R Cerna

D Sarchi

T K Paraïso

G Nardin

Y Léger

Maxime Richard

B Pietka

O El Daif

F Morier-Genoud

V Savona

O El Daif

M T Portella-Oberli

B Deveaud-Plédran

Coherent

Keywords: numbers: 71, 36, +c

published or not. The documents may come

Polaritons in semiconductor microcavities are hybrid quasiparticles consisting of a superposition of photons and excitons. Due to the photon component, polaritons are characterized by a quantum coherence length in the several micron range. Owing to their exciton content, they display sizeable interactions, both mutual and with other electronic degrees of freedom. These unique features have produced striking matter wave phenomena, such as Bose-Einstein condensation [START_REF] Kasprzak | Bose-Einstein condensation of exciton polaritons[END_REF][START_REF] Deng | Spatial coherence of a polariton condensate[END_REF], or parametric processes [START_REF] Stevenson | Continuous wave observation of massive polariton redistribution by stimulated scattering in semiconductor microcavities[END_REF][START_REF] Savvidis | Angle-Resonant Stimulated Polariton Amplifier[END_REF] able to generate quantum entangled polariton states [START_REF] Romanelli | Four wave mixing oscillation in a semiconductor microcavity: Generation of two correlated polariton populations[END_REF][START_REF] Savasta | Quantum complementarity of microcavity polaritons[END_REF]. Recently, several paradigms for spatial confinement of polaritons in semiconductor devices have been established [START_REF] Bajoni | Polariton parametric luminescence in a single micropillar[END_REF][START_REF] Balili | Bose-Einstein condensation of microcavity polaritons in a trap[END_REF][START_REF] Lai | Coherent zero-state and π-state in an exciton-polariton condensate array[END_REF][START_REF] Kaitouni | Engineering the spatial confinement of exciton polaritons in semiconductors[END_REF]. This opens the way to quantum devices in which polaritons can be used as a vector of quantum information [START_REF] Quinteiro | Long-range spin-qubit interaction mediated by microcavity polaritons[END_REF]. An essential element of each quantum device is the quantum state control. Here we demonstrate control of the wave function of confined polaritons, by means of tailored resonant optical excitation. By tuning the energy and momentum of the laser, we achieve precise control of the momentum pattern of the polariton wave function. A theoretical model supports unambiguously our observations.

The key feature of polaritons in respect of confinement is their very small effective mass, which is about 10 5 times smaller than the free electron mass. Hence, confinement over a few microns is enough to produce an atom-like spectrum with discrete energy levels. The photonic component of polaritons also allows control of the electronic excitations by optical means. These features hold great promise for devices where quantum correlations can be generated and controlled over long spatial range [START_REF] Quinteiro | Long-range spin-qubit interaction mediated by microcavity polaritons[END_REF].

The sample we used to carry out our studies consists of a single quantum well (QW) embedded in a planar microcavity (MC) with a mesa pattern on the spacer layer [START_REF] El Daif | Polariton quantum boxes in semiconductor microcavities[END_REF] (see also Fig. 1 and Methods). Patterning the cavity thickness results in a modulation of the photon resonance energy, which corresponds to a potential trap, with finite energy barriers, able to confine polaritons. The confined polariton modes are then characterized by a discrete en- ergy spectrum and spatially confined patterns [START_REF] Kaitouni | Engineering the spatial confinement of exciton polaritons in semiconductors[END_REF][START_REF] Baas | Zero dimensional exciton-polaritons[END_REF]. We excited the sample optically with a cw laser and performed 2D k-space imaging and spatially resolved spectroscopy. All the studies we present here have been carried out in the linear response regime.

Figure 2a shows the spatially resolved photoluminescence spectrum of a 9 µm mesa at negative detuning, under non-resonant excitation. One can clearly distinguish the confined lower (1481.3-1484 meV) and upper (1485.5-1489.5 meV) polariton states in the mesa as well as the continuum states of the polaritons from the surrounding planar MC. This energy spectrum corresponds well to what is expected from theory for a 9 µm trap with cylindrical symmetry [START_REF] Kaitouni | Engineering the spatial confinement of exciton polaritons in semiconductors[END_REF][START_REF] Baas | Zero dimensional exciton-polaritons[END_REF]. Figure 2b shows the emission patterns in the 2D k-space of different confined lower polariton states. For this kind of measurements we excited the states resonantly in energy and k . We controlled the in-plane momentum k = (k x , k y ) transferred to the polaritons by varying the excitation angle θ of the laser beam (k ∝ sin(θ)/λ laser ). The observed emission patterns are proportional to the momentum probability distribution of the excited states thus allowing us to access the squared absolute value |ψ| 2 of their wave functions. For resonant excitation, the emission is dominated by the resonant linear response, while the photoluminescence due to energy relaxation is negligible [START_REF] Hayes | Resonant Rayleigh scattering versus incoherent luminescence in semiconductor microcavities[END_REF]. The eigenstates |n, m can be labeled by a radial quantum number n (1, 2, 3 . . .) and an angular quantum number m (0, ±1, ±2 . . .) with no restriction over m [START_REF] Leyronas | Quantum wells, wires and dots with finite barrier: Analytical expressions for the bound states Solid State[END_REF]. The ±m states are degenerate due to the cylindrical symmetry of the potential. This symmetry should be visible in the observed emission pattern. Figure (2b) shows that the emission at the energies of the states with |m| > 0 and hence the polariton wave functions of these states feature lobes along the angular direction φ. These lobes are evidence of a breaking of the symmetry, which can come either from an anisotropy of the confinement potential or from the incidence angle of the excitation beam. We will see that the anisotropy plays a role only for the m = ±1 doublet of the 3 µm mesa. In 9 µm mesas the anisotropy of the potential can be neglected in first approximation. The specific pattern comes then from the coherent excitation of the ±m states at a k = 0. The phases of the two eigenstates are locked at the laser position in kspace. Due to the angular dependence of the eigenstates wave functions (ψ ±m ∝ e ±miφ ), their phases evolve with opposite signs along φ. Since the observed image is proportional to |ψ +m + ψ -m | 2 , the phase difference of the two components gives rise to positive and negative interferences along φ. The number of nodes on a full circle is hence equal to 2 |m|. The number of lobes in radial direction corresponds to n. The same conclusions hold for all degenerate m-doublets in all investigated mesas. This phase locking effect can be used to manipulate the wave functions of degenerate doublet states. By changing the position of the excitation along φ in k-space, the phase of the ±m components is changed, and thus the interference pattern. This is demonstrated in Fig. 3 for the second excited state (1, ±2) in a 3 µm mesa. The intensity lobes follow the rotation of the laser. This demonstrates control over the probability distribution of polaritons in momentum space and also in real space.

Unlike the larger mesas, the 3 µm mesas have a non negligible ellipticity (b/a ≈ 1.29), which can lift the degeneracy of m = ±1 doublet states. This is the case for the first eigenstate doublet |a and |b , which can be pictured as linear combinations of the m = ±1 states of a cylindrical structure with the same average diameter:

|a = 1 √ 2 (|1, -1 +|1, +1 ) and |b = 1 √ 2 (|1, -1 -|1, +1 ).
Therefore these eigenstates feature lobes along φ and their orientation is fixed by the axis of the ellipse. A degeneracy lift has only been observed for this lowest doublet state, as expected from theory. The splitting of the non-degenerate doublet increases with negative detuning due to the decreasing effective mass and is about 0.11 ± 0.04 meV at zero detuning. This agrees with the Depending on the detuning, the momentum distribution of these split states can nevertheless be manipulated. At negative detuning the energy splitting can be significantly larger than the natural linewidth of the modes (≈ 85 µeV). As the laser spectral width is smaller (< 25 µeV), it is possible to excite only one state, but with different energies. Then the measured momentum pattern is that of the corresponding eigenstate. In the case of a split doublet with sizeable spectral overlap, the laser energy can be tuned between the eigenenergies of the two split states. As a consequence, it is then possible to vary the relative amplitude and phase of the two contributions to the linear response thanks to the different overlap of the laser with the two states. In this case one can rotate the polariton momentum distribution by rotating the laser momentum k as shown before and also by varying the laser energy. This offers additional control: by tuning the laser energy one can control the orientation of the lobes with respect to the laser position in k-space. Figure 4 shows the results of an experiment where we kept the excitation position in k-space constant and changed the excitation energy between two split states with significant spectral overlap.

As the figures 3 and 4 show, the experimental results presented here can be reproduced with a theoretical model which is described into more detail in [START_REF] Sarchi | Polariton parametric photoluminescence in spatially inhomogeneous systems[END_REF]. We start from the bosonic exciton-photon Hamiltonian [START_REF] Sarchi | Spectrum and thermal fluctuations of a microcavity polariton Bose-Einstein condensate[END_REF][START_REF] De-Leon | Exciton-exciton interactions in quantum wells: Optical properties and energy and spin relaxation[END_REF] and we adopt a linearized mean-field approach [START_REF] Carusotto | Spontaneous microcavitypolariton coherence across the parametric threshold: Quantum Monte Carlo studies[END_REF]. In this approach and considering a coher-ent monochromatic optical pump F (r, t) = e -iωt F 0 (r), the coherent exciton and photon fields Φ x,c (r, t) evolve with the frequency of the laser source as Φ x,c (r, t) = e -iωt Φ 0

x,c (r). Their steady-state shape Φ 0 x,c (r) obeys a set of two coupled Schrödinger-like equations for the exciton-photon problem [START_REF] Sarchi | Spectrum and thermal fluctuations of a microcavity polariton Bose-Einstein condensate[END_REF][START_REF] Carusotto | Spontaneous microcavitypolariton coherence across the parametric threshold: Quantum Monte Carlo studies[END_REF]]

- 2 ∇ 2 2m x -ω -iγ x Φ 0 x (r) = Ω R Φ 0 c (r) , (1)
c ( ∇) + U c (r) -ω -iγ c Φ 0 c (r) = Ω R Φ 0 x (r) + F (r) , (2)
where m x is the exciton mass, c (k) is the cavity photon dispersion accounting for the photon-exciton detuning, U c is the photon confining potential, Ω R is the excitonphoton coupling and the decay rates γ x,c account for the finite lifetime of the two species. Here we are omitting the non-linear terms because we are interested in the linear regime (see discussion below). From these equations, it is clear that the phase of the coherent fields Φ x,c (r, t) generated by the coherent pumping is locked to the phase of the laser source. The coherent fields in k-space are obtained from the fields Φ 0

x,c (r) via Fourier transform. From Eqs. ( 1) and ( 2), we see that the exciton coherent field is expected to have the same shape as the photon field, i.e. Φ 0 x (r) ≡ XΦ 0 p (r) and Φ 0 c (r) ≡ CΦ 0 p (r), as a result of the exciton-photon coupling. The relative intensity defines the exciton amount as

| X | 2 = dr | Φ 0 x (r) | 2 dr | Φ 0 p (r) | 2 , (3)
where

|Φ 0 p (r)| 2 =| Φ 0 x (r) | 2 + | Φ 0 c (r) | 2 .
This clearly proves that, since |X| 2 ∼ 0.5, the reported manipulation of the emitted optical field also implies the manipulation of the matter field. For the calculations, we use a confining potential in agreement with AFM images of the mesas, thus accounting for the weak ellipticity of the samples. The decay rates are assumed to be γ c = 0.05 meV, in agreement with the reported photon lifetime τ c = 15 ps, and γ x = 0.01 meV. The other parameters are obtained from the experimental characterization of the sample.

The effect of polariton-polariton interactions could be very important, especially in presence of spatial confinement. By including the non-linear terms in Eqs. (1,2) [START_REF] Carusotto | Spontaneous microcavitypolariton coherence across the parametric threshold: Quantum Monte Carlo studies[END_REF] we have checked that this is not the case for the pump intensities used in this experiment. Therefore, all the reported features are entirely due to the linear response of the polariton system to resonant laser excitation.

METHODS

The investigated sample consists of a single semiconductor quantum well (QW) placed in the middle of a λ microcavity (MC) (see Fig. 1). The QW, which is composed of a 8 nm thick In 0.04 Ga 0.96 As layer sandwiched between two GaAs barrier layers, shows a sharp exciton resonance (500 µeV FWHM) at Ex = 1.4846 eV. The two distributed Bragg reflectors (DBRs) of the MC are made of 21 (top) and 22 (bottom) layer pairs of AlAs/GaAs. The resonance energy of the cavity Ec = ch/λc is given by the cavity spacer thickness λc/n. The sample is wedge-shaped thus allowing to change the energy detuning (δ = Ec -E X ) between the cavity photons and the QW excitons as a function of sampling position. The Rabi splitting (the energy difference between the upper and lower polariton branch at k = 0) has a value of 3.5 meV at zero detuning. Round mesas of different sizes (3, 9 and 19 µm in diameter) have been etched on the top of the spacer before growing the upper DBR [START_REF] El Daif | Polariton quantum boxes in semiconductor microcavities[END_REF]. This leads to a locally lowered resonance energy Ec = ωc of the cavity. The mesas have a height of 6 nm, which corresponds to a potential depth of about 9 meV at the exciton energy [START_REF] Kaitouni | Engineering the spatial confinement of exciton polaritons in semiconductors[END_REF]. Thus the mesas act as a three dimensional trap for MC photons. The confined photon modes couple to the 2D excitons thus creating the upper and lower confined polariton branches [START_REF] Panzarini | Quantum theory of exciton polaritons in cylindrical semiconductor microcavities[END_REF]. The anti-crossing behaviour of all confined polariton modes has been shown in [START_REF] Baas | Zero dimensional exciton-polaritons[END_REF]. When talking about detuning in the context of confined polaritons we always refer to the energy difference between the lowest confined photon mode and the 2D excitons. At zero detuning for example the ground state is half light half matter while the excited states of the confined lower polaritons have an higher excitonic component.

To carry out our study we employed a photoluminescence setup in transmission configuration. The polaritons have been optically excited with a tunable cw Ti:Sa laser. For non resonant excitation the laser has been tuned at the first reflectivity minimum of the DBR's around 784 nm. The laser beam has been focused on the sample with a camera objective (f=55 mm and 1.2 aperture) providing an excitation spot of about 25 to 30 µm for all measurements. This provides a narrow distribution in k-space. The excitation power, which was about 5 mW for non resonant excitation and about 7 µW for resonant excitation, was chosen in such a way that the system would always be in the linear regime. By displacing the excitation beam with a retroreflector parallel to the middle axis of the camera lens we changed the incidence angle and thus the in-plane momentum k = (kx, ky) transferred to the polaritons.

The sample was kept in a continuous flow optical cryostat at around 5 K. On the detection side the photoluminescence light was collected with a microscope objective with high numerical aperture (NA = 0.55). The collimated luminescence light was split by a beam splitter. One part of the beam was dispersed by a spectrometer (25 µeV resolution) and recorded with a nitrogen cooled CCD camera. The second part of the beam was sent to an angle resolved imaging setup in order to measure the in-plane momentum k = (kx, ky) of the resonantly excited polaritons. The emission angle of the luminescence light from a MC is proportional to the in plane momentum of the emitter (polariton) k ∝ sin(θ)/λc.

FIG. 1 :

 1 FIG.1: Sketch of the sample (not to scale). Single quantum well embedded in a λ cavity with the exciton resonance at 1.484 eV. The sample has mesas of three different sizes and is wedge shaped.

FIG. 2

 2 FIG.2: a) real space spectrum of a 9 µm mesa at 2 meV negative detuning (see Methods for detuning definition); upper and lower polariton branches of the 2D and 0D polaritons are clearly visible. b) k-space images of resonantly excited states from the 0D lower polariton branch. The states can be labeled with two quantum numbers n and m. 2 |m| gives the number of nodes along φ on a full circle, while n corresponds to the number of lobes in radial direction.

FIG. 3 :

 3 FIG. 3: Lobes of the degenerate second doublet state in a 3 µm mesa following the rotation of the laser in k-space. The mesa was at zero detuning. The upper part shows the experimental results and the theoretical results from the simulation are shown in the lower part of the figure. The red spot which is visible in each experimental image of this figure is an emission maximum arising from the laser light transmitted directly through the cavity beside the mesa. This makes it easy to observe the excitation angle. In the simulation images the laser position is marked by a circle.

FIG. 4 :

 4 FIG. 4: Manipulation of the wave functions of the split first doublet in a 3 µm mesa (≈ 3 meV negative detuning) by changing the excitation energy while keeping the excitation point constant. Upper part: experiment; Lower part: theoretical results. The polariton lifetime in our sample is about 15 ps, which leads to an energy linewidth of the eigenstates much broader than the spectral width of the laser. The lobes rotate due to the different contributions of the eigenstates to the linear response.

ACKNOWLEDGEMENTS

This work was supported by the Quantum Photonics NCCR and the Swiss National Science Foundation.


K G Lagoudakis

B Pietka

M Wouters

R André

B Deveaud-Plédran

Coherent Oscillations in an Exciton-Polariton Josephson Junction

We report on the observation of spontaneous coherent oscillations in a microcavity polariton bosonic Josephson junction. The condensation of exciton polaritons takes place under incoherent excitation in a disordered environment, where double potential wells tend to appear in the disordered landscape. Coherent oscillations set on at an excitation power well above the condensation threshold. The time resolved population and phase dynamics reveal the analogy with the AC Josephson effect. We have introduced a theoretical two-mode model to describe the observed effects, which allows us to explain how the different realizations of the pulsed experiment have a similar phase relation.

One of the most striking manifestations of the quantum collective behavior of matter is the Josephson effect: it allows for a classically forbidden current to flow without dissipation [START_REF] Barone | Physics and Applications of the Josephson Effect[END_REF]. Traditional Josephson junctions are built by means of superconductors separated by thin non conducting oxide barriers and they have drawn an intense interest for their striking phenomenology. A bosonic analog is the so called Bosonic Josephson junction (BJJ) where two macroscopic populations of bosons are trapped in a double well geometry. Bose Einstein condensation of dilute atomic gases has allowed for the realization of BJJs by trapping two condensates in such double wells [START_REF] Cataliotti | [END_REF], [3]. Although oscillatory behavior of fluids is common in classical systems (i.e. an initially full water container connected to an empty one through a large pipe), the Josephson effect results in a dissipationless flow through classically forbidden barriers and produces an alternating current from a constant potential difference between the two wells. This phenomenon is directly related to one of the most notable properties of quantum fluids: macroscopic phase coherence [START_REF] Pitaevskii | Bose Einstein Condensation[END_REF]. The broken symmetry in such fluids allows for the description of all particles with a common one-body wavefunction and a common phase which in turn allows for a number of quantum effects to occur. The possibility to engineer and dynamically tune atomic condensates, has allowed for all the possible regimes and their respective phenomenology to be observed. More specifically in the Josephson regime, the most remarkable phenomena are the AC and DC Josephson effects [START_REF] Levy | [END_REF], the macroscopic quantum self trapping and the Josephson plasma oscillations [6]. The recent achievement of condensation of exciton polaritons in semiconductor microcavities [7] has provided a new field for the study of quantum phenomena related to the macroscopic phase coherence. Polaritons are Bosonic quasi-particles that occur in semiconductor microcavities as a result of the strong coupling between light (cavity photons) and matter excitations (quantum well excitons). Owing to their half-light nature, polaritons have an exceptionally low mass of the order of 10 -5 the mass of free electrons, which allows for condensation at temperatures easily achievable by cryogenic means. This makes exciton polaritons very attractive in the sense that they are a high temperature quantum fluid. Having a limited lifetime of about 3psec, they leak out of the microcavity and thus have to be continuously replenished, a process that renders polariton condensates out of equilibrium. It is because of this aspect that polariton condensates provide a very rich phenomenology [8]. Another advantage of the system under consideration is that all the condensed polariton properties are imprinted to the luminescence emitted by the condensate so one can gain access to both phase and population by optical imaging and interferometric methods.

Here we have set out to investigate Josephson effects in a BJJ formed by two spatially separated condensates of exciton polaritons in a CdTe microcavity. This work clearly demonstrates AC Josephson oscillations evidenced through relative population and relative phase oscillations between the two trapped condensates. This observation implies the existence of a synchronization mechanism between different experimental realizations which we understand in depth with the two mode model that we have developed to describe the observed phenomenology. The condensate pair is trapped in a naturally occurring disorder double potential well which is set during the sample growth. These structures tend to appear more frequently in linear disorder valleys, see Fig. 1(a). The sample employed here is the same as the one used in our previous studies [9]. We employed pulsed non resonant excitation in conjunction with a streak camera at the detection that provided the necessary temporal resolution. The population oscillations were revealed by forming the image of the real space energy integrated photoluminescence (Fig. 1a) on the entrance slit of the streak camera. The BJJ that we studied here, consisted of two wells of about 2 µm size each, separated by a barrier of approximately 0.5

. From the spectroscopic data we estimated the depth of the confining potential to be approx. 5.6meV with respect to the 2D polariton average energy. Remarkably, the Josephson oscillations only show up above a threshold excitation power that is higher than the power required for condensation . At powers , coherence between the two wells is observed, but the dynamics of the system do not show any interesting features and no population exchange is seen between the two trapped condensates as clearly shown in Fig. 1(b). On the contrary when increasing the excitation power above the oscillation threshold then the behavior changes dramatically and one clearly sees an exchange of particles between the two wells Fig. 1(c). Additionally, to assess that the polariton currents are driven by the phase difference between the two wells, one needs the relative phase difference with

( ) ( ) ( ) L R t t θ θ θ ∆ = - t , L R
θ the absolute condensate phase in the respective wells. This is accessible only by means of interferometric measurements and for this purpose we used the modified Michelson interferometer in a mirror-retroreflector configuration with active stabilization [7] to interfere the luminescence of the left well with that of the right-one. The density of fringes in the interferograms is set by the displacement of the retroreflector which applies an effective angle of the reflected beam when passing through the imaging lens thus giving an interference term of the form ( )

cos I k x t θ ⋅ + ∆ ⎡ ⎤ ⎣ ⎦ , ( ) t θ ∆
being the relative phase difference. One can directly see that, when the two condensates have a constant phase difference, the fringes are straight in time (Fig. 2b) whereas, if their phase increases linearly in time, the fringes will appear tilted (Fig. 2c). By scanning the relative phase of the two interferometer arms over 3 we were able to fit the function π

( ) ( , cos , A x t x t φ θ ⋅ +∆ ⎡ ⎤ ⎣ ⎦ ) ) at each pixel ( ,
( ) ( ) L R L R N N N N N ∆ = - +
are plotted in Fig. 3. Density and phase oscillations show the expected relation: the measurements are consistent with a Josephson current that flows from the well with the lower phase to the one with higher phase and vanishes when the phase difference is zero. We thus observe density oscillations between two macroscopically occupied polariton states driven by their phase difference, i.e. the bosonic Josephson effect [10]. A closer look in the population dynamics reveals that there is a strong imbalance of the population at the first moments of the oscillations (between 35 and 65 psec) which is due to a higher energy level that does not contribute to the beatings and dies out quickly within the same temporal window. Time integrated spectrally resolved studies (not shown) allow to observe the emission lines of the two beating states, their energy difference corresponding to the beating frequency, and the slightly higher energy state that has no oscillatory signature in the temporal response. The relative phase between the two wells is expected to increase linearly in time. Modulo , this should in turn give a perfect saw tooth form of the relative phase evolution from to . Contrary to this, we observe a smoothed saw tooth phase evolution with much smaller amplitude ( to 0.3 ). 2π -π π -0.3π π It is actually not trivial that Josephson oscillations can be observed under the present conditions. Indeed, the pulsed experiment is repeated millions of times and starts every time with an arbitrary relative phase between the condensates in the two disorder minima. One would therefore rather expect to see oscillations in a single run of the experiment, but not in our case of realization averaged signal.

In order to gain more insight in the origin and nature of Josephson oscillations that we observe, we compare our observations with the predictions of a two polariton mode that takes into account the polariton losses and their replenishment from the exciton reservoir [9], [11], [12]:

( ) 2 , , , , 2 ;, , , 2 ,
L R L R L R L R R L R L R R L L R d i i g Rn J dt ψ ε ψ γ ψ ψ ⎧ ⎫ ⎡ ⎤ = + + - + + ⎨ ⎬ ⎣ ⎦ ⎩ ⎭ ξ (1)
Here , L R ε are the energies of the modes in the absence of the tunneling coupling J and γ is the linewidth in the linear regime. The nonresonant excitation enters the polariton dynamics through the stimulated relaxation rate ( )

R L R R L R R R L R R R L R R L R L R dn n n n R n dt r γ γ α ⎛ ⎞ = - - - ⎜ ⎟ ⎝ ⎠ ψ (2)
Here is the steady state ratio between inactive and active excitons. The active exciton density is normalized to be unity at the condensation threshold and 12 r α is the depletion factor of this reservoir due to the scattering into the lower polariton branch. The inactive exciton density dynamics is given by

1 R n 1; , 1; , 12 2; , 1 1; , , 12
R L R R L R R R L R R R L R L R dn n n n dt r γ γ ⎛ ⎞ = - - ⎜ ⎟ ⎝ ⎠ P + (3)
The noise term ξ models the spontaneous decay and scattering events, and can be explicitly computed within the truncated Wigner approximation [11]. The most important stochastic element is however the initial condition of the polariton field: it starts in a vacuum state with random phase. Fig. 4 shows two simulations of the dynamics under pulsed excitation, both above the threshold for condensation and, interestingly, the Josephson oscillations survive the realization averaging only for the higher excitation power (right hand side panels). Our simulations thus reproduce the experimental observation that the threshold for Josephson oscillations is higher than the one for condensation . . Individual realizations of the density-phase evolution clearly show oscillations. In the average over many runs of the experiment (Fig. 4c) these oscillations have however disappeared, because the different realizations are not in phase. Only at higher power (see Fig. 4(d,e,f)), do the oscillations share some common phase and survive the realization averaging. The physical reason is the following. At high excitation intensity, the growth rate is much larger than the tunneling between the two wells and the Josephson currents can be neglected during the exponential growth stage of the condensate formation. This implies that the initial randomness of the relative phase does not affect the density dynamics that is therefore deterministic up to small fluctuations. The dynamics of the relative phase is governed by

R J ( ) ( ) ( ) L R t t θ θ θ ∆ = - t ( ) cos L R R L R R L R L d N N J dt N N θ θ ε ε ⎛ ⎞ ∆ = - ∆ + -+ - ⎜ ⎟ ⎝ ⎠ L g N g N , (4)
Where we have introduced the density and phase of the fields , ,

j i j j N e j L R θ ψ = = .
The emergence of a fixed relative phase shortly after the condensate formation can be easier understood for R L ε ε = . Under asymmetric pumping, one of the two densities is much larger than the other one. Eq. ( 4) then implies that the relative phase will be quickly driven to ∆θ π 2 = , the zero of the cosine. Our simulations show that also when R L ε ε ≠ , a deterministic relative phase is reached in the early stage of the dynamics, irrespective of the initial condition (see Fig. 4d).

A second question we can address with our model is whether our observations are rather related to the Josephson plasma oscillations or the AC Josephson effect. The latter effect is driven by the energy difference between the two potential minima, where the former originates from an initial condition that is different from the steady state. A previous study [13] of plasma oscillations in a nonequilibrium polariton condensate under continuous wave excitation has shown that plasma oscillations feature a damped sinusoidal dynamics, in contrast to the persistent saw-tooth like behavior that is observed in our experiments. The polariton analog of the AC Josephson effect on the other hand occurs when the energy difference between the two wells is too big for a synchronized solution to exist [14], [15]. The beatings between the two modes are responsible for persistent density oscillations and a sawtooth phase profile, as we observe in our experiments. The oscillations between two wells with different frequency condensates are entirely analogous to the AC Josephson oscillations where the difference in chemical potential is imposed by an external voltage.

An unexpected feature of the saw tooth oscillations in Fig. 3 is that their amplitude is much below the expected . This property of the realization averaged phase can be traced back to the fact that the phase evolution is steeper for large 2π θ ∆ (see Fig. 4d). As a consequence large values of θ ∆ do not contribute much to the average phase difference. The physical reason is that large phase differences go together with a low population in one of the two wells. When one of the populations becomes very small, the Josephson current quickly reverts so to replenish it. A further consequence of the rapid phase variation is that the shot to shot variations are more significant at those times. This results in a reduced coherence, seen both in the experiments and theoretical simulations (not shown in the figures).

Finally, our two-mode model also allows us to understand the role of the polariton-polariton interactions on the Josephson oscillations. In the absence of interactions, the oscillation frequency is set by the detuning and tunneling coupling. When the blue shift due to polariton-polariton interactions j j gN µ = becomes comparable to the detuning ε ∆ , interactions make the oscillation frequency increase for increasing pump intensity. In experiments however, no such behavior is observed. Contrary to this, a slight decrease of the frequency is observed. We therefore conclude that polariton-polariton interactions have a negligible effect on the oscillation dynamics and that our polaritonic BJJ is in the so-called Rabi regime [17]. The slight decrease in oscillation frequency can be reproduced in the theory when taking into account the interactions between the polaritons and the exciton reservoir. If the low energy well is pumped with higher intensity, the blue shift due to the excitons will shift it up more than the other one and therefore the oscillation frequency decreases.

In this work we have provided experimental evidence of coherent oscillations in a Bosonic Josephson junction formed by two exciton polariton condensates created under nonresonant pulsed excitation in a double well geometry. We have shown the simultaneous existence of population and phase oscillations and have carefully evaluated their amplitudes and average values. The saw-tooth like oscillations persist for the whole life time of the condensates and are the polariton analog of the AC Josephson regime. Our theoretical analysis shows that the Josephson oscillations survive the realization averaging linked with the random initial phase, though with reduced amplitude, thanks to a spontaneous synchronization mechanism that only appears above a critical pump intensity. The present realization of the Josephson effect relies on the accidental presence of coupled wells in our disordered microcavity. Controlled growth of polariton traps with mesa structures [18] and micropilars [19] will allow for a more systematic analysis of the different regimes of Josephson oscillations, where polaritonpolariton interactions can play a more pronounced role. Also, combining nonresonant with resonant excitation will provide full control over the initial conditions of the Josephson oscillations.

[13] M. The relative population (squares) oscillates symmetrically around zero and the phase (circles) is following the population oscillations. A striking feature is that the phase is not oscillating between and in a saw-tooth manner as expected but rather shows small amplitude sawtooth like oscillations. The region between 5 and 35 psec had a very low overall intensity and thus the relative phase could not be calculated.

Fig. 4 (

 4 a,b) shows multiple realizations of the density/phase evolution for

.

  Numerical results were obtained by Monte Carlo averaging over 1000 realizations over the initial conditions and the noise.

FIG. 1 (FIG. 2 (FIG. 3 (

 123 FIG. 1 (Color online). (a) Time integrated real space image of the luminescence. The dashdotted line is the line selected by the streak camera and the solid lines with dashed edges enclose the linear disorder valley in which the double well potentials are frequent. Here we have chosen the center of the axes as the center of the polariton Josephson junction. (b) Time resolved photoluminescence along the dash-dotted line in (a) for excitation power below the oscillation threshold. (c) Same as in (b) but for an excitation power above the oscillation threshold. One can easily see the striking difference between the case with no population oscillations and the case where oscillatory exchange is strongly pronounced. The incident pulse defines the zero time.

FIG. 4 (

 4 FIG. 4 (Color online). Theoretical simulations: (a) Phase evolution for individual realizations for low excitation powers. (b) Populations in left (red) and right (black) well for the three realizations of (a). (c) Averaged relative phase modulo and normalized populations from (a), (b). (d), (e) and (f) are the same as the panels (a), (b) and (c) only this time for excitation intensity above the oscillation threshold. Both the phase and the populations have very similar behaviors for different realizations demonstrating the effect of the phase locking mechanism. The non negligible fluctuations of the phase are the reason of the smoothed saw tooth phase profile [16]. 2π

  Wouters and I. Carusotto, Phys. Rev. Lett. 99, 140402 (2007) [14] M. Wouters, Phys. Rev. B 77, 115340 (2008) [15] M. O. Borgh, J. Keeling and N. G. Berloff, Arxiv:0911.4468 (2009) [16] In the reported simulations, the parameters were taken

ACKNOWLEDGEMENTS

The authors wish to thank Le Si Dang, Vincenzo Savona, Yoan Leger, Iacopo Carusotto, Davide Sarchi, Maxime Richard and Augustin Baas for continuous stimulating discussions. The work was supported by the Swiss National Research Foundation through NCCR "Quantum Photonics".


Eric Bonnetier
email: eric.bonnetier@univ-grenoble-alpes.fr

AND Mourad Choulli
email: mourad.choulli@univ-lorraine.fr

Faouzi Triki
email: faouzi.triki@univ-grenoble-alpes.fr

STABILITY FOR QUANTITATIVE PHOTOACOUSTIC TOMOGRAPHY REVISITED

Keywords: Mathematics subject classification : 35R30 Elliptic equations, diffusion coefficient, absorption coefficient, stability inequality, multiwave imaging

This paper deals with the issue of stability in determining the absorption and the diffusion coefficients in quantitative photoacoustic imaging. We establish a global conditionnal Hölder stability inequality from the knowledge of two internal data obtained from optical waves, generated by two point sources in a region where the optical coefficients are known.

Define C 1,β (R n ) as the vector space of functions f from C 0,β (R n ) so that ∂ j f ∈ C 0,β (R n ), 1 ≤ j ≤ n. The vector space C 1,β (R n ) equipped with the norm

f C 1,β (R n ) = f C 0,β (R n ) + n j=1 ∂ j f C 0,β (R n ) is a Banach space.
The data in this paper consists in ξ 1 , ξ 2 ∈ R n , Ω R n \ {ξ 1 , ξ 2 } a C 1,1 bounded domain with boundary Γ, 0 < α < 1, 0 < θ < α, λ > 1 and κ > 1. For notational convenience, the set of data will denoted by D. That is D = (n, ξ 1 , ξ 2 , Ω, α, θ, λ, κ).

Denote by D(λ, κ) the set of couples (a, b)

∈ C 1,1 (R n ) × C 0,1 (R n ) satisfying λ -1 ≤ a and a C 1,1 (R n ) ≤ λ, (1.1) κ -1 ≤ b and b C 0,1 (R n ) ≤ κ, (1.2)
Define further the elliptic operator L a,b acting as follows

(1.3) L a,b u(x) = -div(a(x)∇u(x)) + b(x)u(x).
We show in Section 2 that if (a, b) ∈ D(λ, κ) then the operator L a,b admits a unique fundamental solution G a,b satisfying, where This inverse problem is related to photoacoustic tomography (PAT) where optical energy absorption causes thermoelastic expansion of the tissue, which in turn generates a pressure wave [START_REF] Abramowitz | Handbook of mathematical functions: with formulas, graphs, and mathematical tables[END_REF]. This acoustic signal is measured by transducers distributed on the boundary of the sample and it is used for imaging optical properties of the sample. The internal data v 1 (a, b) and v 2 (a, b) are obtained by performing a first step consisting in a linear initial to boundary inverse problem for the acoustic wave equation. Therefore the inverse problem that arises from this first inversion is to determine the diffusion coefficient a and the absorption coefficient b from the internal data v 1 (a, b) and v 2 (a, b) that are proportional to the local absorbed optical energy inside the sample. This inverse problem is known in the literature as quantitative photoacoustic tomography [START_REF] Alessandrini | Stability for quantitative photoacoustic tomography with well chosen illuminations[END_REF]4,2,3,8,[START_REF] Bal | Multi-source quantitative photoacoustic tomography in a diffusive regime[END_REF][START_REF] Naetar | Quantitative photoacoustic tomography with piecewise constant material parameters[END_REF][START_REF] Choulli | Some stability inequalities for hybrid inverse problems[END_REF].

ξ ∈ R n , G a,b (•, ξ) ∈ C 2,α loc (R n \ {ξ}), L a,b G a,b (•, ξ) = 0 in R n \ {ξ}, and, for any f ∈ C ∞ 0 (R n ), u = ˆRn G a,b (•, ξ)f (ξ)dξ
Photoacoustic imaging provides in theory images of optical contrasts and ultrasound resolution [START_REF] Abramowitz | Handbook of mathematical functions: with formulas, graphs, and mathematical tables[END_REF]. Indeed, the resolution is mainly due to the small wavelength of acoustic waves, while the contrast is somehow related to the sensitivity of optical waves to absorption and scattering properties of the medium in the diffusive regime. However, in practice, it has been observed in various experiments that the imaging depth, i.e. the maximal depth of the medium at which structures can be resolved at expected resolution, of (PAT) is still fairly limited, usually on the order of millimeters. This is mainly due to the fact that optical waves are significantly attenuated by absorption and scattering. In fact the generated optical signal decays very fast in the depth direction. This is indeed a well known faced issue in optical tomography [24]. In most physicists works dealing with quantitative (PAT), the absorption b > 0 is assumed to be constant and the optical wave is simplified to Ce -bz , as a function of the depth z, which is known as the Beer-Lambert-Bouguer law [12]. Recently in [START_REF] Ammari | Mathematical modeling in photoacoustic imaging of small absorbers[END_REF], assuming that medium is layered, the authors derived a stability estimate that shows that the reconstruction of the optical coefficients is stable in the region close to the optical illumination source and deteriorate exponentially far away.

Stability inequalities for this inverse problem were first obtained in [START_REF] Bal | Multi-source quantitative photoacoustic tomography in a diffusive regime[END_REF]8] under a strong nondegeneracy assumption. Later in [START_REF] Alessandrini | Stability for quantitative photoacoustic tomography with well chosen illuminations[END_REF], the authors improved these results by removing the nondegeneracy assumption for well-chosen boundary conditions (Definition 2.3).

Assuming that the optical waves are generated by two point sources δ ξi , i = 1, 2, we aim to derive a stability estimate for the recovery of the optical coefficients from internal data. We point out that taking the optical wave generated by a point source outside the sample seems to be more realistic than assuming a boundary condition.

In the statement of Theorem 1.

1 below C = C(D) > 0 and 0 < γ = γ(D) < 1 are constants. Theorem 1.1. For any (a, b), (ã, b) ∈ D(λ, κ) satisfying (a, b) = (ã, b) on Γ, we have a -ã C 1,α (Ω) + b -b C 0,α (Ω) ≤ C v 1 -ṽ1 C(Ω) + v 2 -ṽ2 C(Ω) γ .
The rest of this text is organized as follows. In section 2 we construct a fundamental solution and give its regularity induced by that of the coefficients of the operator under consideration. We derive pointwise lower and upper bounds for the fundamental solution that are of interest themselves. These bounds show how the optical signal decays fast in the depth direction. We also establish in this section a lower bound of the local L 2 -norm of the gradient of the quotient of two fundamental solutions near one of the point sources. This is the key point for establishing our stability inequality. This last result is then used in Section 3 to obtain a uniform polynomial lower bound of the local L 2 -norm of the gradient in a given region. This polynomial lower bound is obtained in two steps. In the first step we derive, via a three-ball inequality for the gradient, a uniform lower bound of negative exponential type. We use then in the second step an argument based on the so-called frequency function in order to improve this lower bound. In the last section we prove our main theorem following the known method consisting in reducing the original problem to the stability of an inverse conductivity problem.

Fundamental solutions

2.1. Constructing fundamental solutions. In this subsection we construct a fundamental solution of divergence form elliptic operators. Since our construction relies on heat kernel estimates, we first recall some known results.

Consider the parabolic operator P a,b acting as follows

P a,b u(x, t) = -L a,b u(x, t) -∂ t u(x, t)
and set

Q = {(x, t, ξ, τ ) ∈ R n × R × R n × R; τ < t}.
Recall that a fundamental solution of the operator

P a,b is a function E a,b ∈ C 2,1 (Q) verifying P a,b E = 0 in Q and, for every f ∈ C ∞ 0 (R n ), lim t↓τ ˆRn E a,b (x, t, ξ, τ )f (ξ)dξ = f (x), x ∈ R n .
The classical results in the monographs by A. Friedman [14], O. A. Ladyzenskaja, V. A. Solonnikov and N.N Ural'ceva [20] show that P a,b admits a non negative fundamental solution when (a, b) ∈ D(λ, κ).

It is worth mentioning that if a = c, for some constant c > 0, and b = 0 then the fundamental solution E c,0 is explicitly given by

E c,0 (x, t, ξ, τ ) = 1 [4πc(t -τ )] n/2 e -|x-ξ| 2 4c(t-τ ) , (x, t, ξ, τ ) ∈ Q.
Examining carefully the proof of the two-sided Gaussian bounds in [13], we see that these bounds remain valid whenever

a ∈ C 1,1 (R n ) satisfies (2.1) λ -1 ≤ a and a C 1,1 (R n ) ≤ λ.
More precisely we have the following theorem in which

E c (x, t) = c t n/2 e -|x| 2 ct , x ∈ R n , t > 0, c > 0.
Theorem 2.1. There exists a constant c = c(n, λ) > 1 so that, for any a ∈ C 1,1 (R n ) satisfying (2.1), we have

(2.2) E c -1 (x -ξ, t -τ ) ≤ E a,0 (x, t; ξ, τ ) ≤ E c (x -ξ, t -τ ),
for all (x, t, ξ, τ ) ∈ Q.

The relationship between E c and E c,0 is given by the formula

(2.3) E c (x -ξ, t -τ ) = (πc) n/2+1 π E c/4,0 (x, t, ξ, τ ), (x, t, ξ, τ ) ∈ Q.
The following comparison principle will be useful in the sequel.

Lemma 2.1. Let (a, b 1 ), (a, b 2 ) ∈ D(λ, κ) so that b 1 ≤ b 2 . Then E a,b2 ≤ E a,b1 . Proof. Pick 0 ≤ f ∈ C ∞ 0 (R n ). Let u be the solution of the initial value problem P a,b1 u(x, t) = 0 ∈ R n × {t > τ }, u(x, τ ) = f.
We have

(2.4) u(x, t) = ˆRn E a,b1 (x, t; ξ, τ )f (ξ)dξ ≥ 0.
On the other hand, as P a,b1 u(x, t) = 0 can be rewritten as

P a,b2 u(x, t) = [b 1 (x) -b 2 (x)]u(x, t), we obtain u(x, t) = ˆRn E a,b2 (x, t; ξ, τ )f (ξ)dξ (2.5) - ˆt τ ˆRn E a,b2 (x, t; ξ, s)[b 1 (ξ) -b 2 (ξ)]u(ξ, s)dξds.
Combining (2.4) and (2.5), we get

ˆRn E a,b2 (x, t; ξ, τ )f (ξ)dξ ≤ ˆRn E a,b1 (x, t; ξ, τ )f (ξ)dξ,
which yields in a straightforward manner the expected inequality.

Consider, for (a, b) ∈ D(λ, κ), the unbounded operator

A a,b : L 2 (R n ) → L 2 (R n ) defined A a,b = -L a,b , D(A a,b ) = H 2 (R n ).
It

|k a,b (t, x + h, ξ) -k a,b (t, x, ξ)| ≤ e -δt |h| √ t + |x -ξ| η E c (x -ξ, t), (2.7) |k a,b (t, x, ξ + h) -k a,b (t, x, ξ)| ≤ e -δt |h| √ t + |x -ξ| η E c (x -ξ, t), (2.8)
where c = c(n, λ, κ) > 0 and δ = δ(n, λ, κ) > 0 and η > 0 are constants.

From the uniqueness of solutions of the Cauchy problem (2.9)

u (t) = A a,b u(t), t > 0, u(0) = f ∈ C ∞ 0 (R n ), we deduce in a straightforward manner that k a,b (t, x; ξ) = E a,b (x, t; ξ, 0).
Prior to giving the construction of the fundamental solution for the variable coefficients operators, we state a result for operators with constant coefficients. This result is proved in Appendix A.

Lemma 2.2. Let µ > 0 and ν > 0 be two constants. Then the fundamental solution for the operator

-µ∆ + ν is given by G µ,ν (x, ξ) = G µ,ν (x -ξ), x, ξ ∈ R n , with G µ,ν (x) = (2πµ) -n/2 ( √ νµ/|x|) n/2-1 K n/2-1 ( √ ν|x|/ √ µ), x ∈ R n .
Here K n/2-1 is the usual modified Bessel function of second kind. Moreover the following two-sided inequality holds

(2.10)

C -1 e - √ ν|x|/ √ µ |x| n-2 ≤ G µ,ν (x) ≤ C e - √ ν|x|/(2 √ µ) |x| n-2 , x ∈ R n , for some constant C = C(n, µ, ν) > 1.
The main result of this section is the following theorem.

Theorem 2.2. Let (a, b) ∈ D(λ, κ). Then there exists a unique function

G a,b satisfying G a,b (•, ξ) ∈ C(R n \ {ξ}), ξ ∈ R n , G a,b (x, •) ∈ C(R n \ {x}), x ∈ R n , and (i) L a,b G a,b (•, ξ) = 0 in D (R n \ {ξ}), ξ ∈ R n , (ii) for any f ∈ C ∞ 0 (R n ), u(x) = ˆRn G a,b (x, ξ)f (ξ)dξ
belongs to H 2 (R n ) and it is the unique solution of L a,b u = f , (iii) there exist two constants c = c(n, λ) > 1 and C = C(n, λ, κ) > 1 so that

C -1 e -2 √ cκ|x-ξ| |x -ξ| n-2 ≤ G a,b (x, ξ) ≤ C e - |x-ξ| √ cκ |x -ξ| n-2 . (2.11) Proof. Pick s ≥ 1 arbitrary and let f ∈ C ∞ 0 (R n ). Applying Hölder's inequality, we find ˆRn k a,b (t, x, ξ)|f (ξ)|dξ ≤ k a,b (t, x, •) L s (R n ) f L s (R n ) ,
where s is the conjugate exponent of s. But, according to (2.6),

k a,b (t, x, •) s L s (R n ) ≤ c t n/2 s ˆRn e -s|x-ξ| 2 ct dξ.
Next, making the change of variable ξ = ( ct/s)η + x, we get

k a,b (t, x, •) s L s (R n ) ≤ c t n/2 s ct s n/2 ˆRn e -|η| 2 dη. Hence k a,b (t, x, •) L s (R n ) ≤ t n(1/s-1)/2 C s ,
with

C s = c c s n/2 ˆRn e -|η| 2 dη 1/s .
We get, by choosing 1

≤ s < n n-2 < s, ˆ+∞ 0 ˆRn k a,b (t, x, ξ)|f (ξ)|dξdt = ˆ1 0 ˆRn k a,b (t, x, ξ)|f (ξ)|dξdt + ˆ+∞ 1 ˆRn k a,b (t, x, ξ)|f (ξ)|dξdt ≤ C s f L s (R n ) ˆ1 0 t n 2 (1/s-1) dt + C s f L s (R n ) ˆ+∞ 1 t n 2 (1/s-1) dt.
In light of Fubini's theorem we obtain (2.12)

ˆ+∞ 0 ˆRn k a,b (t, x, ξ)f (ξ)dξdt = ˆRn ˆ+∞ 0 k a,b (t, x, ξ)dt f (ξ)dξ. Define G a,b as follows G a,b (x, ξ) = ˆ+∞ 0 k a,b (t, x, ξ)dt.
Then (2.12) takes the form (2.13)

ˆ+∞ 0 ˆRn k a,b (t, x, ξ)f (ξ)dξdt = ˆRn G a,b (x, ξ)f (ξ)dξ.
Noting that A a,b is invertible, we obtain

-A -1 a,b f (x) = ˆ+∞ 0 e tA a,b f dt (x) = ˆ+∞ 0 ˆRn k a,b (t, x, ξ)f (ξ)dξdt, x ∈ R n .
This and (2.13) entail

-A -1 a,b f (x) = ˆRn G a,b (x, ξ)f (ξ)dξ, x ∈ R n .
In other words, u defined by

u(x) = ˆRn G a,b (x, ξ)f (ξ)dξ, x ∈ R n , belongs to H 2 (R n ) and satisfies L a,b u = f . Since, for x = ξ, ˆ+∞ 0 1 t n/2 e -|x-ξ| 2 ct dt = c n/2-1 ˆ+∞ 0 τ n/2-2 e -τ dτ 1 |x -ξ| n-2 ,
we get in light of (2.7)

|G a,b (x + h, ξ) -G a,b (x, ξ)| ≤ C |x -ξ| n-2+η |h| η , x = ξ, |h| ≤ |x -ξ|,
where C = C(n, λ, κ) is a constant. In particular, G a,b (•, ξ) ∈ C(R n \ {ξ}). Similarly, using (2.8) instead of (2.7), we obtain G a,b (x, •) ∈ C(R n \ {x}). More specifically we have

(2.14) |G a,b (x, ξ + h) -G a,b (x, ξ)| ≤ C |x -ξ| n-2+η |h| η , x = ξ, |h| ≤ |x -ξ|. Let ξ ∈ R n and ω R n \ {ξ}, and pick g ∈ C ∞ 0 (ω). Then set w a,b (y) = ˆω G a,b (x, y)g(x)dx, y ∈ B(ξ, dist(ξ, ω)/2).
It follows from (2.14) that, for y ∈ B(ξ, dist(ξ, ω)) and |h| < dist(y, ω), we have

|w a,b (y + h) -w a,b (y)| ≤ C dist(y, ω) n-2+η |h| η . Therefore w a,b ∈ C(B(ξ, dist(ξ, ω)/2). Let M(R n ) be the space of bounded measures on R n . Pick a sequence (f k ) of a positive functions of C ∞ 0 (R n ) converging in M(R n ) to δ ξ and let u k = -A -1 a,b f k . In that case, according to Fubini's theorem, we have ˆω u k (x)g(x)dx = ˆω ˆRn G a,b (x, y)g(x)f k (y)dydx = ˆRn w a,b (y)f k (y)dy -→ w a,b (ξ) = ˆω G a,b (x, ξ)g(x)dx,
where we used that suppf k ⊂ B(ξ, dist(ξ, ω)/2), provided that k is sufficiently large. That is we proved that

u k converges to G a,b (•, ξ) weakly in L 2 loc (R n \ {ξ}) (think to the fact that C ∞ 0 (ω) is dense in L 2 (ω)). Now, as L a,b u k = f k , we find L a,b G a,b (•, ξ) = 0 in R n \ {ξ} in the distributional sense.
The uniqueness of G a,b follows from that of u and, as

κ -1 ≤ b ≤ κ, we deduce from Lemma 2.1 that E a,κ (x, t, ξ, 0) ≤ E a,b (x, t, ξ, 0) ≤ E a,κ -1 (x, t, ξ, 0). But a simple change of variable shows that (2.15) E a,κ -1 (x, t, ξ, 0) = e -κ -1 t E a,0 (x, t, ξ, 0) and (2.16) E a,κ (x, t, ξ, 0) = e -κt E a,0 (x, t, ξ, 0).
Therefore, from Theorem 2.1 and identity (2.3), there exists a constant c = c(n, λ) > 1 so that

e -κt (πc -1 ) n/2+1 π E c -1 /4,0 (x, t, ξ, 0) ≤ E a,b (x, t, ξ, 0) ≤ e -κ -1 t (πc) n/2+1 π E c/4,0 (x, t, ξ, 0),
which, combined with identities (2.15) and (2.16), gives

(πc -1 ) n/2+1 π E c -1 /4,κ (x, t, ξ, 0) ≤ E a,b (x, t, ξ,0) ≤ (πc) n/2+1 π E c/4,κ -1 (x, t, ξ, 0).
From the uniqueness of G a,b , we obtain by integrating over (0, +∞), with respect to t, each member of the above inequalities

(πc -1 ) n/2+1 π G c -1 /4,κ (x, ξ) ≤ G a,b (x, ξ) ≤ (πc) n/2+1 π G c/4,κ -1 (x, ξ).
These two-sided inequalities together with (2.10) yield in a straightforward manner (2.11).

The function G a,b given by the previous theorem is usually called a fundamental solution of the operator L a,b .

Regularity of fundamental solutions

. Let ξ ∈ R n and O O R n \ {ξ} with O of class C 1,1 . As G a,b (•, ξ) ∈ C(∂O ), we get from [17, Theorem 6.18, page 106] (interior Hölder regularity) that G a,b (•, ξ) belongs to C 2,α (O). Proposition 2.1. There exist C = C(n, λ, κ, α) and ν = ν(α) > 2 so that, for any ξ ∈ R n and O R n \ {ξ}, we have (2.17) G a,b (•, ξ) C 2,α (O) ≤ CΛ(d + ) ν max -(2+α) , 1 -n+2 . Here = dist ξ, O , d = diam(O) and Λ(h) = [1 + 2h + 2h 2 + h 3 ]λ, h > 0.
The proof of this proposition is based the following lemma consisting in an adaptation of the usual interior Schauder estimates. The proof of this technical lemma will be given in Appendix A.

Lemma 2.3. There exists two constants C = C(n, α) and ν = ν(α) > 1 with the property that, for any bounded subset

Q of R n , δ > 0 so that Q δ = {x ∈ Q; dist(x, ∂Q) > δ} = ∅, w ∈ C 2,α (Q)∩C Q satisfying L a,b w = 0 in Q and Q ⊂ Q δ , we have (2.18) w C 2,α (Q ) ≤ C max δ -(2+α) , 1 Λ(d) ν w C(Q) ,
where Λ is as in Proposition 2.1 and

d = diam(Q).
Proof of Proposition 2.1. We get, by applying Lemma 2.3 with

Q = O, δ = /2 and Q = x ∈ R n ; dist x, O < /2 , G a,b (•, ξ) C 2,α (O) ≤ CΛ(d + ) ν max δ -(2+α) , 1 G a,b (•, ξ) C(Q) .
This and (2.11) yield

(2.19) G a,b (•, ξ) C 2,α (O) ≤ CΛ(d + ) ν max δ -(2+α) , 1 -n+2 e -/ √ cκ , with C = C(n, λ, κ, α) and c = c(n, λ). It is then clear that (2.19) implies (2.17).
The preceding proposition together with Lemma A.2 enable us to state the following corollary.

Corollary 2.1.

There exist C = C(n, λ, κ, α, θ) and ν = ν(α) > 1 so that, for any ξ ∈ R n and O R n \ {ξ}, we have G a,b (•, ξ) H 2+θ (O) (2.20) ≤ CΛ(d + ) ν max d n/2 , d n/2+α-θ max -(2+α) , 1 -n+2 , where = dist ξ, O , d = diam(O).

Corollary 2.2. There exist

C = C(n, λ, κ, α) and c = c(n, λ, κ, α) so that, for any ξ 1 , ξ 2 ∈ R n and O R n \ {ξ 1 , ξ 2 }, we have (2.21) G a,b (•, ξ 2 ) G a,b (•, ξ 1 ) C 2,α (O) ≤ Ce c(d+ + ) 1 + max -(2+α) - , 1 -n+2 - 4
,

where -= min (dist (ξ 1 , O) , dist (ξ 2 , O)) and + = max (dist (ξ 1 , O) , dist (ξ 2 , O)).
Proof. In this proof C = C(n, λ, κ, α), c = c(n, λ, κ, α) and ν = ν(α) > 2 are generic constants. From Proposition 2.1, we have

(2.22) G a,b (•, ξ j ) C 2,α (O) ≤ CΛ(d + + ) ν max -(2+α) - , 1 -n+2 - , j = 1, 2.
Let C 0 ≥ 1 end c 0 ≥ 1 be the constants in (2.11) and fix 0 < δ 0 ≤ 1. Then the first inequality in

(2.11) gives 1 G a,b (•, ξ 1 ) ≤ C 0 (d + + ) n-2 e 2 √ c0κ(d+ + ) .
This inequality together with Lemma A.1 in Appendix A yield

(2.23) 1 G a,b (•, ξ 1 ) C 2,α (O) ≤ Ce c(d+ + ) 1 + G a,b (•, ξ 1 ) C 2,α (O) 3 .
Then in light of (2.22) and (2.23), we get in a straightforward manner

G a,b (•, ξ 2 ) G a,b (•, ξ 1 ) C 2,α (O) ≤ Ce c(d+ + ) 1 + (1 + d) ν max -(2+α) - , 1 -n+2 - 4 , and hence G a,b (•, ξ 2 ) G a,b (•, ξ 1 ) C 2,α (O) ≤ Ce c(d+ + ) 1 + max -(2+α) - , 1 -n+2 - 4
. This is the expected inequality.

This corollary combined with Lemma A.2 yields the following result.

Corollary 2.3. There exist C = C(n, λ, κ, α, θ) and c = c(n, λ, κ, α, θ) so that, for any

ξ 1 , ξ 2 ∈ R n and O R n \ {ξ 1 , ξ 2 }, we have (2.24) G a,b (•, ξ 2 ) G a,b (•, ξ 1 ) H 2+θ (O) ≤ Ce c(d+ + ) 1 + max -(2+α) - , 1 -n+2 - 4 .
Here ± is the same as in Corollary 2.2.

Gradient estimate of the quotient of two fundamental solutions.

The following result uses the singularity of the Green function near the location of the point source.

Lemma 2.4. There exist

x * ∈ B(ξ 2 , |ξ 1 -ξ 2 |/2) \ {ξ 2 }, C = (n, λ, κ, |ξ 1 -ξ 2 |) > 0 and ρ = ρ(n, λ, κ, |ξ 1 -ξ 2 |) > 0 so that B(x * , ρ) ⊂ B(ξ 2 , |ξ 1 -ξ 2 |/2) \ {ξ 2 } and C ≤ ∇ G a,b (•, ξ 2 ) G a,b (•, ξ 1 ) L 2 (B(x * ,ρ)) . Proof. We set for notational convenience w = G a,b (•, ξ 2 )/G a,b (•, ξ 1 )
. In light of Theorem 2.2, we obtain by straightforward computations the following two-sided inequality

(2.25) C -1 |x -ξ 2 | n-2 ≤ w(x) ≤ C |x -ξ 2 | n-2 , x ∈ B(ξ 2 , |ξ 1 -ξ 2 |/2) \ {ξ 2 }.
Here and until the end of this proof

C = C(n, λ, κ, |ξ 1 -ξ 2 |) is a generic constant. Set t = |ξ 1 -ξ 2 |/4 and define ϕ(t, θ) = w(ξ 2 + tθ), (t, θ) ∈ (0, t] × S n-1 .
According to Corollary 2.2, ϕ ∈ C 2,α loc ((0, t] × S n-1 ) and hence

ϕ( t, θ) -ϕ(t, θ) = ˆt t ∇w(ξ 2 + sθ) • θds, which in turn gives |ϕ( t, θ) -ϕ(t, θ)| 2 ≤ ( t -t) ˆt t |∇w(ξ 2 + sθ)| 2 ds ≤ t ˆt t |∇w(ξ 2 + sθ)| 2 ds ≤ t ˆt t s n-1 t n-1 |∇w(ξ 2 + sθ)| 2 ds, (t, θ) ∈ (0, t] × S n-1 .
Whence, where t ∈ (0, t],

(2.26)

t n-1 ˆSn-1 |ϕ( t, θ) -ϕ(t, θ)| 2 dθ ≤ t ˆCt |∇w(x)| 2 dx.
Here

C t = x ∈ R n ; t < |x -ξ 2 | < t .
On the other hand inequalities (2.25) imply, where (t, θ) ∈ (0, t] × S n-1 ,

C -1 t n-2 ≤ ϕ(t, θ) ≤ C t n-2 .
Let us then choose t 0 ≤ t sufficiently small in such a way that

C -1 t n-2 - C tn-2 > 0, t ∈ (0, t 0 ]. Therefore, for (t, θ) ∈ (0, t 0 ] × S n-1 , we have (2.27) C -1 t n-2 - C tn-2 2 ≤ |ϕ( t, θ) -ϕ(t, θ)| 2 .
We then obtain by combining inequalities (2.26) and (2.27)

|S n-1 | C -1 t n-2 - C tn-2 2 ≤ t ˆCt |∇w(x)| 2 dx, t ∈ (0, t 0 ].
We have in particular

C ≤ ˆCt 0 |∇w(x)| 2 dx. Let ρ = t 0 /4. Then it is straightforward to check that, for any x ∈ C t0 , B(x, ρ) ⊂ {y ∈ R n ; 3t 0 /4 ≤ |y -ξ 2 | ≤ 5 t/4} ⊂ B(ξ 2 , |ξ 1 -ξ 2 |/2) \ {ξ 2 }.
Since C t0 is compact, we find a positive integer

N = N (λ, κ, |ξ 1 -ξ 2 |) and x j ∈ C t0 , j = 1, • • • , N , so that C t0 ⊂ N j=1 B(x j , ρ). Hence C ≤ ˆ∪N j=1 B(xj ,ρ) |∇w(x)| 2 dx. Pick then x * ∈ {x j , 1 ≤ j ≤ N } in such a way that ˆB(x * ,ρ) |∇w(x)| 2 dx = max 1≤j≤N ˆB(xj,ρ) |∇w(x)| 2 dx. Therefore C ≤ ˆB(x * ,ρ) |∇w(x)| 2 dx.
This finishes the proof.

Uniform lower bound for the gradient

Let O be a Lipschitz bounded domain of R n and σ ∈ C 0,1 (O) satisfying

κ -1 ≤ σ and σ C 0,1 (O) ≤ κ, (3.1)
for some fixed constant κ > 1.

We prove in this section a polynomial lower bound of the local L 2 -norm of the gradient of solutions of L σ u = div(σ∇u) = 0 in O. In a first step we establish, via a three-ball inequality for the gradient, a uniform lower bound of negative exponential type. We use then in a second step an argument based on the so-called frequency function in order to improve this lower bound.

3.1. Preliminary lower bound. We need hereafter the following three-ball inequality for the gradient.

Theorem 3.1. Let 0 < k < < m be real. There exist two constants

C = C(n, κ, k, , m) > 0 and 0 < γ = γ(n, κ, k, , m) < 1 so that, for any v ∈ H 1 (O) satisfying L σ v = 0, y ∈ O and 0 < r < dist(y, ∂O)/m, we have C ∇v L 2 (B(y, r)) ≤ ∇v γ L 2 (B(y,kr)) ∇v 1-γ L 2 (B(
O δ = {x ∈ O; dist(x, ∂O) > δ} and χ(O) = sup{δ > 0; O δ = ∅}. Define S (O, x 0 , M, η, δ) = u ∈ H 1 (O); L σ u = 0 in O, (3.3) ∇u L 2 (O) ≤ M, ∇u L 2 (B(x0,δ)) ≥ η , with δ ∈ (0, χ(O)/3), x 0 ∈ O 3δ , η > 0 and M ≥ 1 satisfying η < M .
Lemma 3.2. There exist two constants c = c(n, κ) ≥ 1 and 0 < γ = γ(n, κ) < 1 so that, for any u ∈ S (O, x 0 , M, η, δ) and x ∈ O 3δ , we have

(3.4) e -[ln(cM/η)/γ]e [2n| ln γ|]c|x-x 0 |/δ ≤ ∇u L 2 (B(x,δ)) , with c = c O is as in Lemma 3.1.
Proof. Pick u ∈ S (O, x 0 , M, η, δ). Let x ∈ O 3δ and ψ : [0, 1] → O be a Lipschitz path joining x = ψ(0) to x 0 = ψ(1), so that (ψ) ≤ 2d g (x 0 , x). Here and henceforth, for simplicity convenience, we use

d g (x 0 , x) instead of d O g (x 0 , x). Let t 0 = 0 and t k+1 = inf{t ∈ [t k , 1]; ψ(t) ∈ B(ψ(t k ), δ)}, k ≥ 0.
We claim that there exists an integer N ≥ 1 verifying ψ(1) ∈ B(ψ(t N ), δ). If not, we would have ψ(1) ∈ B(ψ(t k ), δ) for any k ≥ 0. As the sequence (t k ) is non decreasing and bounded from above by 1, it converges to t ≤ 1. In particular, there exists an integer

k 0 ≥ 1 so that ψ(t k ) ∈ B ψ( t), δ/2 , k ≥ k 0 . But this contradicts the fact that |ψ(t k+1 ) -ψ(t k )| ≥ δ, k ≥ 0.
Let us check that N ≤ N 0 , where

N 0 = N 0 (n, |x -x 0 |, c, δ). Pick 1 ≤ j ≤ n so that max 1≤i≤n |ψ i (t k+1 ) -ψ i (t k )| = |ψ j (t k+1 ) -ψ j (t k )| ,
where

ψ i is the ith component of ψ. Then δ ≤ n |ψ j (t k+1 ) -ψ j (t k )| = n ˆtk+1 t k ψj (t)dt ≤ n ˆtk+1 t k | ψ(t)|dt.
Consequently, where t N +1 = 1,

(N + 1)δ ≤ n N k=0 ˆtk+1 t k | ψ(t)|dt = n (ψ) ≤ 2nd g (x 0 , x) ≤ 2nc|x -x 0 |.
Therefore

N ≤ N 0 = 2nc|x -x 0 | δ . Let y 0 = x and y k = ψ(t k ), 1 ≤ k ≤ N . If |z -y k+1 | < δ, then |z -y k | ≤ |z -y k+1 | + |y k+1 -y k | < 2δ.
In other words B(y k+1 , δ) ⊂ B(y k , 2δ). We get from Theorem 3.1

(3.5) ∇u L 2 (B(yj ,2δ)) ≤ C ∇u 1-γ L 2 (B(yj ,3δ)) ∇u γ L 2 (B(yj ,δ)) , 0 ≤ j ≤ N, for some constants C = C(n, κ) > 0 and 0 < γ = γ(n, κ) < 1. Set I j = ∇u L 2 (B(yj ,δ)) , 0 ≤ j ≤ N and I N +1 = ∇u L 2 (B(x0,δ)) . Since B(y j+1 , δ) ⊂ B(y j , 2δ), 1 ≤ j ≤ N -1, estimate (3.5) implies (3.6) I j+1 ≤ CM 1-γ I γ j , 0 ≤ j ≤ N.
Let C 1 = C 1+γ+...+γ N +1 and β = γ N +1 . Then, by a simple induction argument, estimate (3.6) yields

(3.7)

I N +1 ≤ C 1 M 1-β I β 0 .
Without loss of generality, we assume in the sequel that C ≥ 1 in (3.6). Using that N ≤ N 0 , we have

β ≥ β 0 = γ N0+1 , C 1 ≤ C 1 1-γ , I 0 M β ≤ I 0 M β0 .
These estimates in (3.7) give

I N +1 M ≤ C 1 1-γ I 0 M γ N 0 +1
, from which we deduce that

∇u L 2 (B(x0,δ)) ≤ C 1 1-γ M 1-γ N 0 +1 ∇u γ N 0 +1 L 2 (B(x,δ)) . But M ≥ 1. Whence η ≤ ∇u L 2 (B(x0,δ)) ≤ C 1 1-γ M ∇u γ N 0 +1 L 2 (B(x,δ))
. The expected inequality follows readily from this last estimate.

3.2. An estimate for the frequency function. Some tools in the present section are borrowed from [START_REF] Garofalo | Monotonicity properties of variational integrals, Ap weights and unique continuation[END_REF][START_REF] Garofalo | Unique continuation for elliptic operators: a geometric-variational approach, Commun[END_REF][START_REF] Kukavica | Quantitative uniqueness for second-order elliptic operators[END_REF]. Let u ∈ H 1 (O) and σ ∈ C 0,1 (O) satisfying the bounds (3.1). We recall that the usual frequency function, relative to the operator L σ , associated to u is defined by

N (u)(x 0 , r) = rD(u)(x 0 , r) H(u)(x 0 , r) , provided that B(x 0 , r) O, with D(u)(x 0 , r) = ˆB(x0,r) σ(x)|∇u(x)| 2 dx, H(u)(x 0 , r) = ˆ∂B(x0,r) σ(x)u 2 (x)dS(x).
Define also

K(u)(x 0 , r) = ˆB(x0,r) σ(x)u 2 (x)dx.
Prior to studying the properties of the frequency function, we prove some preliminary results. Fix u ∈ H 2 (O) so that L σ u = 0 in O and, for simplicity convenience, we drop in the sequel the dependence on u of N , D, H and K.

Lemma 3.3. For x 0 ∈ O δ and 0 < r < δ, we have

∂ r H(x 0 , r) = n -1 r H(x 0 , r) + H(x 0 , r) + 2D(x 0 , r), (3.8) ∂ r D(x 0 , r) = n -2 r D(x 0 , r) + 1 r D(x 0 , r) + 2 Ĥ(x 0 , r). (3.9) Here H(x 0 , r) = ˆ∂B(x0,r) u 2 ∇σ(x) • ν(x)dS(x), Ĥ(x 0 , r) = ˆ∂B(x0,r) σ(x)(∂ ν u(x)) 2 dS(x), D(x 0 , r) = ˆB(x0,r) |∇u(x)| 2 ∇σ(x) • (x -x 0 )dx.
Proof. Pick x 0 ∈ O δ and 0 < r < δ. A simple change of variable yields

H(x 0 , r) = ˆ∂B(0,1) σ(x 0 + ry)u 2 (x 0 + ry)r n-1 dS(y). Hence ∂ r H(x 0 , r) = n -1 r H(x 0 , r) + ˆ∂B(0,1) ∇(σu 2 )(x 0 + ry) • yr n-1 dS(y) = n -1 r H(x 0 , r) + ˆ∂B(0,1) u 2 (x 0 + ry)∇σ(x 0 + ry) • yr n-1 dS(y) + ˆ∂B(0,1) σ(x 0 + ry)∇(u 2 )(x 0 + ry) • yr n-1 dS(y) = n -1 r H(x 0 , r) + ˆ∂B(x0,r) u 2 (x)∇σ(x) • ν(x)dS(x) + ˆ∂B(x0,r) σ(x)∇(u 2 )(x) • ν(x)dS(x) = n -1 r H(x 0 , r) + H(x 0 , r) + ˆ∂B(x0,r) σ(x)∇(u 2 )(x) • ν(x)dS(x).
Identity (3.8) will follow if we prove

(3.10) 2D(x 0 , r) = ˆ∂B(x0,r) σ(x)∇(u 2 )(x) • ν(x)dS(x).
To this end, we observe that div(σ∇u) = 0 implies div(σ∇(u 2 )) = 2udiv(σ∇u) + 2σ|∇u| 2 = 2σ|∇u| 2 .

We then get by applying the divergence theorem

2D(x 0 , r) = ˆB(x0,r) div(σ(x)∇(u 2 )(x))dx (3.11) = ˆ∂B(x0,r) σ(x)∇(u 2 )(x) • ν(x)dS(x).
This proves (3.10). By a change of variable we have

D(x 0 , r) = ˆr 0 ˆ∂B(0,1) σ(x 0 + ty)|∇u(x 0 + ty)| 2 t n-1 dS(y)dt. Hence ∂ r D(x 0 , r) = ˆ∂B(0,1) σ(x 0 + ry)|∇u(x 0 + ry)| 2 r n-1 dS(y) = ˆ∂B(x0,r) σ(x)|∇u(x)| 2 dS(x) = 1 r ˆ∂B(x0,r) σ(x)|∇u(x)| 2 (x -x 0 ) • ν(x)dS(x).
An application of the divergence theorem then gives

∂ r D(x 0 , r) = 1 r ˆB(x0,r) div(σ(x)|∇u(x)| 2 (x -x 0 ))dx. Therefore ∂ r D(x 0 , r) = 1 r ˆB(x0,r) |∇u(x)| 2 div(σ(x)(x -x 0 ))dx + 1 r ˆB(x0,r) σ(x)(x -x 0 ) • ∇(|∇u(x)| 2 )dx implying ∂ r D(x 0 , r) = n r D(x 0 , r) + 1 r D(x 0 , r) (3.12) + 1 r ˆB(x0,r) σ(x)(x -x 0 ) • ∇(|∇u(x)| 2 )dx.
On the other hand, ˆB(x0,r)

σ(x)(x j -x 0,j )∂ j (∂ i u(x)) 2 dx = 2 ˆB(x0,r) σ(x)(x j -x 0,j )∂ 2 ij u∂ i u(x)dx = -2 ˆB(x0,r) ∂ i [∂ i u(x)σ(x)(x j -x 0,j )] ∂ j u(x)dx + 2 ˆ∂B(x0,r) σ(x)∂ i u(x)(x j -x 0,j )∂ j u(x)ν i (x)dS(x) = -2 ˆB(x0,r) ∂ 2 ii u(x)σ(x)(x j -x 0,j )∂ j u(x)dx -2 ˆB(x0,r) ∂ i u(x)∂ j u(x)∂ i [σ(x)(x j -x 0,j )] dx + 2 ˆ∂B(x0,r) σ(x)∂ i u(x)(x j -x 0,j )∂ j u(x)ν i (x)dS(x).
Thus, taking into account that σ∆u = -∇σ • ∇u,

ˆB(x0,r) σ(x)(x -x 0 ) • ∇(|∇u(x)| 2 )dx = -2 ˆB(x0,r) σ(x)|∇u(x)| 2 dx + 2r ˆ∂B(x0,r) σ(x)(∂ ν u(x)) 2 dS(x).
This identity in (3.12) yields

∂ r D(x 0 , r) = n -2 r D(x 0 , r) + 1 r D(x 0 , r) + 2 Ĥ(x 0 , r).
That is we proved (3.9).

Lemma 3.4. We have

K(x 0 , r) ≤ re rκ 2 H(x 0 , r), x 0 ∈ O δ , 0 < r < δ.
Proof. Taking into account that H(x 0 , r) ≥ 0 and D(x 0 , r) ≥ 0, we obtain from identity (3.8)

∂ r H(x 0 , r) ≥ ˆ∂B(x0,r) ∂ ν σ(x)u 2 (x)dS(x) ≥ ˆ∂B(x0,r) ∂ ν σ(x) σ(x) σ(x)u 2 (x)dS(x) ≥ -κ 2 H(x 0 , r).
Consequently r → e rκ 2 H(x 0 , r) is non decreasing and then

ˆr 0 H(x 0 , t)dt ≤ ˆr 0 e tκ 2 H(x 0 , t)dt ≤ ˆr 0 e rκ 2 H(x 0 , r)dt ≤ re rκ 2 H(x 0 , r).
As K(x 0 , r) = ˆr 0 H(x 0 , t)dt, we end up getting K(x 0 , r) ≤ re rκ 2 H(x 0 , r). This completes the proof. Now straightforward computations yield, for x 0 ∈ O δ and 0 < r < δ,

(3.13) ∂ r N (x 0 , r) N (x 0 , r) = 1 r + ∂ r D(x 0 , r) D(x 0 , r) - ∂ r H(x 0 , r) H(x 0 , r) .
Lemma 3.5. For x 0 ∈ O δ and 0 < r < δ, we have

N (x 0 , r) ≤ e 2κ 2 δ N (x 0 , δ).
Proof. We have from formulas (3.8) and (3.9) and identity (3.13)

∂ r N (x 0 , r) N (x 0 , r) = D(x 0 , r) rD(x 0 , r) - H(x 0 , r) H(x 0 , r) + 2 Ĥ(x 0 , r) D(x 0 , r) -2 D(x 0 , r) H(x 0 , r) (3.14) = D(x 0 , r) rD(x 0 , r) - H(x 0 , r) H(x 0 , r) + 2 Ĥ(x 0 , r)H(x 0 , r) -D(x 0 , r) 2 D(x 0 , r)H(x 0 , r) .
But from (3.11) we have

D(x 0 , r) = ˆ∂B(x0,r) σ(x)u(x)∂ ν u(x)dS(x).
Then we find by applying Cauchy-Schwarz's inequality

D(x 0 , r) 2 ≤ ˆ∂B(x0,r) σ(x)u 2 (x)dS(x) ˆ∂B(x0,r) σ(x)(∂ ν u) 2 (x)dS(x) .
That is

(3.15) D 2 (x 0 , r) ≤ H(x 0 , r) Ĥ(x 0 , r).
This and (3.14) lead

(3.16) ∂ r N (x 0 , r) N (x 0 , r) ≥ D(x 0 , r) rD(x 0 , r) - H(x 0 , r) H(x 0 , r) .
On the other hand

(3.17) H(x 0 , r) ≤ κ ∇a ∞ H(x 0 , r) ≤ κ 2 H(x 0 , r),
and similarly

(3.18) D(x 0 , r) ≤ κ 2 rD(x 0 , r).
In light of (3.16), (3.17) and (3.18), we derive

∂ r N (x 0 , r) N (x 0 , r) ≥ -2κ 2 ,
that is to say ∂ r (e 2κ 2 r N (x 0 , r)) ≥ 0. Consequently N (x 0 , r) ≤ e 2κ 2 (δ-r) N (x 0 , δ) ≤ e 2κ 2 δ N (x 0 , δ), as expected.

Polynomial lower bound.

Lemma 3.6. There exist a universal constant and two constants c = c(n, κ) > 0 and 0 < γ = γ(n, κ) < 1 so that if

C 0 (h) = M κ 4 (1 + d)δ -1 e 3κ 2 δ+[2 ln(cM/η)/γ]e [6n| ln γ|]ch , h > 0, then N (u)(x, •) L ∞ (0,δ) ≤ C 0 (|x -x 0 |/δ), for any u ∈ S (O, x 0 , M, η, δ/3), where c = c O is as in Lemma 3.1. Proof. Pick x ∈ O δ . Then from Lemma 3.2 ∇u L 2 (B(x,δ/3)) ≥ e -[ln(cM/η)/γ]e [6n| ln γ|]c|x-x 0 |/δ ,
for some constant c = c(n, κ) and 0 < γ = γ(n, κ)) < 1.

On the other hand, we establish in a quite classical manner the following Caccioppoli's inequality

∇u 2 L 2 (B(x,δ/3)) ≤ κ 2 (1 + d) δ 2 u 2 L 2 (B(x,δ)) ,
where is a universal constant. Therefore

(3.19) u 2 L 2 (B(x,δ)) ≥ C0 (|x -x 0 |/δ), where (3.20) C0 (h) = δ 2 κ 2 (1 + d) e -[2 ln(cM/η)/γ]e [6n| ln γ|c]h , h > 0. Since K(u)(x, δ) ≥ κ -1 u 2 L 2 (B(x,δ)) , we find (3.21) K(u)(x, δ) ≥ δ 2 κ 3 (1 + d) e -[2 ln(cM/η)/γ]e [6n| ln γ|]c|x-x 0 |/δ .
In light of Lemma 3.4, we derive from (3.21)

(3.22) H(u)(x, δ) ≥ δe -κ 2 δ κ 3 (1 + d) e -[2 ln(cM/η)/γ]e [6n| ln γ|]c|x-x 0 |/δ .
In light of Lemma 3.5, we get

N (x, r) ≤ κe 2κ 2 δ ∇u L 2 (O) H(u)(x, δ) , 0 < r < δ,
This inequality and (3.22) give, where c = c(n, κ) is a constant,

N (x, r) ≤ M κ 4 (1 + d)δ -1 e 3κ 2 δ+[2 ln(cM/η)/γ]e [6n| ln γ|]c|x-x 0 |/δ , 0 < r < δ,
which is the expected inequality.

Proposition 3.1. Let C 0 be as in Lemma 3.6, C0 as in (3.20) and set

C 1 (h) = 2C 0 (h) + n, h > 0, (3.23) C2 (h) = κ -2 e -κ 2 δ C0 (h), h > 0. (3.24) If u ∈ S (O, x 0 , M, η, δ/3) then C2 (|x -x 0 |/δ) r δ C1(|x-x0|/δ) ≤ u 2 L 2 (B(x,r)) , x ∈ O δ , 0 < r < δ.
Proof. Observing that, where

H = H(u), ∂ r ln H(x, r) r n-1 = ∂ r H(x, r) H(x, r) - n -1 r ,
we get from Lemma 3.6, (3.8) and the fact that

| H(x, r)| ≤ κ 2 H(x, r), ∂ r ln H(x, r) r n-1 ≤ κ 2 + 2N (x, r) r ≤ κ 2 + 2C 0 (|x -x 0 |/δ) r , 0 < r < δ, Thus ˆsδ sr ∂ t ln H(x, t) t n-1 dt = ln H(x, sδ)r n-1 H(x, sr)δ n-1 ≤ κ 2 (δ -r)s + 2C 0 (|x -x 0 |/δ) ln δ r , for 0 < s < 1 and 0 < r < δ. Hence H(x, sδ) ≤ e κ 2 δ δ r C1(|x-x0|/δ)-1 H(x, sr),
and then

u 2 L 2 (B(x,δ)) ≤ κδ ˆ1 0 H(x, sδ)ds ≤ κδe κ 2 δ δ r C1(|x-x0|/δ)-1 ˆ1 0 H(x, rs)ds ≤ κ 2 e κ 2 δ δ r C1(|x-x0|/δ) u 2 L 2 (B(x,r)) .
Combined with (3. [START_REF] Kukavica | Quantitative uniqueness for second-order elliptic operators[END_REF] this estimate yields in a straightforward manner

κ -2 e -κ 2 δ C0 (|x -x 0 |/δ) r δ C1(|x-x0|/δ) ≤ u 2 L 2 (B(x,r))
. This is the expected inequality.

For a bounded domain D, we denote the first non zero eigenvalue of the Laplace-Neumann operator on D by µ 2 (D). Since µ 2 (B(x 0 , r)) = µ 2 (B(0, 1))/r 2 , we get by applying Poincaré-Wirtinger's inequality

w -{w} 2 L 2 (B(x,r)) ≤ 1 µ 2 (B(x, r)) ∇w 2 L 2 (B(x,r)) (3.25) ≤ r 2 µ 2 (B(0, 1)) ∇w 2 L 2 (B(x,r)) ,
for any w ∈ H 1 (B(x, r)), where {w} =

C 2 (|x -x 0 |/δ) r δ C1(|x-x0|/δ) ≤ ∇u 2 L 2 (B(x,r)) , x ∈ O δ , 0 < r < δ, with (3.26) C 2 (h) = µ 2 (B(0, 1))δ -2 C2 (h), h > 0,
with C2 as in Proposition 3.1.

It is important to remark that the argument we used to obtain Corollary 3.1 from Proposition 3.1 is no longer valid if we substitute L σ by L σ plus a multiplication operator by a function σ 0 .

The following consequence of the preceding corollary will be useful in the proof of Theorem 1.1.

Lemma 3.7. Let ω O and set δ = dist(ω, ∂O). Let u ∈ S (O, x 0 , M, η, δ/3) and f ∈ C 0,α (O).
Then we have

(3.27) f L ∞ (ω) ≤ Ĉ3 f 1-μ C 0,α (O) f |∇u| 2 μ L 1 (O) , with μ = α max x∈O C 1 (|x -x 0 |/δ) + α , Ĉ3 = max 2δ α (max 1, ( Ĉ2 δ α ) -1 , max 1, M 2 ( Ĉ2 δ α ) -1 , where Ĉ2 = max x∈O C 2 (|x -x 0 |/δ) with C 2 is as in Corollary 3.1.
Proof. By homogeneity it is enough to consider those functions f ∈ C 0,α (O) satisfying f C 0,α (O) = 1. Let C 1 and C 2 be respectively as in (3.23) and (3.26). Let u ∈ S (O, x 0 , M, η, δ/3) and f ∈ C 0,α (O) satisfying f C 0,α (O) = 1. Pick then x ∈ ω. From Corollary 3.1, we have

(3.28) C 2 (|x -x 0 |/δ) r δ C1(|x-x0|/δ) ≤ ∇u 2 L 2 (B(x,r)) , 0 < r < δ. On the other hand, it is straightforward to check that |f (x)| ≤ |f (y)| + r α , y ∈ B(x, r). Whence |f (x)| ˆB(x,r) |∇u(y)| 2 dy ≤ ˆB(x,r) |f (y)||∇u(y)| 2 dy + r α ˆB(x,r) |∇u(y)| 2 dy.
That is we have

|f (x)| ∇u 2 L 2 (B(x,r) ≤ f |∇u| 2 L 1 (B(x,r)) + r α ∇u 2 L 2 (B(x,r)
) . Since u is non constant, by the unique continuation property, we have

∇u 2 L 2 (B(x,r)) = 0, 0 < r < δ. Therefore |f (x)| ≤ f |∇u| 2 L 1 (B(x,r)) ∇u 2 L 2 (B(x,r)) + r α , 0 < r < δ.
This and (3.28) entail

|f (x)| ≤ C 2 (|x -x 0 |/δ) -1 δ r C1(|x-x0|) f |∇u| 2 L 1 (B(x,r)) + r α , 0 < r < δ. Hence |f (x)| ≤ C 2 (|x -x 0 |/δ) -1 1 s C1(|x-x0|) f |∇u| 2 L 1 (O) + δ α s α , 0 < s < 1.
In consequence

f L ∞ (ω) ≤ Ĉ2 1 s α f |∇u| 2 L 1 (O) + δ α s α , 0 < s < 1, where α = max x∈O C 1 (|x -x 0 |/δ).
The expected inequality follows by minimizing the right hand side of the last inequality, with respect to s.

4. Proof Theorem 1.1 Pick (a, b), (ã, b) ∈ D(λ, κ) and let u j = G a,b (•, ξ j ) and ũj = G ã, b(•, ξ j ), j = 1, 2. By simple computations we can check that w = u 2 /u 1 is the solution of the equation div(σ∇w) = 0 in R n \ {ξ 1 , ξ 2 }, with σ = au 2 1 = av 2 1 b 2 . Similarly, w = ũ2 /ũ 1 is the solution of the equation div(σ∇ w) = 0 in R n \ {ξ 1 , ξ 2 }, with σ = ãũ 2 1 = ãṽ 2 1 b2 .
We know from Lemma 2.4 that there exist

x * ∈ B(ξ 2 , |ξ 1 -ξ 2 |/2)\{ξ 2 }, η 0 = η 0 (n, λ, κ, |ξ 1 -ξ 2 |) > 0 and ρ = ρ(n, λ, κ, |ξ 1 -ξ 2 |) > 0 so that B(x * , ρ) ⊂ B(ξ 2 , |ξ 1 -ξ 2 |/2) \ {ξ 2 } and (4.1) η 0 ≤ ∇w L 2 (B(x * ,ρ)) .
Fix then a bounded domain Q of R n \ {ξ 1 , ξ 2 } is such a way that Ω ∪ B(x * , ρ) Q, and set

δ = dist(Ω ∪ B(x * , ρ), ∂Q).
In the rest of this proof d = diam(Q). According to Corollary 2.3

(4.2) ∇w L 2 (Q) ≤ M = Ce c(d+ + ) 1 + max -(2+α) - , 1 -n+2 - 4 , with C = C(n, λ, κ, α, θ) and c = c(n, λ, κ, α, θ), -= min (dist (ξ 1 , Q) , dist (ξ 2 , Q)) and + = max (dist (ξ 1 , Q) , dist (ξ 2 , Q)). Now, since σ C 0,1 (Q) ≤ a C 0,1 (Q) u 1 2 C 0,1 (Q) ,
σ C 0,1 (Q) ≤ C a C 0,1 (Q) u 1 2 C 2,α (Q) , where C = C(n, λ, κ, d, ξ 1 , ξ 2 ) > 0 is a constant. This inequality together with Proposition 2.1 yield (4.3) σ C 0,1 (Q) ≤ C, for some constant C = C(n, λ, κ, d, ξ 1 , ξ 2 ) > 0.
On the other hand, we have from (2.11)

(4.4) C -1 min x∈Q e -2 √ cκ|x-ξ1| |x -ξ 1 | n-2 ≤ u 1 , in Q, with constants c = c(n, λ) > 0 and C = C(n, λ, κ) > 0.
We get by combining (4.3) and (4.4) that there exists κ = κ(n, λ, κ, α, Ω, ξ 1 , ξ 2 ) > 1 so that

κ -1 ≤ σ and σ C 0,1 (Q) ≤ κ.
Next, if ρ ≤ δ/3 then (4.1) implies obviously (4.5)

η 0 ≤ ∇w L 2 (B(x0,δ/3)) ,
with η 0 as in (4.1). When ρ > δ/3 we can use the three-ball inequality in Theorem 3.1 in order to get

C ∇w L 2 (B(x * ,ρ)) ≤ ∇w s L 2 (B(x0,δ/3)) ∇w 1-s L 2 (B(x * ,ρ+δ/3)) , where C = C(n, λ, κ, Ω, ξ 1 , ξ 2 ) and 0 < s = s(n, λ, κ, Ω, ξ 1 , ξ 2 ) < 1 are constants. Whence (4.6) ( Cη 0 ) 1/s M (s-1)/s ≤ ∇w L 2 (B(x0,δ/3)) .
In light of (4.2), (4.5) and (4.6), we can infer that, for some constant η = η(n, λ, κ, Ω, ξ 1 , ξ 2 ) > 0, w ∈ S (Q, x * , M, η, δ/3), where M is as in (4.2) and S (Q, x * , M, η, δ/3) is defined in (3.3).

Lemma 4.1. We have

(4.7) C (σ -σ)|∇w| 2 L 1 (Ω) ≤ w -w θ/(2+θ) L 2 (Ω) + σ -σ L ∞ (Γ) , where C = C(n, λ, κ, Ω, α, θ, ξ 1 , ξ 2 ) > 0 is a constant. Proof. Clearly, if ζ = σ -σ and u = w -w, then div(σ∇u) = div(ζ∇w).
Recall that sgn 0 is the sign function defined on R by: sgn 0 (t) = -1 if t < 1, sgn 0 (0) = 0 and sgn 0 (t) = 1 if t > 0. Since

Thus ˆΩ |ζ||∇w| 2 dx ≤ C u H 2 (Ω) + ζ L ∞ (Γ) .
This, the following interpolation inequality

u H 2 (Ω) ≤ c Ω u θ/(2+θ) L 2 (Ω) u 2/(2+θ) H 2+θ (Ω)
and Corollary 2.3 give (4.7).

We have from (3.27) in Lemma 3.7

σ -σ C(Ω) ≤ Ĉ3 σ -σ 1-μ C 0,α (Ω) (σ -σ)|∇w| 2 μ L 1 (Ω) , from which we obtain σ -σ C(Ω) ≤ Ĉ3 max 1, σ -σ C 0,α (Ω) (σ -σ)|∇w| 2 μ L 1 (Ω) . Combined with Proposition 2.1, this inequality gives σ -σ C(Ω) ≤ C (σ -σ)|∇w| 2 μ L 1 (Ω) .
Here and henceforward, C = C(n, λ, κ, Ω, α, θ, ξ 1 , ξ 2 ) > 0 is a generic constant.

Therefore, we obtain in light of Lemma 4.1

σ -σ C(Ω) ≤ C w -w θ/(2+θ) L 2 (Ω) + σ -σ C(Γ) μ .
Since ã = a and b = b on Γ and regarding the regularity of u i and ũi , i = 1, 2, we finally get

σ -σ C(Ω) ≤ C v 1 -ṽ1 C(Ω) + v 2 -ṽ2 C(Ω) μ0 , (4.9) with μ0 = θ μ 2 + θ .
The following lemma will be used in sequel.

Lemma 4.2. We have

(4.10) u -1 1 -ũ-1 1 C 2,α (Ω) ≤ C v 1 -ṽ1 C(Ω) + v 2 -ṽ2 C(Ω) μ1
, where 0 < μ1 = μ1 (n, Ω, λ, κ, α, θ, ξ 1 , ξ 2 ) < 1 and C = C(n, Ω, λ, κ, α, θ, ξ 1 , ξ 2 ) > 0 are constants.

Proof. In this proof C = C(n, Ω, λ, κ, α, θ, ξ 1 , ξ 2 ) > 0 is a generic constant. It is not hard to check that

-div(σ∇u -1 1 ) = v 1 in Ω, -div(σ∇ũ -1 1 ) = ṽ1 in Ω. Hence -div(σ∇(u -1 1 -ũ-1 1 )) = (v 1 -ṽ1 ) + div((σ -σ)∇ u -1 1 )
in Ω. By the usual Hölder a priori estimate (see [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Theorem 6.6, page 98])

C u -1 1 -ũ-1 1 C 2,α (Ω) ≤ v 1 -ṽ1 C 0,α (Ω) + div((σ -σ)∇ u -1 1 ) C 0,α (Ω) + u -1 1 -ũ-1 1 C 0,α (Γ) . Consequently (4.11) u -1 1 -ũ-1 1 C 2,α (Ω) ≤ C v 1 -ṽ1 C 0,α (Ω) + σ -σ C 1,α (Ω) , where we used u -1 1 -ũ-1 1 C 0,α (Γ) = b(v -1 1 -ṽ-1 1 ) C 0,α (Γ) . On the other hand, since σ -σ C 1,1 (Ω) ≤ C, v 1 -ṽ1 C 1,α (Ω) ≤ C
and Ω is C 1,1 , we get again from the interpolation inequality in [17, Lemma 6.35, page 135] (4.12)

σ -σ C 1,α (Ω) ≤ C σ -σ τ C(Ω) , v 1 -ṽ1 C 0,α (Ω) ≤ C v 1 -ṽ1 τ C(Ω)
, where 0 < τ = τ (Ω, α) < 1 is a constant. Inequality (4.12) in (4.11) yields (4.13)

u -1 1 -ũ-1 1 C 2,α (Ω) ≤ C v 1 -ṽ1 τ C(Ω) + σ -σ τ C(Ω)
. On the other hand, we have from (4.9)

σ -σ C(Ω) ≤ C v 1 -ṽ1 C(Ω) + v 2 -ṽ2 C(Ω) μ0 . (4.14)
Whence, we get in light of inequalities (4.13) and (4.14), where μ1 = τ μ0 ,

u -1 1 -ũ-1 1 C 2,α (Ω) ≤ C v 1 -ṽ1 C(Ω) + v 2 -ṽ2 C(Ω) μ1 .
This is the expected inequality.

Also, since

σ -σ C 1,1 (Ω) ≤ C, v 1 -ṽ1 C 2,α (Ω) ≤ C,
we can proceed as in the preceding proof to get

(4.15) σ -σ C 1,α (Ω) ≤ C σ -σ τ C(Ω) , v 1 -ṽ1 C 1,α (Ω) ≤ C v 1 -ṽ1 τ C(Ω) , the constant 0 < τ = τ (Ω, α) < 1. But a -ã = σu -2 1 -σũ 1 -2 = (σ -σ)u -2 1 + σ(u -2 1 -ũ-2 1 ) = (σ -σ)u -2 1 + σ(u -1 1 + ũ-1 1 )(u -1 1 -ũ-1 1 ). Hence (4.16) a -ã C 1,α (Ω) ≤ C u -1 1 -ũ-1 1 C 1,α (Ω) + σ -σ C 1,α (Ω)
. This inequality together with (4.9), (4.10) and (4.15) imply (4.17)

a -ã C 1,α (Ω) ≤ C v 1 -ṽ1 C(Ω) + v 2 -ṽ2 C(Ω) μ1 . We proceed similarly for b -b. Since b -b = v 1 u -1 1 -ṽ1 ũ-1 1 = (v 1 -ṽ1 )u -1 1 + ṽ1 (u -1 1 -ũ-1 1 ), we have (4.18) b -b C 0,α (Ω) ≤ C v 1 -ṽ1 C(Ω) + v 2 -ṽ2 C(Ω) μ1 .
The expected inequality follows by putting together (4.17) and (4.18).

Appendix A. Proof of technical lemmas Proof of Lemma 2.2. In this proof C = C(n, µ, ν) > 1 is a generic constant. It is well known that G 1,ν , ν > 0, the fundamental solution of the operator -∆ + ν, is given by G 1,ν (x, ξ) = G 1,ν (x -ξ), x, ξ ∈ R n , with G 1,ν (x) = (2π) -n/2 ( √ ν/|x|) n/2-1 K n/2-1 ( √ ν|x|).
In the particular case n = 3, we have K 1/2 (z) = π/(2z)e -z and therefore

G 1,ν (x) = e - √ ν|x|
4π|x| .

Let f ∈ C ∞ 0 (R n ), µ > 0 and ν > 0 be two constants, and denote by u the solution of the equation

(-µ∆ + ν)u = f in R n . Then (A.1) u(x) = ˆRn G µ,ν (x, ξ)f (ξ)dξ, x ∈ R n . We remark that v(x) = u( √ µx), x ∈ R n satisfies (-∆ + ν)v = f ( √ µ •). Whence u( √ µx) = v(x) = ˆRn G 1,κ (x -ξ)f ( √ µξ)dξ = µ -n/2 ˆRn G 1,ν (x -ξ/ √ µ)f (ξ)dξ, x ∈ R n . Hence (A.2) u(x) = µ -n/2 ˆRn G 1,ν ((x -ξ)/ √ µ)f (ξ)dξ, x ∈ R n .
Comparing (A.1) and (A.2) we find

G µ,ν (x, ξ) = µ -n/2 G 1,ν ((x -ξ)/ √ µ), x, ξ ∈ R n . Consequently G µ,ν (x, ξ) = G µ,ν (x -ξ) with (A.3) G µ,ν (x) = (2πµ) -n/2 ( √ νµ/|x|) n/2-1 K n/2-1 ( √ ν|x|/ √ µ), x ∈ R n .
By the usual asymptotic formula for modified Bessel functions of the second kind (see for instance [5, 9.7.2, page 378]) we have, when |x| → ∞,

K n/2-1 ( √ ν|x|/ √ µ) = π √ µ 2 √ ν|x| 1/2 e - √ ν|x|/ √ µ (1 + O(1/|x|)) ,
where O(1/|x|) only depends on n, µ and ν.

Consequently, there exits R = R(n, µ, ν) > 0 so that

(A.4) C -1 e - √ ν|x|/ √ µ |x| 1/2 ≤ K n/2-1 ( √ ν|x|/ √ µ) ≤ C e - √ ν|x|/ √ µ |x| 1/2 , |x| ≥ R.
Substituting if necessary R by max(R, 1), we have

(A.5) 1 |x| n/2-1 ≤ 1 |x| 1/2 , |x| ≥ R. Moreover, we have e - √ ν|x|/ √ µ |x| 1/2 = |x| (n-3)/2 e - √ ν|x|/(2 √ µ) e - √ ν|x|/(2 √ µ) |x| n/2-1 , |x| ≥ R. Since the function x → |x| (n-3)/2 e - √ ν|x|/(2 √ µ) is bounded in R n , we deduce (A.6) e - √ ν|x|/ √ µ |x| 1/2 ≤ C e - √ ν|x|/(2 √ µ) |x| n/2-1 , |x| ≥ R.
Using (A.5) and (A.6) in (A.4) in order to obtain

(A.7) C -1 e - √ ν|x|/ √ µ |x| n/2-1 ≤ K n/2-1 ( √ ν|x|/ √ µ) ≤ C e - √ ν|x|/(2 √ µ) |x| n/2-1 , |x| ≥ R.
We now establish a similar estimate when |x| → 0. To this end, we recall that according to formula [5, 9.6.9, page 375] we have

K n/2-1 (ρ) ∼ 1 2 Γ(n/2 -1) 2 ρ n/2-1
as ρ → 0, from which we deduce in a straightforward manner that there exists 0 < r ≤ R so that

(A.8) C -1 e - √ ν|x|/ √ µ |x| n/2-1 ≤ K n/2-1 ( √ ν|x|/ √ µ) ≤ C e - √ ν|x|/(2 √ ν) |x| n/2-1 , |x| ≤ r.
The expected two sided inequality (2.10) follows by combining (A.4), (A.7) and (A.8).

Proof of Lemma

2.3. Let Q be an open subset of R n , set d = diam(Q), d x = dist(x, ∂Q) and d x,y = min(d x , d y ).
We introduce the following weighted Hölder semi-norms and Hölder norms, where σ ∈ R, 0 < γ ≤ 1, and k is non-negative integer,

[w] (σ) k,0;Q = [w] (σ) k,Q = sup x∈Q, |β|=k d k+σ x |∂ β w(x)|, [w] (σ) k,γ;Q = sup x,y∈Q, |β|=k d k+γ+σ x,y |∂ β w(y) -∂ β w(x)| |y -x| γ , |w| (σ) k;Q = k j=0 [w] (σ) j;Q , |w| (σ) k,γ;Q = |w| (σ) k;Q + [w] (σ)
k,γ;Q . In term of these notations, we have

|a| (0) 0,α;Q = sup x∈Q |a(x)| + sup x,y∈Q d α x,y |a(y) -a(x)| |y -x| α ≤ (1 + d)λ, |∂ j a| (1) 0,α;Q = sup x∈Q d x |∂ j a(x)| + sup x,y∈O d 1+α x,y |∂ j a(y) -∂ j a(x)| |y -x| α ≤ (d + d 2 )λ, |b| (2) 0,α;Q = sup x∈O d 2 x |b(x)| + sup x,y∈Q d 2+α x,y |b(y) -b(x)| |y -x| α ≤ (d 2 + d 3 )λ.
In consequence

(A.9) |a| (0) 0,α;Q + |∂ j a| (1) 0,α;Q + |b| (2) 0,α;Q ≤ Λ(d) = [1 + 2d + 2d 2 + d 3 ]λ.
Following [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] we define also

[w] * k,0;Q = [w] * k,O = sup x∈Q, |β|=k d k x |∂ β w(x)|, [w] * k,γ;Q = sup x,y∈Q, |β|=k d k+α x,y |∂ β w(y) -∂ β w(x)| |y -x| γ , |w| * k;Q = k j=0 [w] * j;Q , |w| * k,γ;Q = |w| * k;Q + [w] * k,γ;O .
From [17, Lemma 6.32, page 130] and its proof we have the following interpolation inequalities: suppose that j and k, non negative integers, and 0 ≤ β, γ ≤ 1 are so that j + β < k + γ. Then there exist C = C(n, α, β) > 0 and ϑ = ϑ(α, β) so that, for any w ∈ C k,α (Q) and > 0, we have Checking carefully the proof of interior Schauder estimates in [17, Theorem 6.2, page 90], we get, taking into account inequalities (A.9)-(A.11), the following result: there exist a constant C = C(n) > 0 and τ = τ (α) so that, for any 0 < µ ≤ 1/2 and w ∈ C k,α (Q) satisfying L a,b w = 0 in Q, we have (A.12)

[w] * 2,α,Q ≤ CΛ(d) µ -τ |w| 0;Q + µ α [w] * 2,α,Q .

Substituting in (A.12) C by max(C, 2 α-1 ), we may assume in (A.12) that C = C(n, α) ≥ 2 α-1 .

Bearing in mind that Λ(d) > 1, we can take in (A.12), µ = (2CΛ(d)) -1/α . We find

1 f 2 (y) - 1 f 2 (x) = 1 f (x)f 2 (y) + 1 f (x) 2 f (y) (f (x) -f (y)), 1 f 3 (y) - 1 f 3 (x) = 1 f (x)f 3 (y) + 1 f 2 (x)f 2 (y) + 1 f (x) 3 f (y) (f (x) -f (y)),
we easily get (A.16) [1/f j ] α;K ≤ 3c 4 + [f ] α;K , j = 2, 3. Also, we have

∂ i f (y)∂ j f (x) f 3 (y) - ∂ i f (y)∂ j f (x) f 3 (x) = ∂ i f (y) f 3 (y) (∂ j f (y) -∂ j f (x)) + ∂ j f (x) f 3 (y) (∂ i f (y) -∂ i f (x)) + 1 f 3 (y) - 1 f 3 (x) (∂ i f (y)∂ j f (x)).
In light of (A.16), this identity yields

[∂ i f ∂ j f /f 3 ] α;K ≤ c 4 + ([∂ i f ] α;K |∂ j f | 0;K (A.17) +[∂ j f ] α;K |∂ i f | 0;K + [f ] α;K |∂ i f | 0;K |∂ j f | 0;K ) .
On the other hand, since

∂ 2 ij f (y) f 2 (y) - ∂ 2 ij f (x) f 2 (x) = 1 f 2 (y) (∂ 2 ij f (y) -∂ 2 ij f (x)) + 1 f 2 (y) - 1 f 2 (y) ∂ 2 ij f (x),
we find, by using again (A. Recall that 0 < θ < α < 1. for some constant C = C(n, α -θ) > 0. This is the expected inequality

  belongs to H 2 (R n ) and it is the unique solution of L a,b u = f . We deal in the present work with the problem of reconstructing (a, b) ∈ D(λ, κ) from energies generated by two point sources located at ξ 1 and ξ 2 . Precisely, if u j (a, b) = G a,b (•, ξ j ), j = 1, 2, we want to determine (a, b) from the internal measurements v j (a, b) = bu j (a, b) in Ω, j = 1, 2.

  y,mr)) . A proof of this theorem can be found in[9] or[START_REF] Choulli | Boundary value problems for elliptic partial differential equations[END_REF]. Define the geometric distance d D g on the bounded domainD of R n by d D g (x, y) = inf { (ψ); ψ : [0, 1] → D Lipschitz path joining x to y} , where (ψ) = ˆ1 0 | ψ(t)|dt is the length of ψ. Note that according to Rademacher's theorem any Lipschitz continuous function ψ : [0, 1] → D is almost everywhere differentiable with | ψ(t)| ≤ k a.e. t ∈ [0, 1], where k is the Lipschitz constant of ψ. Lemma 3.1. Let D be a bounded Lipschitz domain of R n . Then d D g ∈ L ∞ (D × D) and there exists a constant c D > 0 so that |x -y| ≤ d D g (x, y) ≤ c D |x -y|, x, y ∈ D. (3.2) We refer to [23, Lemma A3] for a proof. In this subsection we use the following notations

1 |B 1 Corollary 3 . 1 .

 1131 (x,r)| ´B(x,r) w(x)dx. Noting that S (O, x 0 , M, η, δ/3) is invariant under the transformation u → u -{u}, we can state the following consequence of Proposition 3.With the notations of Proposition 3.1, if u ∈ S (O, x 0 , M, η, δ/3) then

  div(|ζ|∇w) = ∇|ζ| • ∇w + |ζ|∆w = sgn 0 (ζ)∇ζ • ∇w + sgn 0 (ζ)ζ∆w = sgn 0 (ζ)div(ζ∇w) = sgn 0 (ζ)div(σ∇u),we get by integrating by parts ˆΩ |ζ||∇w| 2 dx = -ˆΩ div(|ζ|∇w)wdx + ˆΓ |ζ|w∂ ν wdS(x) (4.8) = -ˆΩ sgn 0 (ζ)div(σ∇u)wdx + ˆΓ |ζ|w∂ ν wdS(x).

  [w] * j,β;Q ≤ C -ϑ |w| 0;Q + [w] * k,γ;Q , (A.10) |w| * j,β;Q ≤ C -ϑ |w| 0;Q + [w] * k,γ;Q . (A.11) Here |w| 0;Q = sup x∈Q |w(x)|.

  ,Q ≤ CΛ(d) κ |w| 0;Q , for some constants C = C(n, α) > 0 and κ = κ(α) > 1.Using again interpolation inequalities (A.10) and (A.11), we deduce that(A.14) |w| * 2,α,Q ≤ CΛ(d) κ |w| 0;Q . Let δ > 0 be so that Q δ = {x ∈ Q; dist(x, ∂Q) > δ} is nonempty. If Q is an open subset of Q δ then (A.14) yields in a straightforward manner w C 2,α (Q ) ≤ C max δ -(2+α) , 1 Λ(d) κ |w| 0;Q .This is the expected inequality.Lemma A.1. Let K be a compact subset of R n and f ∈ C 2,α (K) satisfying min K |f | ≥ c -> 0. Then (A.15) 1/f C 2,α (K) ≤ Cc 4 + 1 + f C 2,α (K) 3,where c + = max(1, c -1 -) and C = C(diam(K)) is a constant. Proof. Let x, y ∈ K. Using |1/f | 0;K ≤ c + and the following identities

Lemma A. 2 .

 2 C 2,α (O) is continuously embedded in H 2+θ (O). Furthermore, there exists C = C(n, α -θ) so that, for any w ∈ C 2,α (O), we have(A.20) w H 2+θ (O) ≤ C max d n/2 , d n/2+α-θ w C 2,α (O),where d = diam(O). Proof. Let w ∈ C 2,α (O) and, for fixed 1 ≤ i, j ≤ n, set g = ∂ 2 ij w. Then ˆO ˆO |g(x) -g(y)| 2 |x -y| n+2θ dxdy ≤ [g] 2 α;O ˆO ˆO 1 |x -y| n-2(α-θ) dxdy.In light of [10, Lemma A3, page 246], this inequality yieldsˆO ˆO |g(x) -g(y)| 2 |x -y| n+2θ dxdy ≤ |S n-1 ||O|d 2(α-θ) 2(α -θ) [g] 2 α;O , But |O| ≤ |B(0, d)|. Hence (A.21) ˆO ˆO |g(x) -g(y)| 2 |x -y| n+2θ dxdy ≤ |S n-1 | 2 d n+2(α-θ) 2(α -θ) [g] 2 α;O .Using (A.21) and the inequalityh 2 L 2 (O) ≤ |S n-1 |d n |h| 0,O , h ∈ C(O), we get from the definition of the norm of H s -spaces in [18, formula (1.3.2.2), page 17] w H 2+θ (O) ≤ C max d n/2 , d n/2+α-θ w C 2,α (O) ,

  is well known that A a,b generates an analytic semigroup e tA a,b . Therefore in light of [6, Theorem 4 on page 30, Theorem 18 on page 44 and the proof in the beginning of Section 1.4.2 on page 35] k a,b (t, x; ξ), the Schwarz kernel of e tA a,b , is Hölder continuous with respect to x and ξ, satisfies

	(2.6)		|k a,b (t, x, ξ)| ≤ e -δt E c (x -ξ, t)
	and, for |h| ≤	√	t + |x -ξ|,

  we get, similarly to the end of the proof of Corollary 2.3, from[START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] Lemma 6.35, page 135]

  ] α;K + [f ] α;K |∂ 2 ij f | 0,K . Inequalities (A.17), (A.18), the identity ∂ 2 ij (1/f ) = 2∂ i f ∂ j f /f 3 -∂ 2 ij f /f 2 andthe interpolation inequality [17, Lemma 6.35, page 135] (by proceeding as in Corollary 2.2) imply (A.19) [∂ 2 ij (1/f )] α,K ≤ Cc 4 + 1 + f C 2,α (K) 3 , with C = C(diam(K)) is a constant.The other terms for 1/f appearing in the norms • C 2,α (K) can be estimated similarly to the semi-norm in (A.[START_REF] Kukavica | Quantitative uniqueness for second-order elliptic operators[END_REF]). Inequality (A.15) then follows.

		16),
	(A.18)	[∂ 2 ij f /f 2 ] α;K ≤ 3c 4 + [∂ 2 ij f

Introduction

Throughout this text n ≥ 3 is a fixed integer. If 0 < β ≤ 1 we denote by C 0,β (R n ) the vector space of bounded continuous functions

) is then a Banach space when it is endowed with its natural norm

The authors were supported by the grant ANR-17-CE40-0029 of the French National Research Agency ANR (project MultiOnde).