experiment. L 1 : Laser above the transparency, L 2 : Laser below the transparency, BS: beam splitter. Detection of L 1 (resp. L 2 ) includes a CCD camera monitoring the near field of L 1 (resp. L 2 ) and a fast detector to monitor the local temporal behavior.

FIG. 2 :

 2 FIG.2:Local intensity output emit by the system when we scan I L 1 for all the other parameters constant (I L 1 is drive very slowly to keep the temperature controller stable). A) below threshold, B)lasing by feedback, C)Absorption by L 2 , bistable behavior, D) pattern formation.

Fig. 2 )

 2 the power output of the compound system saturates and then decreases for increasing values of I L 1 . This happens when the longitudinal resonances of both cavities match and absorption in device L 2 takes place. The fact that the resonance frequencies of both devices match only for certain current values is due to the linear shift experienced by the laser frequency as a function of the pumping current due to Joule heating. The absorption by laser L 2 in region C of Fig. 2 has been verified by performing the same measurement with laser L 2 unpumped, which allows to verify the presence of a light induced current through the device, which is absent in regions A or B. Increasing further I L 1 , the intensity output remains constant at a low value. The optical spectra of this low intensity emission shows a broad band peak indicating that this state corresponds to spontaneous emission since absorption cut the feedback from L 2 . Further increase of I L 1 above a critical value I c L 1 the local intensity jumps up to a high value. The optical spectrum associated to this state shows a well pronounced peak red-detuned with respect the spontaneous emission peak observed in the low level state. In the near field profile this local transition of the intensity output corresponds to the formation of a bright single peak structure inside the monitored area. For larger I L 1 (region D of Fig. 2) the local intensity keeps increasing, while the near field profile reveals the formation of multi peaked structures and extended patterns around the monitored region. If I L 1 is decreased the local intensity shows hysteresis demonstrating bistability between a low and a high emitted intensity state in the region monitored by the detector.

FIG. 3 :

 3 FIG. 3: Examples of near field of both devices. Dark areas correspond to high intensities. a)Near field of the high current laser (L 1 ) before the interaction, b) Near field of the low current laser (L 2 ) before the interaction (I L 1 = 180 mA), c,d) Near field of L 1 NF and L 2 in the absorption zone (I L 1 = 358 mA), e,f) Near field of L 1 and L 2 when the pattern is developed (I L 1 = 365 mA). L 2 is slightly shifted on the left.

FIG. 4 :

 4 FIG. 4: L 1 near field intensity distribution. Dark areas correspond to high intensities. Sequence of successive switching of two independent structures with an incoherent WB tuned at the CS wavelength with all parameter fixed. a) Both structures are off, b) The injection of WB switches one structure, c) the structure remains on after the WB is blocked, d) the injection of the WB at a second location ignites a second structure, e) when the WB is blocked, both structures are on, f) reapplying the WB in the vicinity of the second structure attracts it to a slightly different location, g) when the WB is blocked the structure switches off, h) the WB is applied in the vicinity of the first structure to attract it to a slightly different location, i) when the WB is blocked, both structures are off.

This work was supported by the FET Open Project FunFACS (www.funfacs.org). We are grateful to L. Gil, L. Columbo and G. Tissoni for many useful discussions.


Frédéric Magoulès

Thi Phuong

Kieu Nguyen

Pascal Omnes

Anna Rozanova-Pierrat
email: anna.rozanova-pierrat@centralesupelec.fr

Optimal absorption of acoustical waves by a boundary

Keywords: Absorbing wall, wave propagation, shape optimization, Helmholtz equation, sound absorption, Robin boundary condition

Introduction

The diffraction and absorption of waves by a system with both absorbing properties and irregular geometry is an open physical problem. This has to be solved to understand why anechoic chambers (electromagnetic or acoustic) do work better with irregular absorbing walls. The first studies relating irregular geometry and absorption are performed numerically in [START_REF] Félix | Localization and increased damping in irregular acoustic cavities[END_REF]. Therefore there is a question about the existence of an optimal shape of an absorbent wall (for a fixed absorbing material), optimal in the sense that it is as dissipative as possible for a large range of frequencies, and at the same time that such a wall could effectively be constructed. In the framework of the propagation of acoustic waves, the acoustic absorbent material of the wall is considered as a porous medium. In this article, for a fixed frequency of the sound wave, we solve the shape optimization problem that consists in minimizing the acoustic energy for a frequency model with a damping on the boundary. Then we extend this method in order to find an efficient shape for a finite range of frequencies.

In the area of optimization of acoustic performances of non absorbing walls, Duhamel [START_REF] Duhamel | Calcul de murs antibruit et control actif du son[END_REF][START_REF] Duhamel | Shape optimization of noise barriers using genetic algorithms[END_REF] studies sound propagation in a two-dimensional vertical cut of a road wall and uses genetic algorithms to obtain optimal shapes (some of them are however not connected and thus could not be easily manufactured). The author also uses a branch and bound (combinatorial optimization) type linear programming in order to optimize the sensors' positions that allow an active noise control, following former work introduced by Lueg [START_REF] Guicking | On the invention of active noise control by Paul Lueg[END_REF] in 1934. Abe et al. [START_REF] Abe | A BE-based shape optimization method enhanced by topological derivative for sound scattering problems[END_REF] consider a boundary elements based shape optimization of a non absorbing two-dimensional wall in the framework of a two-dimensional sound scattering problem for a fixed frequency (for the Helmholtz equation) using a topological derivative with the principle that a new shape or topology is obtained by nucleating small scattering bodies. Also for the Helmholtz equation for a fixed frequency, using the shape derivative of a functional representing the acoustical energy, Cao and Stanescu [START_REF] Cao | Shape optimization for noise radiation problems[END_REF] consider a two-dimensional shape design problem for a non-absorbing part of the boundary to reduce the amount of noise radiated from aircraft turbofan engines. For the same problem, Farhadinia [START_REF] Farhadinia | An Optimal Shape Design Problem for Fan Noise Reduction[END_REF] developed a method based on measure theory, which does not require any information about gradients and the differentiability of the cost function.

On the other hand, for shape optimization problems there are theoretical results, reviewed in Refs. [START_REF] Allaire | Conception optimale de structures, 58 Mathématiques et Applications[END_REF][START_REF] Mohammadi | Applied shape optimization for fluids[END_REF], which rely on the topological derivatives of the cost functional to be minimized, with numerical application of the gradient method in both two and three dimensional cases (in the framework of solid mechanics). In particular, Achdou and Pironneau [START_REF] Achdou | Optimization of a photocell[END_REF] considered the problem of optimization of a photocell, using a complex-valued Helmholtz problem with periodic boundary conditions with the aim to maximize the solar energy in a dissipative region. For acoustic waves in the two-dimensional case, optimization of the shape of an absorbing inclusion placed in a lossless acoustic medium was considered in Refs. [START_REF] Münch | Optimal Internal Dissipation of a Damped Wave Equation Using a Topological Approach[END_REF][START_REF] Münch | Optimal design of the damping set for the stabilization of the wave equation[END_REF]. The considered model is the linear damped wave equation [START_REF] Cox | The rate at which energy decays in a damped string[END_REF][START_REF] Asch | The Spectrum of the Damped Wave Operator for a Bounded Domain in R 2[END_REF]. Using the topology derivative approach, Münch et al. consider in [START_REF] Münch | Optimal Internal Dissipation of a Damped Wave Equation Using a Topological Approach[END_REF][START_REF] Münch | Optimal design of the damping set for the stabilization of the wave equation[END_REF] the minimization of the acoustic energy of the solution of the damped wave equation at a given time T > 0 without any geometric restrictions and without the purpose of the design of an absorbent wall. See also [START_REF] Antil | Shape Optimization of Shell Structure Acoustics[END_REF] for the shape optimization of shell structure acoustics.

In this article, we study the two-dimensional shape optimization problem for a Helmholtz equation with a damping on the boundary, modeled by a complex-valued Robin boundary condition. The shape of the damping boundary is to be found, in the aim to minimize the total acoustical energy of the system. In Section 2, we introduce the frequency model and its time-dependent analogue with a dissipation on the boundary. We analyze its dissipative properties and give the well-posedness results, due to [START_REF] Bardos | Variational algorithms for the Helmholtz equation using time evolution and artificial boundaries[END_REF][START_REF] Gander | An optimized Schwarz method with two-sided Robin transmission conditions for the Helmholtz equation[END_REF] for at least Lipschitz boundaries, but we generalize the results for the Helmholtz problem in a larger class of domains with d-set boundaries using [START_REF] Arfi | Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets[END_REF][START_REF] Pierrat | Generalization of Rellich-Kondrachov theorem and trace compacteness in the framework of irregular and fractal boundaries[END_REF] (see Appendix A). This class, named in [START_REF] Arfi | Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets[END_REF] "admissible domains" and containing for instance the Von Koch fractals, is composed of all Sobolev extension domains, thanks to results of [START_REF] Hajłasz | Sobolev embeddings, extensions and measure density condition[END_REF], with boundaries on which it is possible to define a surjective linear continuous trace operator with linear continuous right inverse. However, for the shape optimization problem, only the Lipschitz boundary case is considered here.

We compare the frequency model with dissipation by the boundary to the corresponding model with a dissipation in the volume. Dissipation in the volume is described by a damped wave equation in which the values of the coefficients for a given porous medium are given as functions of its macroscopic parameters (as porosity, tortuosity and resistivity to the passage of air), as proposed by [START_REF] Hamet | Acoustical characteristics of porous pavements: a new phenomenological model[END_REF]. In particular, in Theorem 4 we propose a possible way to find the complex parameter in the Robin boundary condition of the former model that best approximates the latter. In Section 3, for the case of uniform Lipschitz boundaries satisfying a uniform ε-cone property for a fixed ε > 0 and in addition having a uniform upper bound for their lengths inside non trivial balls (and hence which have boundaries with uniformly bounded lengths), we introduce the class of (shape) admissible domains, adding, as in the classical framework of shape optimization, the assumptions that all moving parts of the boundary belong to a compact set and that all domains having a fixed volume are included in a fixed bounded open set. In this class of admissible domains, for any fixed frequency we obtain the existence of an optimal shape in the sense that there exists a positive measure µ * on the optimal shape Γ, equivalent to the usual Lebesgue measure λ, such that µ * ≥ λ and such that the weak solution of the corresponding Helmholtz problem realizes the infimum of the acoustical energy, the latter being defined using λ on the boundary. In the case where µ * = λ (this depends on the properties of the minimizing sequence which we don't know in advance), then it is also the minimum. Moreover, we notice that in order to have the existence of an optimal shape in a higher dimensional case (for instance in R 3 or simply in R N ) it is sufficient to replace everywhere the N -1-dimensional Lebesgue measure of the boundary by the N -1-dimensional Hausdorff measure, since in that case the Lebesgue measure of the N -1-dimensional boundary is not equal to the Hausdorff measure as for one dimensional curves, but proportional to it (see [START_REF] Falconer | The Geometry of Fractal Sets[END_REF]Thrm. 1.12,p. 13], for the optimization in R 3 the Lebesgue measure of the boundary is equal to π/4 times the Hausdorff measure). See also Ref. [START_REF] Bucur | Shape optimization problems with Robin conditions on the free boundary[END_REF] for a free discontinuity approach to a class of shape optimization problems involving a Robin condition on a free boundary.

To summarize, the rest of the paper is organized as follows. Section 2 introduces a damped acoustical propagation model in which damping occurs through the boundary. It is described by the Helmholtz problem with a Robin boundary condition with a complex coefficient α, for which we give a well-posedness result on an admissible domain in R N in the sense of Ref. [START_REF] Arfi | Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets[END_REF]. The existence of an optimal shape in the introduced acoustical framework is proved in Section 3. We recall useful results from [START_REF] Arfi | Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets[END_REF] in Appendix A. Appendix B details how to obtain the damping parameter α in the Robin boundary condition that best approximates a given model with dissipation in the volume.

The model: motivation and known properties

To describe the acoustic wave absorption by a porous medium, there are two possibilities. The first one is to consider wave propagation in two media, typically air and a wall, which corresponds to a damping in the volume. The most common mathematical model for this is the damped wave equation [START_REF] Asch | The Spectrum of the Damped Wave Operator for a Bounded Domain in R 2[END_REF]. The second one is to consider only one lossless medium, air, and to model energy dissipation by a damping condition on the boundary. In both cases, we need to ensure the same order of energy damping corresponding to the physical characteristics of the chosen porous medium as its porosity φ, tortuosity α h and resistivity to the passage of air σ [START_REF] Hamet | Acoustical characteristics of porous pavements: a new phenomenological model[END_REF].

Thanks to Ref. [START_REF] Hamet | Acoustical characteristics of porous pavements: a new phenomenological model[END_REF], we can define the coefficients in the damped wave equation (damping in volume) as functions of the above mentioned characteristics. More precisely, for a regular bounded domain Ω ⊂ R 2 (for instance ∂Ω ∈ C 1 ) composed of two disjoint parts Ω = Ω 0 ∪ Ω 1 of two homogeneous media, air in Ω 0 and a porous material in Ω 1 , separated by an internal boundary Γ, we consider the following boundary value problem (for the pressure of the wave)

   ξ(x)∂ 2 t u + a(x)∂ t u -∇ • (η(x)∇u) = 0 in Ω, ∂u ∂n | Rt×∂Ω ≡ 0, [u] Γ = [η∇u • n] Γ = 0, u| t=0 = u 0 1 Ω0 , ∂ t u| t=0 = u 1 1 Ω0 , (1)
with ξ(x) = 1 c 2 0 , a(x) = 0, η(x) = 1 in air, i.e., in Ω 0 , and

ξ(x) = φγ p c 2 0 , a(x) = σ φ 2 γ p c 2 0 ρ 0 α h , η(x) = φ α h
in the porous medium, i.e., in Ω 1 . The external boundary ∂Ω is supposed to be rigid, i.e., Neumann boundary condition are applied, and on the internal boundary Γ we have no-jump conditions on u and η∇u • n, where n denotes the normal unit vector to Γ. Here, c 0 and ρ 0 denote respectively the sound velocity and the density of air, whereas γ p = 7/5 denotes the ratio of specific heats. But instead of energy absorption in volume, we can also consider the following frequency model of damping by the boundary. Let Ω be a connected bounded domain of R 2 with a Lipschitz boundary ∂Ω. We suppose that the boundary ∂Ω is divided into three parts ∂Ω = Γ D ∪ Γ N ∪ Γ (see Fig. 1 for an example of Ω, chosen for the numerical calculations) and consider

u + ω 2 u = f (x), x ∈ Ω, u = g(x) on Γ D , ∂u ∂n = 0 on Γ N , ∂u ∂n + α(x)u = Trh(x) on Γ, (2)
where α(x) is a complex-valued regular function with a strictly positive real part (Re(α) > 0) and a strictly negative imaginary part (Im(α) < 0).

Remark 1. This particular choice of the signs of the real and the imaginary parts of α are needed for the well-posedness properties [START_REF] Gander | An optimized Schwarz method with two-sided Robin transmission conditions for the Helmholtz equation[END_REF] and the energy decay of the corresponding time-dependent problem. In addition, as the frequency ω > 0 is supposed to be fixed, α can contain a dependence on ω, i.e., α ≡ α(x, ω).

Problem ( 2) is a frequency version of the following time-dependent wave propagation problem with U (t, x) = e -iωt u(x), considered in Ref. [START_REF] Bardos | Variational algorithms for the Helmholtz equation using time evolution and artificial boundaries[END_REF] for g = 0 on Γ D :

∂ 2 t U -U = -e -iωt f (x), (3)
U | t=0 = U 0 , ∂ t U | t=0 = U 1 , (4)
U | Γ D = g, ∂U ∂n Γ N = 0, (5)
∂U ∂n - Im(α(x)) ω ∂ t U + Re(α(x))U | Γ = 0. (6)
To show the energy decay, we follow [START_REF] Bardos | Variational algorithms for the Helmholtz equation using time evolution and artificial boundaries[END_REF] and introduce the Hilbert space X 0 (Ω), defined as the Cartesian product of the set of functions u ∈ H 1 (Ω), which vanish on Γ D with the space L 2 (Ω). The equivalent norm on X 0 (Ω) is defined by

(u, v) 2 X0(Ω,µ) = Ω |∇ x u| 2 + |v| 2 dx + Γ Re(α(x))|u| 2 dµ
with the corresponding inner product

(u 1 , u 2 ), (v 1 , v 2 ) = Ω (∇ x u 1 ∇ x v 1 + u 2 v 2 ) dx + Γ Re(α(x))u 1 v 1 dµ. (7)
Here µ is a Radon positive measure on Γ, which in the case of a regular Γ (at least Lipschitz), if there are no specific assumptions, is equal to the Lebesgue measure on Γ, and in this case is denoted by λ.

The advantage of this norm is that the energy balance of the homogeneous problem (3)-( 6) has the form

∂ t (U, ∂ t U ) 2 X0(Ω,µ) = 2 ω Γ Im(α(x))|∂ t U | 2 dµ.
Therefore, for Im(α) < 0 on Γ, the energy decays in time. For the case of a smooth boundary ∂Ω (at least Lipschitz), we have the well-posedness of both models. Thanks to [START_REF] Bardos | Variational algorithms for the Helmholtz equation using time evolution and artificial boundaries[END_REF], for all f ∈ L 2 (Ω), (U 0 , U 1 ) ∈ X 0 (Ω) there exists a unique solution (U, U t ) ∈ C(]0, ∞[, X 0 (Ω)) of system (3)-( 6) under the assumption that Re(α(x)) > 0 and Im(α(x)) < 0 are continuous functions.

For the frequency model (2) it is possible to generalize the weak well-posedness result in domains with Lipschitz boundaries [START_REF] Gander | An optimized Schwarz method with two-sided Robin transmission conditions for the Helmholtz equation[END_REF] to domains with a more general class of boundaries, named Ahlfors d-regular sets or simply d-sets [START_REF] Jonsson | Function spaces on subsets of R n[END_REF] (see Appendix A), using functional analysis tools on "admissible domains" developed in [START_REF] Arfi | Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets[END_REF]. The interest of this generalization is that this class of domains is optimal in the sense that it is the largest possible class [START_REF] Arfi | Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets[END_REF] which keeps the Sobolev extension operators, for instance H 1 (Ω) to H 1 (R N ), continuous. In what follows, for our well-posedness result we take Ω ⊂ R N , N ≥ 2.

We use, as in Refs. [START_REF] Arfi | Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets[END_REF][START_REF] Pierrat | Generalization of Rellich-Kondrachov theorem and trace compacteness in the framework of irregular and fractal boundaries[END_REF], the existence of a d-dimensional (0 < d ≤ N , d ∈ R) measure µ equivalent or equal to the Hausdorff measure m d on ∂Ω (see Definition 1) and a generalization of the usual trace theorem [START_REF] Jonsson | Function spaces on subsets of R n[END_REF] (see Appendix A) and the Green formula [START_REF] Lancia | A Transmission Problem with a Fractal Interface[END_REF][START_REF] Arfi | Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets[END_REF] in the sense of the Besov space B 2,2 β (∂Ω) with β = 1 -N -d 2 > 0 constructed on ∂Ω with the help of the measure µ (for the definition of the Besov spaces on d-sets see Ref. [START_REF] Jonsson | Function spaces on subsets of R n[END_REF] p.135 and Ref. [START_REF] Wallin | The trace to the boundary of Sobolev spaces on a snowflake[END_REF]). Note that for d

= N -1, one has β = 1 2 and B 2,2 1 2
(∂Ω) = H 1 2 (∂Ω) as usual in the case of a Lipschitz boundary. In what follows we write L 2 (∂Ω, µ) to specify that the space is defined with respect to the measure µ. Some main elements of functional analysis on d-sets are presented in Appendix A.

Moreover, we stress that once a measure µ is fixed on the boundary ∂Ω, it modifies the meaning of the Green formula in the following sense: for all u and v from H 1 (Ω) with ∆u ∈ L 2 (Ω) the normal derivative of u is understood as the linear continuous functional on the Besov space B 2,2 β (∂Ω) constructed by µ according to the definition

∂u ∂ν , Trv ((B 2,2 β (∂Ω)) ,B 2,2 β (∂Ω)) := Ω v∆udx + Ω ∇v • ∇udx.
Considering the Helmholtz problem (2), we introduce the Hilbert space

V (Ω) = {u ∈ H 1 (Ω)| u = 0 on Γ D } (8)
equipped with the norm (equivalent to the usual norm • H 1 (Ω) ) thanks to Theorem 3

u 2 V (Ω,µ) = Ω |∇u| 2 dx + Γ Re(α)|u| 2 dµ,
and obtain the following well-posedness result:

Theorem 1. Let Ω ⊂ R N be a bounded admissible domain with a compact d-set boundary (N -2 < d < N ) with a d-measure µ in the sense of Theorem 3, ∂Ω = Γ D ∪ Γ N ∪ Γ and β = 1 -(N -d)/2 > 0. Let Γ D be also a d-set with the same properties as ∂Ω itself. Let in addition Re(α(x)) > 0, Im(α(x)) < 0 be smooth functions (at least continuous) on Γ. Then for all f ∈ L 2 (Ω), g ∈ B 2,2 β (Γ D ), h ∈ H 1 (Ω)
, and ω > 0 there exists a unique solution u of the Helmholtz problem [START_REF] Achdou | Optimization of a photocell[END_REF], such that (u -g) ∈ V (Ω) (where g is a lifting in H 1 (Ω) of the boundary data g) in the following sense: for all v ∈ V (Ω)

Ω ∇u • ∇vdx -ω 2 Ω uvdx + Γ α Tr u Tr v dµ = - Ω f vdx + Γ Tr h Tr v dµ. (9)
Moreover, the solution of problem (2) u ∈ H 1 (Ω), continuously depends on the data: there exists a constant C > 0, depending only on α, ω and on C P (Ω), such that

u H 1 (Ω) ≤ C f L2(Ω) + g B 2,2 β (Γ D ) + h H 1 (Ω) , (10)
where C P (Ω) is the Poincaré constant associated to Ω. In particular, taking g = 0, for all fixed ω > 0 the operator

B : L 2 (Ω) × H 1 (Ω) → V (Ω), defined by B(f, h) = u
with u, the weak solution of (9), is a linear compact operator.

In addition, if, for m ∈ N * , ∂Ω ∈ C m+2 (hence µ is an N -1-measure on ∂Ω), f ∈ H m (Ω) and g ∈ H m+ 3 2 (Γ D ), then the solution u belongs to H m+2 (Ω).
Proof. Let us start with g = 0 and in addition suppose that α is a constant (the generalization for α(x) is straightforward). By the linearity of the Helmholtz problem (2), we set u = u f + u h , where u f is the solution of the Helmholtz problem with Tr h = 0 on Γ and u h is the solution of the Helmholtz problem with f = 0. When h = 0, the variational formulation for u f becomes: for all φ ∈ V (Ω)

(u f , φ) V (Ω,µ) -ω 2 (u f , φ) L2(Ω) + i Im α(Tr u f , Tr φ) L2(Γ,µ) = -(f, φ) L2(Ω) ,
where we have defined the following equivalent inner product on V (Ω):

∀(v, w) ∈ V (Ω) × V (Ω) (v, w) V (Ω,µ) = (∇v, ∇w) L2(Ω) + Re α(Tr v, Tr w) L2(Γ,µ) .
Hence, the Riesz representation Theorem ensures the existence of a linear bounded operator

A : L 2 (Ω) → V (Ω) such that for v ∈ L 2 (Ω) ∀φ ∈ V (Ω) (v, φ) L2(Ω) = (Av, φ) V (Ω,µ) (11)
and in addition, by the Poincaré inequality,

Av V (Ω,µ) = sup φ V (Ω,µ) =1 |(v, φ) L2(Ω) | ≤ sup φ V (Ω,µ) =1 v L2(Ω) φ L2(Ω) ≤ C P (Ω) sup φ V (Ω,µ) =1 v L2(Ω) φ V (Ω,µ) = C P (Ω) v L2(Ω)
ensuring that A ≤ C P (Ω).

In the same way, using the Riesz representation Theorem we also define a linear bounded operator  :

L 2 (Γ, µ) → V (Ω) such that for w ∈ L 2 (Γ, µ) ∀φ ∈ V (Ω) (w, Tr φ) L2(Γ,µ) = ( Âw, φ) V (Ω,µ) . Indeed, it is sufficient to notice that for a fixed w ∈ L 2 (Γ, µ) the form : φ ∈ V (Ω) → (φ) = (w, Tr φ) L2(Γ,µ) ∈ C is linear and continuous on V (Ω): (w, Tr φ) L2(Γ,µ) ≤ w L2(Γ,µ) Tr φ L2(Γ,µ) ≤ C φ V (Ω,µ) ,
thanks to the continuity and the linearity of the trace from

V (Ω) to L 2 (Γ, µ). Moreover, Â ≤ C(Re(α)) since Âw V (Ω,µ) = sup φ V (Ω,µ) =1 |(w, Tr φ) L2(Γ,µ) | ≤ sup φ V (Ω,µ) =1 w L2(Γ,µ) Tr φ L2(Γ,µ) ≤ C(Re(α)) sup φ V (Ω,µ) =1 w L2(Γ,µ) φ V (Ω,µ) = C(Re(α)) w L2(Γ,µ) .
Thus, denoting by S the compact embedding operator of V (Ω) in L 2 (Ω) (by the Poincaré inequality it holds that S ≤ C P (Ω)), the variational formulation can be rewritten in the following form:

∀φ ∈ V (Ω) (Id -ω 2 A • S + i Im α Â • Tr)u f , φ V (Ω,µ) = (-Af, φ) V (Ω,µ) . (12)
Thanks to the compactness of the trace operator Tr :

V (Ω) → L 2 (∂Ω) [7] (with Tr ≤ C(Re(α))), the operator T = A•S -i Im α ω 2 •Tr : V (Ω) → V (Ω)
is compact as a composition of continuous and compact operators (with T ≤ C(ω, α, C P (Ω))). Thanks to the Fredholm alternative, it is then sufficient to prove that for (h, f ) = (0, 0), then the unique solution is u = 0, and this will allow us to conclude to the well-posedness of [START_REF] Capitanelli | Asymptotics for mixed Dirichlet-Robin problems in irregular domains[END_REF]. Setting f = 0 in [START_REF] Capitanelli | Asymptotics for mixed Dirichlet-Robin problems in irregular domains[END_REF], choosing φ = u f and separating real and imaginary parts of the equality, we first obtain that Tr u f = 0 on Γ (since | Im α| > 0). By the Robin boundary condition on Γ, we then obtain that ∂u ∂n = 0 on Γ (in the sense of a continuous linear functional on B 2,2 β (Γ)). Then, u f = 0 in Ω follows by the uniqueness of the solution to the Cauchy problem for ∆ + ω 2 Id in the connected domain Ω with Cauchy data on Γ (see for example [17, Theorems 1.1 and 1.2], which can be directly adapted to the case of a domain Ω with a d-set boundary satisfying the conditions of Theorem 1 thanks to Theorem 3). The operator (Id -ω 2 T ) -1 is thus well defined and is also a linear continuous operator, by the Fredholm alternative theorem. Thus, we obtain

u f V (Ω,µ) ≤ A Id -ω 2 T f L2(Ω) ≤ C(ω, α, C P (Ω)) f L2(Ω) .
In the same way, when f = 0, the solution u h in V (Ω) satisfies the following variational formulation:

∀φ ∈ V (Ω) (Id -ω 2 A • S + i Im α Â • Tr)u f , φ V (Ω,µ) = ( Â • Tr h, φ) V (Ω,µ) .
Hence, as previously, we have

u h V (Ω,µ) ≤ Â Tr Id -ω 2 T h H 1 (Ω,µ) ≤ C(ω, α, C P (Ω)) h H 1 (Ω,µ) .
Consequently, we have proved the well-posedness and estimate [START_REF] Bucur | Shape optimization problems with Robin conditions on the free boundary[END_REF] for g = 0. Hence, by the standard lifting procedure, we can for instance chose g ∈ H 1 (Ω) as the unique weak solution of (2) with ω = f = 0, with the homogeneous Neumann boundary conditions on Γ N and Γ and with g| Γ D = g, to obtain the result with g = 0.

To prove that the regularity of the boundary improves the regularity of the solution, we follow the classical approach, explained for elliptic equations in [START_REF] Evans | Partial Differential Equations[END_REF] Theorem 5 p. 323.

The linearity and the continuity of B are evident and equivalent to estimate [START_REF] Bucur | Shape optimization problems with Robin conditions on the free boundary[END_REF]. Let us prove that for any fixed ω > 0, B is also compact (see also Ref. [START_REF] Arfi | Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets[END_REF] for the real Robin boundary condition). Indeed, let

(f j , h j ) (f, h) in L 2 (Ω) × H 1 (Ω).
Taking for all j ∈ N, u j = B(f j , h j ) and u = B(f, h), by the linearity and the continuity of B it follows that

u j V (Ω) u. Knowing in addition that Tr : V (Ω) → L 2 (Γ, µ) and the inclusion of H 1 (Ω)
in L 2 (Ω) are compact (see Ref. [START_REF] Arfi | Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets[END_REF]) we have that Tr u j → Tr u in L 2 (Γ, µ) and u j → u in L 2 (Ω). Choosing v = u j in the variational formulation ( 9) we find

u j 2 V (Ω,µ) = ω 2 u j 2 L2(Ω) -i Γ Im α| Tr u j | 2 dµ - Ω f j u j dx + Γ Tr h j Tr u j dµ, (13)
and hence,

lim j→+∞ u j 2 V (Ω,µ) =ω 2 u 2 L2(Ω) -i Γ Im α| Tr u| 2 dµ - Ω f udx + Γ Tr hTr udµ = u 2 V (Ω,µ) .
Having both u j u in V (Ω) and u j V (Ω,µ) → u V (Ω,µ) implies that u j → u in V (Ω) and hence B is compact. Since the norm u 2 V (Ω,µ) on V (Ω) is equivalent [START_REF] Arfi | Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets[END_REF] to the norm

u 2 J = A u 2 L2(Ω) + B ∇u 2 L2(Ω) + C u 2 L2(Γ,µ) ,
the operator B is also compact with respect to this norm.

In order to relate the model with a damping on the boundary and the model with a damping in the volume, we propose in Appendix B a new theorem to identify the parameter α in the Robin boundary condition (Theorem 4). This parameter provides the best approximation (in some error minimizing sense) of the latter model by the former, in the case of a flat boundary Γ.

Shape design problem

We consider the two dimensional shape design problem, which consists in optimizing the shape of Γ with the Robin dissipative condition in order to minimize the acoustic energy of system (2). The boundaries with the Neumann and Dirichlet conditions Γ D and Γ N are supposed to be fixed.

We also define a fixed open set D with a Lipschitz boundary which contains all domains Ω. Actually, as only a part of the boundary (precisely Γ) changes its shape, we also impose that the changing part always lies inside of the closure of a fixed open set G with a Lipschitz boundary: Γ ⊂ G. The set G forbids Γ to be too close to Γ D , making the idea of an acoustical wall more realistic.

To introduce the class of admissible domains, on which we minimize the acoustical energy of system (2), we define Lip as the class of all domains Ω ⊂ D for which 1. there exists a fixed ε > 0 such that all domains Ω ∈ Lip satisfy the ε-cone property [START_REF] Agmon | Lectures on elliptic boundary value problems[END_REF][START_REF] Chenais | On the existence of a solution in a domain identification problem[END_REF]: for all x ∈ ∂Ω, there exists ξ x ∈ R 2 with ξ x = 1 such that for all y ∈ Ω ∩ B(x, ε)

C(y, ξ x , ε) = {z ∈ R 2 |(z -y, ξ x ) ≥ cos(ε) z -y and 0 < z -y < ε} ⊂ Ω.
2. there exists a fixed ĉ > 0 such that for any Ω ∈ Lip and for all x ∈ Γ we have

Γ∩B(x,r) dλ ≤ ĉr, (14)
where B(x, r) is the open Euclidean ball centered in x with radius r and λ is the usual one-dimensional Lebesgue measure on Γ. [START_REF] Chenais | On the existence of a solution in a domain identification problem[END_REF] implies that all Γ for Ω ∈ Lip have uniform length: there exists M > 0 depending on the chosen ĉ > 0 such that for all Ω ∈ Lip it holds Vol(∂Ω) = ∂Ω dλ ≤ M .

The constant M (and hence initially ĉ) can be chosen arbitrary large but finite. We denote by Ω 0 ∈ Lip and Γ 0 ⊂ G the "reference" domain and the "reference" boundary respectively (actually ∂Ω 0 = Γ D ∪ Γ N ∪ Γ 0 ) corresponding to the initial shape before optimization.

Thus, the admissible class of domains can be defined as

U ad (Ω 0 , ε, ĉ, G) = {Ω ∈ Lip | Γ D ∪ Γ N ⊂ ∂Ω, Γ ⊂ G, M 0 ≤ Γ dλ ≤ M (ĉ), Ω dx = Vol(Ω 0 )}, ( 15
)
where ĉ is given sufficiently large in the aim to have a sufficiently large constant M > 0 in the sense that it is not less than M 0 > 0, which is the length of the straight line boundary. Moreover the case when M is equal to the length of the plane boundary M 0 is the trivial case when U ad (Ω 0 , ε, ĉ, G) contains only one unique domain with the plane boundary, which hence is trivially optimal. Therefore the problem becomes interesting for a sufficiently large M .

In what follows we denote by λ the 1-dimensional Lebesgue measure on the Lipschitz boundary Γ, by m 1 the 1-dimensional Hausdorff measure (which is equal to λ on Γ) and we denote by u(Ω, µ) the weak solution of the Helmholtz problem on Ω satisfying (9) with 1-dimensional Radon measure µ.

We define

J(Ω, u(Ω, µ), λ) = A Ω |u(Ω, µ)| 2 dx + B Ω |∇u(Ω, µ)| 2 dx + C Γ |u(Ω, µ)| 2 dλ ( 16
)
for given µ and λ and with A ≥ 0, B ≥ 0, C ≥ 0 positive constants for any fixed ω > 0.

Ideally we would like to minimize J(Ω, u(Ω, λ), λ) on U ad (Ω 0 , ε, ĉ, G), however we are able to prove the existence of Ω opt in U ad (Ω 0 , ε, ĉ, G) with a 1-measure µ * , equivalent to λ, satisfying µ * (Γ opt ) ≥ λ(Γ opt ) on its boundary Γ opt , such that J(Ω, u(Ω, µ * ), λ) realizes the infimum of J(Ω, u(Ω, λ), λ) on U ad (Ω 0 , ε, ĉ, G). So, if µ * (Γ opt ) = λ(Γ opt ), i.e. µ * = λ, then Ω opt realizes the minimum of J(Ω, u(Ω, λ), λ).

In order to keep the volume constraint, instead of Eq. ( 16) we can also consider the objective function

J 1 (Ω, u(Ω, λ), λ) =A Ω |u(Ω, λ)| 2 dx + B Ω |∇u(Ω, λ)| 2 dx + C Γ |u(Ω, λ)| 2 dλ + κ(Vol(Ω) -Vol(Ω 0 )) 2 , ( 17
)
where κ is some (large) positive constant penalizing the volume variation. First of all let us prove the following lemma:

Lemma 1. Let U ad (Ω 0 , ε, ĉ, G) be the class of admissible domains defined in [START_REF] Cox | The rate at which energy decays in a damped string[END_REF]. Then the following statements hold:

1. U ad (Ω 0 , ε, ĉ, G) is closed with respect to the Hausdorff convergence, in the sense of characteristic functions in L 1 (D) and in the sense of compacts.

If

(Ω n ) n∈N * ⊂ U ad (Ω 0 , ε, ĉ, G) then there exists a subsequence (Ω n k ) k∈N * ⊂ (Ω n ) n∈N * and a domain Ω ∈ U ad (Ω 0 , ε, ĉ, G) such that (Ω n k ) k∈N * converges
to Ω with respect to these three types of convergences and in addition Ω n k and Γ n k converge in the sense of Hausdorff respectively to Ω and Γ.

3. Let (Ω n ) n∈N * ⊂ U ad (Ω 0 , ε, ĉ, G) be a sequence converging to Ω in U ad (Ω 0 , ε, ĉ, G) in
(i.e. if (Ω n ) n∈N * ⊂ U ad (Ω 0 , ε, ĉ, G) and d H (D \ Ω n , D \ Ω) → 0 for n → +∞, which means that Ω n → Ω in the sense of Hausdorff, then Ω ∈ U ), but also in the sense of characteristic functions, (i.e. if (Ω n ) n∈N * ⊂ U ad (Ω 0 , ε, ĉ, G) and 1 Ωn → 1 Ω for n → +∞ in L p loc (R 2 ) for all p ∈ [1, ∞[, then Ω ∈ U )
and also in the sense of compacts (if for all K compact in Ω it follows that K ⊂ Ω n and for all O compact in D \ Ω it follows that O ∈ D \ Ω n for a sufficiently large n, then Ω ∈ U ). Consequently, if (Ω n ) n∈N * ⊂ U ad (Ω 0 , ε, ĉ, G) ⊂ U converges in these three senses to Ω ∈ U , then, since for all n the domains Ω n are subsets of a fixed open set D which is bounded in R 2 , the sequence 1 Ωn → 1 Ω for n → +∞ in L 1 (D). This strong convergence of the characteristic functions [29, Prop. 2.3.6, p.47] implies that the perimeter and volume are respectively lower semicontinuous and continuous functions of Ω n . Hence,

Vol(Ω) = lim n→+∞ Vol(Ω n ) = Vol(Ω 0 )
and in addition, since Γ N and Γ D are fixed and all Γ n ⊂ G,

Γ dλ ≤ lim inf n→+∞ Γn dλ. (18)
In particular Eq. ( 18) ensures that as soon as for all Γ n , ( 14) holds with ĉ > 0 independent of n, then it holds also for Γ and in the same way Γ dλ ≤ M . As Γ is Lipschitz and in particular is given by a continuous curve, we also have Γ dλ ≥ M 0 . Thus, we conclude that

U ad (Ω 0 , ε, ĉ, G) is closed. Point 2.
For the second statement it is sufficient to notice that Theorem 2.4.10 [29, p. 56] holds on U . As U is compact with respect to the three considered convergences and as U ad (Ω 0 , ε, ĉ, G) is its closed subset, then it is compact too.

Point 3. Let us prove the third point. We take a sequence (Ω n ) n∈N * in U ad (Ω 0 , ε, ĉ, G) which converges to Ω ∈ U ad (Ω 0 , ε, ĉ, G) in all senses of point 2. By Lemma 3.5 [START_REF] Falconer | The Geometry of Fractal Sets[END_REF] the Lipschitz curves are 1-sets (see Definition 1). We take the 1-dimensional Hausdorff measures η n on D in R 2 having their supports on Γ n . The sequence of measures As Γ n k converges in the sense of Hausdorff to Γ then the support of µ * is equal to Γ:

(η n ) n∈N * ⊂ M b (D) is
D ψdµ * = Γ ψdµ * , and hence ∀ψ ∈ C 0 (D) Γn k ψdλ → Γ ψdµ * . (19)
But the restriction of µ * to Γ is not necessarily equal to the Hausdorff measure on Γ, we only can prove that it is an equivalent 1-dimensional measure to the usual Lebesgue measure on Γ by [30, Prop. 1, p. 30] (see also Theorem 1 on p. 32). Indeed, taking in the definition of the weak * limit a particular ψ equal to 1 for all x ∈ G and knowing that all Γ n k and Γ itself are subsets of G, we find by the lower semicontinuity of perimeters that

Γ dµ * = lim k Γn k dλ = lim inf k Γn k dλ ≥ Γ dλ.
For instance, if Γ n are boundaries with a constant length which oscillate around a plane segment that has a length twice smaller than Γ n , and such that Γ n → Γ in the sense of Hausdorff, it easy to see that

λ(Γ n ) → µ * (Γ) = 2λ(Γ) > λ(Γ).
In addition, once again by the definition of G containing all Γ n and Γ in D, the weak * limit ( 19) also holds for all ψ ∈ C(D). Consequently, if η is the 1-dimensional Hausdorff measure with support equal to Γ, then µ * ≥ η and thus there exists c 1 > 0 such that

c 1 r ≤ η(B(x, r)) ≤ µ * (B(x, r)) x ∈ Γ, 0 < r ≤ 1.
To show that µ * is also a 1-measure we need to have the upper bound too. Since B(x, r) is open, it holds, thanks to [START_REF] Chenais | On the existence of a solution in a domain identification problem[END_REF] and to the uniform bound of lengths

M 0 ≤ η n k (Γ n k ) = λ(Γ n k ) ≤ M , 1 µ * (Γ) µ * (B(x, r)) ≤ lim inf k 1 Γn k dλ η n k (B(x, r)) ≤ 1 M 0 lim inf k η n k (B(x, r)) ≤ 1 M 0 ĉr
by the Portmanteau theorem (see for instance Theorem 11.1.1 in Dudley [START_REF] Dudley | Real Analysis and Probability, The Wadsworth & Brooks Cole Mathematics Series[END_REF]). Hence, µ * (B(x, r)) ≤ c 2 r with the constant c 2 = µ * (Γ) M0 ĉ > 0 independent of n k . Then we conclude that µ * is 1-measure on Γ and thus equivalent to η by [START_REF] Jonsson | Function spaces on subsets of R n[END_REF]Prop. 1,p. 30].

Let us now prove the existence of an optimal shape in a certain sense: Theorem 2. Let Ω 0 ⊂ D be a domain of the class Lip with a Lipschitz boundary ∂Ω 0 of bounded length, such that Γ D ∪ Γ N ⊂ ∂Ω 0 and Γ 0 = ∂Ω 0 \ (Γ D ∪ Γ N ) ⊂ G, U ad be defined by [START_REF] Cox | The rate at which energy decays in a damped string[END_REF] and ω > 0 be fixed. For the objective function J(Ω, u(Ω, λ), λ), defined in ( 16) and constructed with the weak solution of the Helmholtz problem (2) for some fixed

α ∈ C(G), f ∈ L 2 (D), g ∈ H 1 2 (Γ D ) and h ∈ H 1 (D)
, there exists Ω opt ∈ U ad (Ω 0 , ε, ĉ, G) and there exists a finite valued 1-dimensional positive measure µ * on its boundary Γ opt equivalent to λ such that

Γopt dµ * ≥ Γopt dλ ( 20
)
and

J(Ω opt , u(Ω opt , µ * ), λ) ≤ inf Ω∈U ad (Ω0,ε,ĉ,G) J(Ω, u(Ω, λ), λ) = J(Ω opt , u(Ω opt , µ * ), µ * ). ( 21
)
If µ * = λ, then J(Ω opt , u(Ω opt , λ), λ) is the minimum on U ad (Ω 0 , ε, ĉ, G).

Proof. Let us start by noticing that, since all Ω ∈ U ad (Ω 0 , ε, ĉ, G) are included in the same domain D, it follows that the Poincaré constants in Theorem 1 can all be bounded by the same constant C(Vol(D)) depending only on the volume of D.

Let (Ω n ) n∈N * ⊂ U ad (Ω 0 , ε, ĉ, G)
be a minimizing sequence of the functional J(Ω) (this minimizing sequence exists since J(Ω) ≥ 0).

Hence, by Lemma 1 there exist a domain Ω with Γ and a subsequence (Ω n k ) k∈N converging in these three senses to Ω and such that Ω n k and Γ n k converge in the sense of Hausdorff to Ω and Γ respectively (all other boundary parts are the same as the sequence is in U ad (Ω 0 , ε, ĉ, G)). In the aim to abbreviate the notations, in what follows the index n k is changed to n.

By point 3 of Lemma 1 there exist a subsequence of boundaries (Γ n k ) k∈N * ⊂ (Γ n ) n∈N * and a positive measure µ * on Γ, equivalent to λ, such that Eq. ( 20) holds and

∀ψ ∈ C(D) Γn k ψdλ → Γ ψdµ * . ( 22
)
Therefore, all norms of L 2 (Γ) and V (Ω) computed with this measure on Γ are equivalent to the corresponding norms computed with the Lebesgue measure on Γ.

To avoid complicated notations we denote again Γ n k by Γ n . Let us consider now the solutions (u n ) n∈N * of the Helmholtz problem on (Ω n ) n∈N * . Since domains of U ad (Ω 0 , ε, ĉ, G)) have Lipschitz boundaries with finite perimeters, we use (see [START_REF] Chenais | On the existence of a solution in a domain identification problem[END_REF]) the fact that the norm of the extension operator E :

H 1 (Ω) → H 1 (R 2 ) is bounded on such a family of domains i.e there exists a constant C E > 0 independent of n such that Eu n | D H 1 (D) ≤ C E u n H 1 (Ωn) , (23)
to deduce that the sequence

(Eu n | D ) n∈N * is bounded in H 1 (D): Eu n | D H 1 (D) ≤ C E u n H 1 (Ωn) ≤ C E C(α, ω, C p (Ω n )) f L2(Ωn) + g H 1 2 (Γ D ) + h H 1 (Ωn) ≤ C E C(α, ω, Vol(D)) f L2(D) + g H 1 2 (Γ D ) + C E h H 1 (D) ,
which means that there exists a constant C > 0 independent of n such that

Eu n | D H 1 (D) ≤ C for all n.
Here, in addition to the uniform boundedness of the extension operators, we have also used Eq. ( 10) and the fact that Γ D is the same for all n. Consequently, there exists

u * ∈ H 1 (D) such that Eu n | D u * in H 1 (D)
. By compactness of the trace operator Tr Γ :

H 1 (D) → L 2 (Γ) and of the inclusion of H 1 (D) in L 2 (D) (see Theorem 3), we directly have that Tr Γ (Eu n | D ) → Tr Γ (u * ) in L 2 (Γ) and Eu n | D → u * in L 2 (D).
Let us show that u * is equal to the weak solution u(Ω, µ * ) of ( 9) on Ω in the sense of the variational formulation ( 9) considered with µ * .

From the variational formulation ( 9), taking f ∈ L 2 (D) and h ∈ H 1 (D), let us consider linear functionals defined for a fixed v ∈ V (D) and for all w n and w in V (D)

F n [w n , v] =(∇w n , ∇v) L2(Ωn) -ω 2 (w n , v) L2(Ωn) + (αw n , v) L2(Γn,λ) + (f, v) L2(Ωn) -(Tr Γn h, v) L2(Γn,λ) , F [w, v] =(∇w, ∇v) L2(Ω) -ω 2 (w, v) L2(Ω) + (αw, v) L2(Γ,µ * ) + (f, v) L2(Ω) -(Tr Γ h, v) L2(Γ,µ * ) .
We start by showing that as soon as w n w in V (D)

∀v ∈ V (D) F n [w n , v] → F [w, v] for n → +∞. ( 24
)
Thus we consider

|F n [w n , v] -F [w, v]| ≤ |(∇w n , 1 Ωn ∇v) L2(D) -(∇w, 1 Ω ∇v) L2(D) | +ω 2 |(w n , 1 Ωn v) L2(D) -(w, 1 Ω v) L2(D) | + (αw n , v) L2(Γn,λ) -(αw, v) L2(Γ,µ * ) + |(f, (1 Ωn -1 Ω )v) L2(D) | +|(Tr Γn h, v) L2(Γn,λ) -(Tr Γ h, v) L2(Γ,µ * ) |.
Since Ω n → Ω in the sense of characteristic functions and v ∈ H 1 (D), we directly have that 1 Ωn ∇v → 1 Ω ∇v in L 2 (D), which with w n w in V (D) gives that

(∇w n , 1 Ωn ∇v) L2(D) → (∇w, 1 Ω ∇v) L2(D) for n → +∞.
By the compactness of the inclusion of

H 1 (D) in L 2 (D), w n → w in L 2 (D)
and by the convergence of the characteristic functions

1 Ωn v → 1 Ω v in L 2 (D)
, hence we also have

(w n , 1 Ωn v) L2(D) → (w, 1 Ω v) L2(D)
and similarly, (f, (1

Ωn -1 Ω )v) L2(D) → 0. Let us prove that ∀v ∈ C(D) ∩ V (D) (αw n , v) L2(Γn,λ) → (αw, v) L2(Γ,µ * ) . (25)
Thanks to [START_REF] Maz'ja | Sobolev Spaces[END_REF] Theorem 1.1.6/2, for all domains Ω ∈ U ad (Ω 0 , ε, ĉ, G) and D itself, the space C ∞ (Ω) ∩ V (Ω) is dense in V (Ω). Thus there exists a sequence (φ m ) m∈N ⊂ C ∞ (D) ∩ V (D) converging strongly to w ∈ V (D). Therefore, following [START_REF] Capitanelli | Asymptotics for mixed Dirichlet-Robin problems in irregular domains[END_REF] we have

Γn Trw n Trvdλ - Γ TrwTrvdµ * ≤ ≤ Γn |Trw n -Trw||Trv|dλ + Γn |Trw -Trφ m ||Trv|dλ + Γn Trφ m Trvdλ - Γ Trφ m Trvdµ * + Γ |Trφ m -Trw||Trv|dµ * . (26)
We start by estimating the first term in Eq. ( 26): control it with the Cauchy-Schwarz inequality

Γn |Trw n -Trw||Trv|dλ ≤ Tr(w n -w) L2(Γn,λ) Trv L2(Γn,λ) .
Moreover, there exists a positive constant C σ > 0 independent of n such that for

1 2 < σ ≤ 1 it holds ∀w ∈ H σ (R 2 ) Tr Γn w 2 L2(Γn,λ) ≤ C σ w 2 H σ (R 2 ) . (27)
It is a direct corollary of the proof of [START_REF] Capitanelli | Asymptotics for mixed Dirichlet-Robin problems in irregular domains[END_REF] Theorem 5.3 and the fact that the lengths of Γ and all Γ n are finite and bounded by a constant, denoted by M . In addition,

Trv L2(Γn,λ) ≤ λ(Γ n ) v L ∞ (D) ≤ M v L ∞ (D) .
Moreover, by Theorem 5.8 [START_REF] Capitanelli | Asymptotics for mixed Dirichlet-Robin problems in irregular domains[END_REF], for D (but also for all domains in U ad (Ω 0 , ε, ĉ, G)) there exists a bounded linear extension operator

E σ : H σ (D) → H σ (R 2 ), 1 2 < σ ≤ 1, with E σ v H σ (R 2 ) ≤ C D v H σ (D) . (28)
Hence, applying Eq. ( 27) and Eq. ( 28), we obtain that

Tr(w n -w) L2(Γn,λ) ≤ C σ E σ (w n -w) H σ (R 2 ) ≤ C σ C D w n -w H σ (D) ,
from where, by the compactness of the embedding of H 1 (D) in H σ (D) for 1 2 < σ < 1, we finally have that w n -w H σ (D) → 0 for n → +∞ and consequently the first term in Eq. ( 26) converges to 0 for n → +∞.

For the second term (and in the same way the last term) in Eq. ( 26), as previously we directly find

Γn |Trw -Trφ m ||Trv|dλ ≤ C w -φ m H 1 (D) → 0 for m → +∞
with a constant C > 0 independent of n. For the last term we simply replace Γ n by Γ, knowing that Ω ∈ U ad (Ω 0 , ε, ĉ, G). Hence, for all ε > 0 there exists k ∈ N (uniformly on n) such that

∀m ≥ k max Γn |Trw -Trφ m ||Trv|dλ , Γ |Trφ m -Trw||Trv|dµ * < ε.
Thus, let us fix such an m.

Finally, for the third term in Eq. ( 26), we use Eq. ( 22) which, by the continuity and the boundedness of φ m v in D with a standard density argument, implies

Γn |Trφ m Trv|dλ → Γ |Trφ m Trv|dµ * for n → +∞. (29)
Therefore, for the sufficiently large m that we have fixed, we also have that

∀ε > 0 ∃p ∈ N : ∀n ≥ p Γn |Trφ m Trv|dλ - Γ |Trφ m Trv|dµ * < ε.
Putting all results together for the four terms of Eq. ( 26), we obtain Eq. ( 25), which by the density of C(D) ∩ V (D) in V (D), also holds for all v ∈ V (D). Consequently, we also have, as

h ∈ V (D) ∀v ∈ V (D) (Tr Γn h, v) L2(Γn,λ) → (Tr Γ h, v) L2(Γ,µ * ) for n → +∞.
This concludes the proof of Eq. ( 24).

Hence, taking w n = Eu n | D ∈ V (D), i.e. the extensions of solutions on Ω n , which are uniformly bounded and weakly converge to u * ∈ V (D), we find that for all v ∈ V (D)

0 = F n [Eu n | D , v] → F [u * , v] = 0 for n → +∞.
This means that u * is a weak solution on Ω in the sense of the variational formulation (9) considered with µ * , and by the uniqueness of the weak solution on Ω, u * | Ω = u(Ω, µ * ). In order to conclude that the infimum of J is realized, we shall prove that inf

Ω∈U ad (Ω0,ε,ĉ,G) J(Ω, u(Ω, λ), λ) = lim n→+∞ J(Ω n , u n (Ω n , λ), λ) = J(Ω, u(Ω, µ * ), µ * ).

Let us start by showing that

(1 Ωn Eu n | D ) n∈N * converges strongly to 1 Ω Eu| D in V (D).
Firstly, we find in the same way as previously (see [START_REF] Capitanelli | Robin boundary condition on scale irregular fractals[END_REF]) that

Γn |Trw n | 2 dλ - Γ |Trw| 2 dµ * → 0 for n → +∞. (30)
Then, once again, by the fact that the weak convergence of (Eu

n | D ) n∈N * to Eu| D in V (D)
implies the strong convergence in L 2 (D) and by Eqs. ( 25), [START_REF] Jonsson | Function spaces on subsets of R n[END_REF], we obtain

lim n→+∞ 1 Ωn Eu n | D 2 V (D,λ) = lim n→+∞ ω 2 1 Ωn Eu n | D 2 L2(D) -i Γn Im α| Tr u n | 2 dλ - D 1 Ωn f Eu n | D dx + Γn Tr hTr u n dλ = ω 2 1 Ω Eu| D 2 L2(D) -i Γ Im α| Tr u| 2 dµ * - D 1 Ω f Eu| D dx + Γ Tr hTr udµ * = u 2 V (Ω,µ * ) .
Since we have at the same time the weak convergence and the convergence of norms, it implies the strong convergence in V (D) of (1

Ωn Eu n | D ) n∈N * to 1 Ω Eu| D .
Hence, as the functional J, which can be considered as an equivalent norm on V (D), is continuous:

lim n→+∞ J(Ω n , u n (Ω n , λ), λ) = J(Ω, u(Ω, µ * ), µ * ),
i.e., as Ω n is a minimizing sequence of J, J(Ω, u(Ω, µ * ), µ * ) is the infimum for all Ω ∈ U ad (Ω 0 , ε, ĉ, G). By the relation Γ dµ * ≥ Γ dλ we directly have

J(Ω, u(Ω, µ * ), µ * ) ≥ J(Ω, u(Ω, µ * ), λ). (31)
If we have µ * = λ, then J(Ω, u(Ω, λ), λ) is the minimum.

Conclusion

Started by the well-posedness result on a large class of domains with d-set boundaries including even fractal boundaries, we showed that the problem of finding an optimal shape for the Helmholtz problem with a dissipative boundary has at least one solution in the sense of a suitable measure µ * .

A d-sets and trace theorems on a d-set

Let us define the main notions which we use in Theorem 1.

Definition 1 (Ahlfors d-regular set or d-set [START_REF] Jonsson | Function spaces on subsets of R n[END_REF]). Let F be a closed Borel non-empty subset of R N . The set F is is called a d-set (0 < d ≤ N ) if there exists a d-measure µ on F , i.e. a positive Borel measure with support F (supp µ = F ) such that there exist constants c 1 ,

c 2 > 0, c 1 r d ≤ µ(B(x, r)) ≤ c 2 r d , for ∀ x ∈ F, 0 < r ≤ 1,
where B(x, r) ⊂ R N denotes the Euclidean ball centered at x and of radius r.

As [30, Prop. 1, p 30] all d-measures on a fixed d-set F are equivalent, it is also possible to define a d-set by the d-dimensional Hausdorff measure m d :

c 1 r d ≤ m d (F ∩ B(x, r)) ≤ c 2 r d , for ∀ x ∈ F, 0 < r ≤ 1
which in particular implies that F has Hausdorff dimension d in the neighborhood of each point of F [30, p.33].

If the boundary ∂Ω is a d-set endowed with a d-measure µ, then we denote by L 2 (∂Ω, µ) the Lebesgue space defined with respect to this measure with the norm

u L2(∂Ω,µ) = ∂Ω |u| 2 dµ 1 2
.

In particular, N -sets (d-set with d = N ) satisfy

∃c > 0 ∀x ∈ Ω, ∀r ∈]0, δ[∩]0, 1] λ(B(x, r) ∩ Ω) ≥ Cλ(B(x, r)) = cr N ,
where λ(A) denotes the Lebesgue measure of a set A of R N . This property is also called the measure density condition [START_REF] Hajłasz | Sobolev embeddings, extensions and measure density condition[END_REF]. Let us notice that an N -set Ω cannot be "thin" close to its boundary ∂Ω.

The trace operator on a d-set is understood in the following way:

Definition 2 (Trace operator). For an arbitrary open set Ω of R N , the trace operator Tr is defined [START_REF] Jonsson | Function spaces on subsets of R n[END_REF] for u ∈ L 1 loc (Ω) by

Tr u(x) = lim r→0 1 λ(Ω ∩ B(x, r)) Ω∩B(x,r) u(y)dy,
where λ denotes the Lebesgue measure. The trace operator Tr is considered for all x ∈ Ω for which the limit exists.

Hence, the following Theorem (see Ref. [START_REF] Arfi | Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets[END_REF] Section 2) generalizes the classical results [START_REF] Lions | Non-Homogeneous Boundary Value Problems and Applications[END_REF][START_REF] Marschall | The trace of Sobolev-Slobodeckij spaces on Lipschitz domains[END_REF] for domains with the Lipschitz boundaries ∂Ω: Theorem 3. Let Ω be an admissible domain in R N in the sense of Ref. [START_REF] Arfi | Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets[END_REF], i.e. Ω is an N -set, such that its boundary ∂Ω is a compact d-set with a d-measure µ, N -2 < d < N , and the norms f

H 1 (Ω) and f C 1 2 (Ω) = f L2(Ω) + f 1,Ω L2(Ω) with f 1,Ω (x) = sup r>0 r -1 inf c∈R 1 λ(B(x, r)) B(x,r)∩Ω |f (y) -c|dy,
where λ is the N -dimensional Lebesgue measure, are equivalent on H 1 (Ω). Then, 4. the Green formula holds for all u and v from H 1 (Ω) with ∆u ∈ L 2 (Ω):

1. H 1 (Ω) is compactly embedded in L loc 2 (Ω) or in L 2 (Ω) if Ω is bounded; 2. Tr Ω : H 1 (R N ) → H 1 (Ω)
Ω v∆udx + Ω ∇v.∇udx = ∂u ∂ν , Trv ((B 2,2 β (∂Ω)) ,B 2,2 β (∂Ω)) , (32)
where the dual Besov space (B 2,2 β (∂Ω)) = B 2,2 -β (∂Ω) is introduced in Ref. [START_REF] Jonsson | The dual of Besov spaces on fractals[END_REF].

5. the usual integration by parts holds for all u and v from H 1 (Ω) in the following weak sense

uν i , v (B 2,2 β (∂Ω)) ,B 2,2 β (∂Ω)) := Ω ∂u ∂x i vdx + Ω u ∂v ∂x i dx i = 1, . . . , N, (33)
where by uν i is denoted the linear continuous functional on B 2,2 β (∂Ω).

6. u H 1 (Ω) is equivalent to u Tr = Ω |∇u| 2 dx + ∂Ω |Tru| 2 dµ 1 2 .
Theorem 3 is a particular case of the results proven in Ref. [START_REF] Arfi | Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets[END_REF]. We also notice that in the framework of the Sobolev space H 1 and the Besov spaces B 2,2 β with β < 1, as here, we do not need to impose Markov's local inequality on ∂Ω, as it is trivially satisfied (see Ref. [START_REF] Jonsson | Boundary value problems and brownian motion on fractals[END_REF] p. 198). To prove formula [START_REF] Lancia | A Transmission Problem with a Fractal Interface[END_REF] we follow the proof of formula (4.11) of Theorem 4.5 in [START_REF] Creo | Magnetostatic problems in fractal domains[END_REF] using the existence of a sequence of domains (Ω m ) m∈N * with Lipschitz boundaries such that

Ω m ⊂ Ω m+1 and Ω = ∪ ∞ m=1 Ω m .
B Approximation of the damping parameter α in the Robin boundary condition by a model with dissipation in the volume boundary, denoted by Γ, which is the straight line starting in (0, -) and ending in (0, ). In addition let Ω 1 be the supplementary domain of Ω 0 in Ω, so that Γ is the common boundary of Ω 0 and Ω 1 . The length L is supposed to be large enough.

Let the original problem (the frequency version of the wave damped problem (1)) be

-∇ • (η 0 ∇u 0 ) -ω 2 ξ 0 u 0 = 0 in Ω 0 , (34)
-∇ • (η 1 ∇u 1 ) -ω 2 ξ1 u 1 = 0 in Ω 1 , (35)
with ξ1 = ξ 1 1 + ai ξ 1 ω ,
together with boundary conditions on Γ

u 0 = u 1 and η 0 ∇u 0 • n = η 1 ∇u 1 • n, (36)
and the condition on the left boundary

u 0 (-L, y) = g(y), (37)
and some other boundary conditions. Let the modified problem be

-∇ • (η 0 ∇u 2 ) -ω 2 ξ 0 u 2 = 0 in Ω 0 (38)
with boundary absorption condition on Γ

η 0 ∇u 2 • n + αu 2 = 0 (39)
and the condition on the left boundary

u 2 (-L, y) = g(y). (40)
Let u 0 , u 1 , u 2 and g be decomposed into Fourier modes in the y direction, denoting by k the associated wave number. Then the complex parameter α, minimizing the following expression

A||u 0 -u 2 || 2 L2(Ω0) + B||∇(u 0 -u 2 )|| 2 L2(Ω0)
can be found from the minimization of the error function e(α) :=

k= nπ L ,n∈Z e k (α),
where e k are given by

e k (α) = (A + B|k| 2 ) 1 2λ 0 |χ| 2 [1 -exp(-2λ 0 L)] +|γ| 2 [exp(2λ 0 L) -1] + 2LRe (χγ) + B λ 0 2 |χ| 2 [1 -exp(-2λ 0 L)] + |γ| 2 [exp(2λ 0 L) -1] -2Bλ 2 0 LRe (χγ) if k 2 ≥ ξ0 η0 ω 2 or e k (α) = (A + B|k| 2 ) L(|χ| 2 + |γ| 2 ) + i λ 0 Im {χγ [1 -exp(-2λ 0 L)]} +BL|λ 0 | 2 |χ| 2 + |γ| 2 + iBλ 0 Im {χγ [1 -exp(-2λ 0 L)]} if k 2 < ξ0 η0 ω 2 , in which f (x) = (λ 0 η 0 -x) exp(-λ 0 L) + (λ 0 η 0 + x) exp(λ 0 L), χ(k, α) = g k λ 0 η 0 -λ 1 η 1 f (λ 1 η 1 ) - λ 0 η 0 -α f (α) , γ(k, α) = g k λ 0 η 0 + λ 1 η 1 f (λ 1 η 1 ) - λ 0 η 0 + α f (α) , where    λ 0 = k 2 -ξ0 η0 ω 2 if k 2 ≥ ξ0 η0 ω 2 , λ 0 = i ξ0 η0 ω 2 -k 2 if k 2 ≤ ξ0 η0 ω 2 . ( 41
)
Proof. First of all, e(α)

:= A||u 0 -u 2 || 2 L2(Ω0) + B||∇(u 0 -u 2 )|| 2 L2(Ω0)
can be decomposed as a sum of e k (α) e(α) :=

k= nπ L ,n∈Z e k (α), with e k (α) = A||u 0,k -u 2,k || 2 L2(]-L,0[) + B||∇(u 0,k -u 2,k )|| 2 L2(]-L,0[) ,
where we have decomposed decomposed u 0 , u 1 and u 2 into modes in the y direction, denoting by k the associated wave number.

The mode u 0,k solves

∂ xx u 0,k -k 2 - ξ 0 η 0 ω 2 u 0,k = 0, and thus u 0,k (x) = A 0 exp(λ 0 x) + B 0 exp(-λ 0 x), (42)
where λ 0 is given in Eq. ( 41). The mode u 1,k solves

∂ xx u 1,k -k 2 - ξ1 η 1 ω 2 u 1,k = 0, and thus u 1,k (x) = A 1 exp(λ 1 x) + B 1 exp(-λ 1 x), (43)
where

λ 2 1 = k 2 -1 + ai ξ 1 ω ξ 1 η 1 ω 2 , so that λ 1 = 1 √ 2 k 2 - ξ 1 η 1 ω 2 + k 2 - ξ 1 η 1 ω 2 2 + aω η 1 2 - i √ 2 ξ 1 η 1 ω 2 -k 2 + k 2 - ξ 1 η 1 ω 2 2 + aω η 1 2 .
For large L, since Re(λ 1 ) > 0, the value of A 1 tend to 0, so that we may neglect the first contribution in the right-hand side of Eq. ( 43). Consequently we consider the expression

u 1,k (x) = B 1 exp(-λ 1 x). (44)
Continuity conditions [START_REF] Maz'ja | Sobolev Spaces[END_REF] and expressions (42) and (44) imply the following relations

A 0 + B 0 = B 1 , η 0 λ 0 (A 0 -B 0 ) = -η 1 λ 1 B 1 ,
from which we infer that B 0 = λ 0 η 0 + λ 1 η 1 λ 0 η 0 -λ 1 η 1 A 0 , and thus u 0,k (x) = A 0 exp(λ 0 x) + λ 0 η 0 + λ 1 η 1 λ 0 η 0 -λ 1 η 1 exp(-λ 0 x) .

The decomposition of the boundary condition [START_REF] Mohammadi | Applied shape optimization for fluids[END_REF] into Fourier modes implies that u 0,k (-L) = g k , which gives the final expression u 0,k (x) = g k [(λ 0 η 0 -λ 1 η 1 ) exp(λ 0 x) + (λ 0 η 0 + λ 1 η 1 ) exp(-λ 0 x)] [(λ 0 η 0 -λ 1 η 1 ) exp(-λ 0 L) + (λ 0 η 0 + λ 1 η 1 ) exp(λ 0 L)] .

Let us now turn to the expression of u 2,k . Since equation [START_REF] Münch | Optimal Internal Dissipation of a Damped Wave Equation Using a Topological Approach[END_REF] is the same as that verified by u 0,k , both solutions have the same general form:

Using ( 45) and ( 46), we have that

(u 0,k -u 2,k )(x) = χ(k, α) exp(λ 0 x) + γ(k, α) exp(-λ 0 x), (47)
where the coefficients χ and γ are computed from (45) and (46). In order to compute the L 2 norm of this expression, we must first compute the square of its modulus (by γ is denoted the complex conjugate of γ): Now, we also have to compute the L 2 norm of the gradient of (u 0,k -u 2,k ). Noting that

|u 0,k -u 2,k | 2 (x) =
∇(u 0,k -u 2,k ) = ∂ x (u 0,k -u 2,k ) ik(u 0,k -u 2,k ) , it holds that |∇(u 0,k -u 2,k )| 2 = |k| 2 |u 0,k -u 2,k | 2 + |∂ x (u 0,k -u 2,k )| 2 .
With expression (47), it follows that Therefore, we can find α as the solution of the mentioned minimization problem. Since the minimization will be done numerically and since the sequence (z, -z, zz, • • • ) = z(exp(i(j∆x)/∆x)) is the highest frequency mode that can be reached on a grid of size ∆x, then, in practice, the sum may be truncated to e ∆x (α) := k= nπ L ,n∈Z,-L ∆x ≤n≤ L ∆x e k (α).

For the equations ( 34)- [START_REF] Marschall | The trace of Sobolev-Slobodeckij spaces on Lipschitz domains[END_REF], we use the same coefficients as for problem [START_REF] Abe | A BE-based shape optimization method enhanced by topological derivative for sound scattering problems[END_REF] and take the values corresponding to a porous medium, called ISOREL, using in building insulation. More precisely we assume: φ = 0.7, γ p = 1.4, σ = 142300N.m -4 .s, ρ 0 = 1.2kg/m 3 , α h = 1.15, c 0 = 340m.s -1 . We could find the value of α presented in Fig. 2.

Remark 2. Fig. 2 allows us to compare the difference between two considered time-dependent models for the damping in the volume and for the damping on the boundary. We see that Re(α) is not a constant in general, but for ω → +∞ Im(α) is a linear function of ω. In this sense, the damping properties of two models are almost the same, but the reflection is more accurately considered by the damping wave equation in the volume.

Figure 1 :

 1 Figure 1: Example of a domain Ω in R 2 with three types of boundaries: Γ D and Γ N are fixed and Γ can be changed in the restricted area G. Here Ω ∪ G = D and obviously Ω ⊂ D.

the sense of point 2 .

 2 Then there exists a subsequence (Ω n k ) k∈N * with boundaries (Γ n k ) k∈N * and there exists a positive Radon 1-measure µ * with support on Γ, equivalent to the one-dimensional Hausdorff measure on it or simply to its Lebesgue measure λ, such that (a) ∀ψ ∈ C(D) Γn k ψdλ → Γ ψdµ * , (b) Γ dλ ≤ Γ dµ * or equivalently µ * (Γ) ≥ λ(Γ). Proof. Point 1. Let us denote by U = {domains Ω ⊂ D| Ω satisfies the ε -cone property and Γ D ∪ Γ N ⊂ ∂Ω, Γ ⊂ G}, which obviously contains U ad (Ω 0 , ε, ĉ, G), then thanks to [29] Theorem 2.4.10 p. 56 (see also p. 145), the set U is closed with respect to the Hausdorff convergence

  uniformly bounded on D (for all n η n 1 = η n (D) = Γn dλ ≤ M ). Then by [5, Thm. 1.59, p 26] or [29, Prop. 2.3.9,p. 48] there exits a weakly * convergent subsequence and there exists a Radon positive measure µ * ∈ M b (D) such that η n k converges weakly * to µ * ∀ψ ∈ C 0 (D) D ψdη n k = Γn k ψdλ → D ψdµ * .

  is a linear continuous and surjective operator with linear bounded inverse (the extension operatorE Ω : H 1 (Ω) → H 1 (R N )); 3. for β = 1 -(N -d)/2 > 0 the operators Tr : H 1 (R N ) → L 2 (∂Ω),and Tr ∂Ω : H 1 (Ω) → L 2 (∂Ω) are linear compact operators with dense image Im(Tr) = Im(Tr ∂Ω ) = B 2,2 β (∂Ω) and with linear bounded right inverse (the extension operators) E : B 2,2 β (∂Ω) → H 1 (R N ) and E ∂Ω : B 2,2 β (∂Ω) → H 1 (Ω);

Theorem 4 .

 4 Let Ω =] -L, L[ × ] -, [ be a domain with a simply connected sub-domain Ω 0 , whose boundaries are ] -L, 0[ ×{ }, {-L} × ] -, [, ] -L, 0[ ×{-} and another

u 2 ,

 2 k (x) = A 2 exp(λ 0 x) + B 2 exp(-λ 0 x).The Robin boundary condition[START_REF] Münch | Optimal design of the damping set for the stabilization of the wave equation[END_REF] on Γ implies thatη 0 λ 0 (A 2 -B 2 ) + α(A 2 + B 2 ) = 0, which means that u 2,k (x) = A 2 exp(λ 0 x) + λ 0 η 0 + α λ 0 η 0 -α exp(-λ 0 x) .Application of the boundary condition[START_REF] Pierrat | Generalization of Rellich-Kondrachov theorem and trace compacteness in the framework of irregular and fractal boundaries[END_REF] implies the final expressionu 2,k (x) = g k [(λ 0 η 0 -α) exp(λ 0 x) + (λ 0 η 0 + α) exp(-λ 0 x)] [(λ 0 η 0 -α) exp(-λ 0 L) + (λ 0 η 0 + α) exp(λ 0 L)] .

1 2λ0 |χ| 2 [ 1 -

 121 |χ| 2 | exp(λ 0 x)| 2 + |γ| 2 | exp(-λ 0 x)| 2 + 2Re χγ exp(λ 0 x)exp(-λ 0 x) .Note that, according to the values of k, the expression above may be simplified into|u 0,k -u 2,k | 2 (x) = |χ| 2 exp(2λ 0 x) + |γ| 2 exp(-2λ 0 x) + 2Re (χγ) , if k 2 ≥ ξ0 η0 ω 2 , or |u 0,k -u 2,k | 2 (x) = |χ| 2 + |γ| 2 + 2Re (χη exp(2λ 0 x)) , if k 2 < ξ0 η0 ω 2 . Thus, we have for k 2 ≥ ξ0 η0 ω 2 0 -L |u 0,k -u 2,k | 2 (x)dx = exp(-2λ 0 L)] + |γ| 2 [exp(2λ 0 L) -1] +2LRe (χγ) or, for k 2 < ξ0 η0 ω 2 , 0 -L |u 0,k -u 2,k | 2 (x)dx = L(|χ| 2 + |γ| 2 ) + i λ 0 Im {χγ [1 -exp(-2λ 0 L)]} .

2 |χ| 2 [ 1 - 1 ] -2λ 2 0

 22112 |∂ x (u 0,k -u 2,k )| 2 = |λ 0 | 2 |χ| 2 exp(2λ 0 x) + |γ| 2 exp(-2λ 0 x) -2Re (χγ) , if k 2 ≥ ξ0 η0 ω 2 , or |∂ x (u 0,k -u 2,k )| 2 = |λ 0 | 2 |χ| 2 + |γ| 2 -2Re (χγ exp(2λ 0 x)) , if k 2 < ξ0 η0 ω 2 , and thus 0 -L |∂ x (u 0,k -u 2,k )| 2 (x)dx = λ 0 exp(-2λ 0 L)] + |γ| 2 [exp(2λ 0 L) -LRe (χγ) , if k 2 ≥ ξ0 η0 ω 2 , or, if k 2 < ξ0 η0 ω 2 , 0 -L |∂ x (u 0,k -u 2,k )| 2 (x)dx = L|λ 0 | 2 |χ| 2 + |γ| 2 + iλ 0 Im {χγ [1 -exp(-2λ 0 L)]} .

Figure 2 :

 2 Figure 2: The real (top left) and imaginary (top right) parts of α and the sum of the errors e ∆x (in the bottom) as function of frequencies ω ∈ [600, 30000] calculated for the ISOREL porous material.

Acknowledgments

We would like to thank for fruitful discussions and for a strong support of our work Profs. M.R. Lancia, M. Hinz and especially A. Teplyaev for his helpful remarks and advice, crucial in the writing of the article. We also thank our anonymous referee for pointing the interest and main difficulties in the developing of our shape optimal existence theory.