aa r X i v : . [ m a t h . A P ] D ec SOME STABILITY INEQUALITIES FOR HYBRID INVERSEPROBLEMS
MOURAD CHOULLI
Abstract.
We study some hybrid inverse problems associated to BVP’s forSchrödinger and Helmholtz type equations. The inverse problems we considerconsist in the determination of coefficients from the knowledge of internal en-ergies. We establish local Lipschitz stability inequalities as well as conditionalHölder stability inequalities. Introduction
Coupled-physics or hybrid inverse problems have attracted many researchersfrom the inverse problem community during the last decades.Among results concerning these hybrid inverse problems we quote those estab-lished by Bal and Uhlmann [7] in the isotropic case. Precisely, they considered thequantitative photoacoustic tomography. They showed that there exists an opensubset of illuminations for which we have Lipschitz stability for the determinationof two medium parameters from 2 n boundary measurements in a n -dimensionalspace. They also proved that two measurements are sufficient provided that thedomain under consideration satisfies an extra geometric condition. Using a differentmethod, Alessandrini, Di Cristo, Francini and Vessella [3] proved a Hölder stabilityestimate in the case of two well chosen illuminations. Recently, the author, Bon-netier and Triki [8] established also a Hölder stability estimate in the case of twoarbitrary pointwise sources generating two illuminations. This situation is moresuitable for real physical problems. The determination of the absorption coefficientfrom a single measurement was already studied by the author and Triki [10, 11].For this problem we got a weighted Hölder stability estimate.We just quote few results for the quantitative photoacoustic tomography. Werefer to [1] and references therein for a complete overview concerning recent progressdealing with hybrid inverse problems for both isotropic and anisotropic cases.We discuss in the present work the stability issue for various kind of hybridinverse problems that lead to the same BVP. For possible applications we list belowfour examples.1.1. Quantitative photoacoustic tomography.
It is a hybrid imaging modal-ity where high frequency electromagnetic waves are combined with ultrasounds.Precisely, the medium is illuminated by high frequency electromagnetic wave (e.g.
Mathematics Subject Classification.
Key words and phrases. quantitative photoacoustic tomography, quantitative dynamic elas-tography, microwave imaging by elastic deformation, acousto-optic imaging, Lipschitz stabilityestimate, Hölder stability estimate.The author is supported by the grant ANR-17-CE40-0029 of the French National ResearchAgency ANR (project MultiOnde). laser). A part of the electromagnetic radiation is then absorbed by the tissues andtherefore transformed into heat. The increase of temperature produces an expansionof the medium and in consequence creates acoustic waves. From the mathematicalpoint of view, the first step consists in determining the absorbed electromagneticenergy H from boundary measurements. It turns out that H is the initial con-dition in an acoustic wave equation. This is typically a control problem which isalready solved and many results can be found in the literature devoted to controltheory. Once we recover H , the objective is then the determination of the diffusioncoefficient or the diffusion matrix a and the absorption coefficient q from the energy H ( x ) = G ( x ) q ( x ) u ( x ) , x ∈ Ω . where Ω is the domain occupied by the medium, G is the Grüneisen parameter and u is the light intensity. In the diffusive regime, it is shown that u is the solution ofthe BVP(1.1) (cid:26) − div( a ∇ u ) + q u = 0 in Ω ,u | Γ = f. Here Γ is the boundary of Ω, a is the diffusion coefficient in the isotropic case orthe diffusion matrix in the anisotropic case and f represents the illumination.We assume in the present work that the parameter G is known. In that case itis usual to take G identically equal to 1. This is what we assume in the sequel.In fact, several energies H may be necessary to recover a and q . Each energycorresponds to a different illumination.1.2. Quantitative dynamic elastography.
In the quantitative dynamic elastog-raphy we want to recover the tissue parameters from the tissue displacement. Inthe simplified elastic scalar model u , one component of the displacement, is thesolution of the BVP (1.1) in which a is the shear modulus and q = − ρk , where ρ is the density and k is the frequency. We assume that k > u corresponding todifferent values of the boundary data f .1.3. Microwave imaging by elastic deformation.
The construction of the con-ductivity in the context of electrical impedance tomography from boundary mea-surements is very known to be severelly ill-conditioned. It is shown in [4] thatthe boundary measurements with simultaneous localized ultrasonic perturbationsallow the recovery of the conductivity with good resolution. In that case the elec-trical impedance tomography is substituted by the problem of reconstructing theconductivity (and permittivity) from internal electrical energies. In the microwaveregime, this problem leads again to the BVP (1.1), where a is the conductivityand q = − k p , k > p is the permittivity. The internaldata is given by various electrical energies of the form H = p u or H = a |∇ u | ,corresponding to several boundary data f .In the preceding two examples, we assume that the frequency k is known. Wetake for simplicity k = 1.1.4. Acousto-optic imaging.
When a medium is exited by an acoustic radiationthen its optical properties are modified. It is known that in this case the scatteredfield carries informations about the medium. This principle was the basis for thedevelopment of a hybrid imaging modality, known as acousto-optic imaging. In
YBRID INVERSE PROBLEMS 3 the simplified model, the electromagnetic energy density u solves the BVP (1.1) inwhich a is the diffusion coefficient and q is the absorption coefficient (e.g. [6]). Themeasured internal energy for the actual inverse problem is given by H = q u .1.5. Main notations and definitions.
In the rest of this text we use the followingnotations. The norm of a Banach space E is denoted by k·k E . The ball, of a Banachspace E , of center x ∈ E and radius r > B E ( x , r ). If E and F are two Banach spaces, the norm of B ( E, F ), the Banach space of linear boundedoperators between E and F , will denoted by k · k op .Unless otherwise specified Ω is a C , bounded domain of R n ( n ≥
2) withboundary Γ. We denote by λ (Ω) the first eigenvalue of the Laplace operator on Ωwith Dirichlet boundary condition.For µ ≥ ≤ λ , define A µ = (cid:8) a = ( a kℓ ) ∈ C , (cid:0) Ω , R n × n (cid:1) ; a is symmetric and µ − | ξ | ≤ ( a ξ | ξ ) ≤ µ | ξ | for all ξ ∈ R n (cid:9) , A sµ = { a ∈ C , (Ω); a I n ∈ A µ } , Q λ = { q ∈ L ∞ (Ω); q ≥ − λ } , Q − λ = { q ∈ L ∞ (Ω); 0 ≥ q ≥ − λ } . Here ( ·|· ) is the Euclidian scalar product on R n and I n is the n × n identity matrix.Let J = { ( µ, λ ) ∈ [1 , ∞ ) × [0 , ∞ ); µλ < λ (Ω) } . Define, for all ( µ, λ ) ∈ J , D µ,λ = A µ × Q λ , D • µ,λ = A sµ × Q − λ and D = [ ( µ,λ ) ∈J D µ,λ . Local Lipschitz stability.
Let r ≥
2. We prove in Corollary 2.1 (Subsection2.1) that, for any ( a , q ) ∈ D and f ∈ W − /r,r (Γ), the BVP (1.1) has a uniquesolution u a , q ∈ W ,r (Ω). Theorem 1.1.
Suppose that, for some < θ < , Ω is of class C ,θ , a ∈ C ,θ (Ω , R n × n ) ∩ A µ , for some µ ≥ , and f ∈ C ,θ (Γ) satisfies f > . Let j = 1 or j = 2 . Then there exists U , a neighborhood of in L ∞ (Ω) , and a constant C = C ( n, Ω , µ, f, r ) so that, for any q , ˜ q ∈ U , we have (1.2) k q − ˜ q k L ∞ (Ω) ≤ C k q u j q − ˜ q u j ˜ q k L ∞ (Ω) , where u q = u a , q and u ˜ q = u a , ˜ q . In this theorem the case q ≥
0, ˜ q ≥ j = 1 corresponds to a stabilityinequality for the problem that consists in the determination of the absorptioncoefficient q in the quantitative photoacoustic tomography, from the energy H = q u q . The case q ≥
0, ˜ q ≥ j = 2 gives a stability inequality for the problemof determining the absorption coefficient in the acousto-optic imaging, from theenergy H = q u q . While the case q ≤
0, ˜ q ≤ j = 2 contains a stabilityresult for the determination of the permittivity in the microwave imaging, from theenergy H = q u q .Next, we state a local Lipschitz stability estimate that applies to the problemof determining the absorption coefficient in the acousto-optic imaging from the MOURAD CHOULLI knowledge of an internal data. Prior to doing that, we introduce a definition. Fix0 < θ < µ ≥
1, 0 < q < q < µ − λ (Ω) and define Q = { q ∈ C ,θ (Ω); q ≤ q ≤ q } . Theorem 1.2.
Assume that Ω is of class C ,θ , f ∈ C ,θ (Γ) satisfies f > on Γ . Let a ∈ C ,θ (Ω , R n × n ) ∩ A µ . For all q , ˜ q ∈ Q , we have, with u q = u a , q and u ˜ q = u a , ˜ q , k q − ˜ q k L (Ω) ≤ C k q u q − ˜ q u q k L (Ω) , where C = C ( n, Ω , µ, q , q , min f ) . Note that according to the usual Hölder regularity (e.g. [14, Theorem 6.14, page107]) u q and u ˜ q belong to C ,θ (Ω).1.7. Conditional Hölder stability.
Pick two constants 0 < q − ≤ q + < λ (Ω)and set Q = { q ∈ L ∞ (Ω); q − ≤ − q ≤ q + } . Fix ̺ > Q ̺ = { q ∈ Q ∩ C ,α (Ω); k q k C ,α (Ω) ≤ ̺ } 6 = ∅ . Let q ∈ Q and r ≥
2. Noting that ( I n , q ) ∈ D , we deduce from Corollary 2.1(Subsection 2.1) that the BVP (1.1), in which we take a = I n , has unique solution u q ∈ W ,r (Ω). Theorem 1.3.
Suppose that r = 2 if n = 2 , and n/ < r < n if n ≥ , and that f > on Γ . Then, for any q , ˜ q ∈ Q ̺ , we have (1.3) k q − ˜ q k L r (Ω) ≤ C̺ − γ k u q − u ˜ q k γW ,r (Ω) , where C = C ( n, Ω , r, q − , q + , f ) > and < γ = γ ( n, Ω , r, q − , q + , f ) < areconstants. Theorem 1.4.
Assume that r > n and f ∈ W − /r,r (Γ) is non constant. Let ω ⋐ Ω , ( λ, µ ) ∈ J and ̺ > . Then, for any ( a , q ) , (˜ a , q ) ∈ D • λ,µ so that k a k C , (Ω) ≤ ̺ , k ˜ a k C , (Ω) ≤ ̺ and a = ˜ a on Γ , we have, with u a = u a , q and u ˜ a = u ˜ a , q , k a − ˜ a k C ( ω ) ≤ C̺ − γ k u a − u ˜ a k γL (Ω) , where C = C ( n, Ω , ω, λ, µ, f ) and < γ = γ ( n, Ω , ω, λ, µ, f ) are constants. The preceding two theorems can be used to obtain stability inequalities forthe quantitative dynamic elastography (at least for the simplified model describedabove). Theorem 1.3 can be interpreted as conditional stability estimate of theproblem of recovering the density from the knowledge of tissue displacement, as-suming that the shear modulus is known and it is identically equal to 1. WhileTheorem 1.4 gives an interior Hölder stability estimate of recovering the shearmodulus when the density is supposed to be known. Here again the internal dataconsists in the tissue displacement.Fix 0 < β < < κ ≤ Λ, define then D κ , Λ as the set of couples ( a , q )satisfying a ∈ C ,β (Ω), q ∈ C ,β (Ω) and q ≥ , a ≥ κ and k a k C ,β (Ω) + k q k C ,β (Ω) ≤ Λ . YBRID INVERSE PROBLEMS 5
Suppose that f ∈ C ,β (Γ). Then, in light of [14, Theorem 6.6, page 98 andTheorem 6.14, page 107], the BVP (1.1) admits a unique solution u a , q ( f ) ∈ C ,β (Ω)so that(1.4) k u a , q ( f ) k C ,β (Ω) ≤ K, for all ( a , q ) ∈ D κ , Λ , where K = K ( n, Ω , β, κ , Λ , f ) is a constant. Theorem 1.5.
Let f , f ∈ C ,β (Γ) with f > . Assume that Ω is of class C ,β , h = f /f is non constant and the set of critical points of h consists of itsextrema. For all ( a , q ) , (˜ a , ˜ q ) ∈ D κ , Λ satisfying ( a , q ) = (˜ a , ˜ q ) on Γ , we have, with u j = u a , q ( f j ) and ˜ u j = u ˜ a , ˜ q ( f j ) , j = 1 , , k a − ˜ a k C ,β (Ω) + k q − ˜ q k C ,β (Ω) ≤ (cid:16) k u − ˜ u k C (Ω) + k u − ˜ u k C (Ω) (cid:17) γ , where C = C ( n, Ω , β, κ , Λ , f , f ) > and < γ = γ ( n, Ω , β, κ , Λ , f , f ) < areconstants. In the case of dimension two functions called almost two-to-one, as it is definedin [15], have no other critical points than their extrema. A larger class of functionsadmitting such property consists in quantitatively unimodal functions. (e.g. [2] fora precise definition).Theorem 1.5 establishes conditional Hölder stability estimate of the quantitativephotoacoustic tomopography consisting in determining simultaneously the diffusionand the absorption coefficients from two internal energies, corresponding to twowell-chosen illuminations. This theorem was already established in [8] when thetwo illuminations are generated from two point sources located outside the medium.We point out that a similar result was obtained in [3] under the assumptions that h is quantitatively unimodal and Ω is diffeomorphic to the unit ball.The rest of this text is devoted to the proof of the results stated in this intro-duction. The local Lipschitz stability inequalities are proved in Section 2. Whilethe proofs of the conditional Hölder stability inequalities are given in Section 3.2. Local Lipschitz stability inequalities
Solvability of the BVP (1.1) in W ,r (Ω) . It is is contained in the followingtheorem. Henceforward κ will denote a generic universal constant. Theorem 2.1.
Let r ≥ .(i) For any ( a , q ) ∈ D , the linear mapping P a , q : u ∈ W ,r (Ω) ( − div( a ∇ u ) + q u, u | Γ ) ∈ L r (Ω) × W − /r,r (Γ) is an isomorphism.(ii) Let ( a , q ) ∈ D . Then there exists a constant δ = δ ( a , q ) > so that, forany q ∈ B L ∞ (Ω) (0 , δ ) , P a , q + q : u ∈ W ,r (Ω) ( − div( a ∇ u ) + q u, u | Γ ) ∈ L r (Ω) × W − /r,r (Γ) is an isomorphism with kP − a , q + q k op ≤ kP − a , q k op , for all q ∈ B L ∞ (Ω) (0 , δ ) . Proof. (i) Follows by modifying slightly the proof of [9, Theorem 4.2].(ii) Pick (
F, f ) ∈ L r (Ω) × W − /r,r (Γ) and define the mapping T as follows T : W ,r (Ω) → W ,r (Ω) : u
7→ P − a , q ( − q u + F, f ) . MOURAD CHOULLI
Clearly, we have T ( u ) − T ( u ) = P − a , q ( − q ( u − u ) , . Hence k T ( u ) − T ( u ) k W ,r (Ω) ≤ kP − a , q k op k q k L ∞ (Ω) k u − u k W ,r (Ω) . Let δ = 1 / (cid:2) kP − a , q k op (cid:3) . Whence, if q satisfies k q k L ∞ (Ω) < δ then(2.1) k T ( u ) − T ( u ) k W ,r (Ω) ≤ (1 / k u − u k W ,r (Ω) . According to Banach’s fixed point theorem, T admits a unique fixed point u ∗ ∈ W ,r (Ω). In other words, we proved that there exists a unique u ∗ ∈ W ,r (Ω)satisfying P a , q + q u ∗ = ( F, f ). Furthermore, we get in light of (2.1) kP − a , q + q ( F, f ) k W ,r (Ω) = k u ∗ k W ,r (Ω) ≤ k T ( u ∗ ) − T (0) k W ,r (Ω) + k T (0) k W ,r (Ω) ≤ (1 / k u ∗ k W ,r (Ω) + kP − a , q k op k ( F, f ) k L r (Ω) × W − /r,r (Γ) ≤ (1 / kP − a , q + q ( F, f ) k W ,r (Ω) + kP − a , q k op k ( F, f ) k L r (Ω) × W − /r,r (Γ) and then kP − a , q + q k op ≤ kP − a , q k op . The proof is then complete. (cid:3)
Corollary 2.1.
For any ( a , q ) ∈ D and f ∈ W − /r,r (Γ) , the BVP (1.1) has aunique solution u = P − a , q (0 , f ) . Differentiability properties.
Fix ( a , q ) ∈ D and f ∈ W − /r,r (Γ) nonidentically equal to zero. Let δ = δ ( a , q ) be as in Theorem 2.1 (ii). For notationalconvenience we use in the sequel the notations S q = P − a , q + q , for each q ∈ B L ∞ (Ω) (0 , δ )and ̟ = 2 kP − a , q k op . That is we have, according to Theorem 2.1 (ii),(2.2) k S q k op ≤ ̟, for each q ∈ B L ∞ (Ω) (0 , δ ) . Define Ψ : B L ∞ (Ω) (0 , δ ) → W ,r (Ω) : q S q (0 , f ) . We claim that the mapping Ψ is Lipschitz continuous. Indeed, for q , q ∈ B L ∞ (Ω) (0 , δ ), we have S q (0 , f ) − S q (0 , f ) = S q ( F, , with F = ( q − q ) S q (0 , f ) . We find by applying twice inequality (2.2) k S q (0 , f ) − S q (0 , f ) k W ,r (Ω) ≤ ̟ k f k W − /r,r (Γ) k q − q k L ∞ (Ω) . That is we have(2.3) k Ψ( q ) − Ψ( q ) k W ,r (Ω) ≤ c k q − q k L ∞ (Ω) , where c = ̟ k f k W − /r,r (Γ) .Let q ∈ B L ∞ (Ω) (0 , δ ) and consider the linear map L q : p ∈ L ∞ (Ω) S q ( − p Ψ( q ) , ∈ W ,r (Ω) . YBRID INVERSE PROBLEMS 7
In light of (2.2) we have k L q ( p ) k W ,r (Ω) ≤ c k p k L ∞ (Ω) , implying that L q is bounded.Next, let p ∈ L ∞ (Ω) so that p + q ∈ B L ∞ (Ω) (0 , δ ). Then it is not difficult tocheck that Ψ( q + p ) − Ψ( q ) − L q ( p ) = S q ( − p [Ψ( q + p ) − Ψ( q )] , . We get by applying inequality (2.2) and then inequality (2.3) k Ψ( q + p ) − Ψ( q ) − L q ( p ) k W ,r (Ω) ≤ c κ k p k L ∞ (Ω) . This shows that Ψ is Fréchet differentiable at q . The differential of Ψ at q , denotedby Ψ ′ ( q ), is then given by(2.4) Ψ ′ ( q )( p ) = S q ( − p Ψ( q ) , , for all p ∈ L ∞ (Ω) . Let us now prove thatΨ ′ : B L ∞ (Ω) (0 , δ ) → B ( L ∞ (Ω) , W ,r (Ω))is continuous. To this end, let q ∈ B L ∞ (Ω) (0 , δ ) and ˆ q ∈ L ∞ (Ω) so that q + ˆ q ∈ B L ∞ (Ω) (0 , δ ). We get in light of formula (2.4), where p ∈ L ∞ (Ω),Ψ ′ ( q + ˆ q )( p ) − Ψ ′ ( q )( p ) = S q +ˆ q ( − p Ψ( q + ˆ q ) , − S q ( − p Ψ( q ) , S q ( − p [Ψ( q + ˆ q ) − Ψ( q )] ,
0) + S q ( − ˆ q Ψ( q + ˆ q ) , . We can proceed as before to derive, with the aid of inequalities (2.2) and (2.3),the following estimate k Ψ ′ ( q + ˆ q ) − Ψ ′ ( q ) k op ≤ C k ˆ q k L ∞ (Ω) , where C > q . This shows that Ψ ′ is continuousat q . In other words, we proved that Ψ in continuously Fréchet differentiable in B L ∞ (Ω) (0 , δ ).2.3. Proof of Theorem 1.1.
We give the proof for j = 1. That for j = 2 is quitesimilar.According to C ,θ -Hölder regularity, we get S (0 , f ) ∈ C ,θ (Ω). Furthermore,in light of the strong maximum principle (e.g. [14, Theorem 3.5, page 35]), we have S (0 , f ) > min Γ f in Ω. That is we have Ψ(0) > q = 0 and introduce the mappingΦ : B L ∞ (Ω) (0 , δ ) → L ∞ (Ω) : q Φ( q ) = q Ψ( q ) . Since Ψ is continuously Fréchet differentiable then so is Φ. We have in additionΦ ′ ( q )( p ) = p Ψ( q ) + q Ψ ′ ( q )( p ) , for all p ∈ L ∞ (Ω) . In particular Φ ′ (0)( p ) = p Ψ(0) , for all p ∈ L ∞ (Ω) . We define the linear map ℓ : L ∞ (Ω) → L ∞ (Ω) by ℓ ( h ) = [Ψ(0)] − h , h ∈ L ∞ (Ω) . Clearly, ℓ is bounded with k ℓ k op ≤ k [Ψ(0)] − k L ∞ (Ω) . MOURAD CHOULLI
We can check that ℓ is the inverse of Φ ′ (0). Therefore, according to the inversefunction theorem Φ is a diffeomophism from a neighborhood U of 0 in L ∞ (Ω) ontoa neighborhood V of 0 in L ∞ (Ω). Whence the expected inequality follows.2.4. Proof of Theorem 1.2.
The proof is inspired by that of [5, Theorem 3.1].Pick q , ˜ q ∈ Q , and set u = u q , ˜ u = u ˜ q , v = q u and ˜ v = ˜ q ˜ v .Let m = min Γ f . In light of [14, Corollary 3.2, page 33] we have(2.5) u ≥ m, ˜ u ≥ m on Ω . On the other hand straightforward computations givediv( a ∇ ( u − ˜ u )) + p q ˜ q ( u − ˜ u ) = ( √ q + p ˜ q )( √ v − √ ˜ v ) . Taking into account that u − ˜ u ∈ H (Ω), we find by applying Green’s formula ˆ Ω a ∇ ( u − ˜ u ) · ∇ ( u − ˜ u ) dx − ˆ Ω p q ˜ q ( u − ˜ u ) dx (2.6) = ˆ Ω ( √ q + p ˜ q )( √ ˜ v − √ v )( u − ˜ u ) dx. But(2.7) ˆ Ω p q ˜ q ( u − ˜ u ) dx ≤ q ˆ Ω ( u − ˜ u ) dx, and, using that a ∈ A µ and Poincaré’s inequality, we find(2.8) ˆ Ω a ∇ ( u − ˜ u ) · ∇ ( u − ˜ u ) dx ≥ µ − λ (Ω) ˆ Ω ( u − ˜ u ) dx. Therefore (2.7) and (2.8) in (2.6) yield( µ − λ (Ω) − q ) ˆ Ω ( u − ˜ u ) dx ≤ ˆ Ω ( √ q + p ˜ q )( √ ˜ v − √ v )( u − ˜ u ) dx, which, combined with Cauchy-Schwarz’s inequality, entails(2.9) k u − ˜ u k L (Ω) ≤ √ q µλ (Ω) − µ q k√ ˜ v − √ v k L (Ω) . Also, elementary calculations enable us to establish the following identity p ˜ q − √ q = √ q ˜ q √ v ( u − ˜ u ) + √ q √ v ( √ ˜ v − √ v ) . Whence(2.10) k p ˜ q − √ q k L (Ω) ≤ q √ q m k u − ˜ u k L (Ω) + √ q √ q m k√ ˜ v − √ v k L (Ω) . Let C = 2 q √ q µ √ q m ( λ (Ω) − µ q ) + √ q √ q m . Then (2.9) together with (2.10) imply k p ˜ q − √ q k L (Ω) ≤ C k√ ˜ v − √ v k L (Ω) . YBRID INVERSE PROBLEMS 9
To complete the proof it is sufficient to use the following inequalities k ˜ q − q k L (Ω) = k ( p ˜ q + √ q )( p ˜ q − √ q ) k L (Ω) ≤ p q k p ˜ q − √ q k L (Ω) , k√ ˜ v − √ v k L (Ω) = (cid:13)(cid:13)(cid:13)(cid:13) ˜ v − v √ ˜ v + √ v (cid:13)(cid:13)(cid:13)(cid:13) L (Ω) ≤ √ q m k ˜ v − v k L (Ω) . Remark 2.1.
Let us observe that the preceding proof can be adapted to obtain auniqueness result. Let q , ˜ q ∈ Q so that q u q = ˜ q u q and 0 is not an eigenvalue of theoperator A = − div( a ∇· ) + √ q ˜ q with domain D ( A ) = H (Ω) ∩ H (Ω). Thendiv( a ∇ ( u q − u ˜ q )) + p q ˜ q ( u q − u ˜ q ) = 0and, since u q − u ˜ q ∈ D ( A ), we derive that u q = u ˜ q , from which we get in astraightforward manner that q = ˜ q .3. Conditional Hölder Stability inequalities
Proof of Theorem 1.3.
Let 0 < γ ≤
1. We say
W ⊂ L (Ω) = { w ∈ L ∞ (Ω); w ≥ } is a uniform set of weights for the interpolation inequality(3.1) k f k L ∞ (Ω) ≤ C k f k − µC ,γ (Ω) k f w k µL (Ω) , if the constants C > < µ < w ∈ W and f ∈ C ,γ (Ω).We note that our choice of r guarantees that W ,r (Ω) is continuously embeddedin C (Ω). Let then ˜ e denotes the norm of this embedding.Fix 0 < m ≤ ˜ e M and set S = { u ∈ W ,r (Ω); − ∆ u + q u = 0 in Ω , for some q ∈ Q , k u k W ,r (Ω) ≤ M and | u | Γ | ≥ m } . Minors modifications in the proof of [11, Theorem 2.2] yield the following result.
Theorem 3.1. W = { w = u ; u ∈ S } is a uniform set of weights for theweighted interpolation inequality (3.1) , with C = C ( n, Ω , r, q − , q + , m, M ) and µ = µ ( n, Ω , r, q − , q + , m, M ) . We know that, according to [9, Theorem 4.2], there exists a constant M = M ( n, Ω , r, f, q − , q + ) so that(3.2) k u q k W ,r (Ω) ≤ M, for all q ∈ Q . In light of Theorem 3.1 we have the following consequence.
Corollary 3.1.
Let W = { w = u q ; q ∈ Q} . Suppose that f > on Γ . Then W is a uniform set of weights for the weighted interpolation inequality (3.1) , with C = C ( n, Ω , r, q − , q + , f ) and µ = µ ( n, Ω , r, q − , q + , f ) . We are now ready to complete the proof of Theorem 1.3. We pick q , ˜ q ∈ Q ̺ and,for sake of simplicity, we set u = u q and ˜ u = u ˜ q .Using the identity ( q − ˜ q ) u = ∆( u − ˜ u ) + ˜ q (˜ u − u ) , we find k ( q − ˜ q ) u k L r (Ω) ≤ k u − ˜ u k W ,r (Ω) + q + k ˜ u − u k L r (Ω) . Hence(3.3) k ( q − ˜ q ) u k L r (Ω) ≤ c k u − ˜ u k W ,r (Ω) , with c = 1 + q + .From Corollary 3.1 there exist two constants C = C ( n, Ω , r, q − , q + , f ) > < µ = µ ( n, Ω , r, q − , q + , f ) < k q − ˜ q k L r (Ω) ≤ C k q − ˜ q k − µC ,α (Ω) k ( q − ˜ q ) u k µL (Ω) . Hence k q − ˜ q k L r (Ω) ≤ C̺ − µ k ( q − ˜ q ) u k µL r (Ω) k u k µL r ∗ (Ω) , where r ∗ is the conjugate exponent of r .Using (3.2) and the fact that W ,r (Ω) is continuously embedded in L r ∗ (Ω) inorder to get k q − ˜ q k L r (Ω) ≤ C̺ − µ k ( q − ˜ q ) u k µL r (Ω) . This estimate together with (3.3) give k q − ˜ q k L r (Ω) ≤ C̺ − µ k u − ˜ u k µW ,r (Ω) as expected.3.2. Proof of Theorem 1.4.
In this proof sgn denotes the sign function given by:sgn ( t ) = − t <
0, sgn (0) = 0 and sgn ( t ) = 1 if t >
0. Let ( a , q ) , (˜ a , q ) ∈ D • λ,µ with a = ˜ a on Γ and(3.4) k a k C , (Ω) ≤ ̺, k ˜ a k C , (Ω) ≤ ̺. For notational convenience we set u = u a and ˜ u = u ˜ a .We obtain after some straightforward computationsdiv( | a − ˜ a |∇ u ) = sgn ( a − ˜ a )[ q ( u − ˜ u ) + div(˜ a ∇ ( u − ˜ u ))] . Whence ˆ Ω div( | a − ˜ a |∇ u ) udx = ˆ Ω sgn ( a − ˜ a )[ q ( u − ˜ u ) + div(˜ a ∇ ( u − ˜ u ))] udx. Taking into account that a = ˜ a on Γ, we get by applying Green’s formula to theleft hand side of the last identity(3.5) ˆ Ω | a − ˜ a ||∇ u | dx = ˆ Ω sgn ( a − ˜ a )[ q ( u − ˜ u ) + div(˜ a ∇ ( u − ˜ u ))] udx. A very standard argument consisting in reducing the BVP satisfied by u to aBVP with zero Dirichlet boundary condition, combined with elementary estimatesgive(3.6) k u k H (Ω) ≤ C. Here and henceforward C = C ( n, Ω , µ, λ, ̺, f ) is a generic constant.We use (3.6) and (3.5) in order to get(3.7) k| a − ˜ a ||∇ u | k L (Ω) ≤ C k u − ˜ u k H (Ω) . As (3.6) remains valid when u is substituted by ˜ u , we obtain by using an inter-polation inequality k u − ˜ u k H (Ω) ≤ C k u − ˜ u k / L (Ω) . YBRID INVERSE PROBLEMS 11
This inequality in (3.7) entails(3.8) k| a − ˜ a ||∇ u | k L (Ω) ≤ C k u − ˜ u k / L (Ω) . On the other hand, we can proceed similarly to the proof of [8, Lemma 3.7], byapplying [13, Theorem 2.1] instead of [8, Lemma 3.6], in order to obtain(3.9) k a − ˜ a k C ( ω ) ≤ C k a − ˜ a k − γC , (Ω) k| a − ˜ a ||∇ u | k γL (Ω) , where 0 < γ = γ ( n, Ω , ω, µ, λ, ̺, f ) < C = C ( n, Ω , ω, µ, λ, ̺, f ) are constants.We end up getting the expected inequality by putting together (3.7) and (3.9).3.3. Uniform lower bound for the gradient at the boundary.
Fix 0 < γ <ν < < σ ≤ σ , setΣ = n σ ∈ C ,ν (Ω); σ ≥ σ and k σ k C ,ν (Ω) ≤ σ o . Pick ϕ ∈ C ,γ (Γ) be non constant so that its critical points are its extrema. Con-sider then the BVP(3.10) (cid:26) div( σ ∇ u ) = 0 in Ω ,u | Γ = ϕ. As C ,ν (Ω) is continuously embedded in C ,γ (Ω), with reference to [14, Theorem6.6, page 98 and Theorem 6.14, page 107] we deduce that, for any σ ∈ Σ, the BVP(3.10) has a unique solution u σ ∈ C ,γ (Ω) so that(3.11) k u σ k C ,γ (Ω) ≤ C, where C = C ( n, Ω , ϕ, σ , σ , γ, ν ) is a constant. Proposition 3.1.
There exists a constant η = η ( n, Ω , ϕ, σ , σ , γ, ν ) > so that (3.12) |∇ u σ ( x ) | ≥ η for all ( x, σ ) ∈ Γ × Σ . Proof.
Let σ ∈ Σ. We first note that, according to the strong maximum principle, u σ achieves both its maximum and its minimum on Γ. That is the maximumand and the minimum of u σ coincide with those of ϕ . But according to Hopf’slemma (e.g. [14, Lemma 3.4, page 34]), if x ∈ Γ is an extremum point then | ∂ ν u ( x ) | >
0. On the other hand, according to the assumption on ϕ , we have |∇ τ u σ ( x ) | = |∇ τ ϕ ( x ) | > x is not an extremum point of ϕ , where ∇ τ stands forthe tangential gradient. In consequence |∇ u σ ( x ) | > x ∈ Γ.Let Σ + = { σ ∈ C ,γ (Ω); σ > } and consider the mapping T : ( x, σ ) ∈ Γ × Σ + → [0 , ∞ ) : ( x, σ )
7→ |∇ u σ | . Fix σ ∈ Σ + . Let σ ′ ∈ C ,γ (Ω) so that k σ ′ k C ,γ (Ω) ≤ σ + σ ′ > min σ/
2. Wehave (cid:26) div( σ ∇ ( u σ − u σ + σ ′ )) = div( σ ′ ∇ u σ + σ ′ ) in Ω , ( u σ − u σ + σ ′ ) | Γ = 0 , We can apply twice [14, Theorem 6.6, page 98 ] in order to get(3.13) k u σ − u σ + σ ′ k C ,γ (Ω) ≤ C k σ ′ k C ,γ (Ω) , where C = C (Ω , min σ ; k σ k C ,γ (Ω) + 1). Therefore, for any x, x ′ ∈ Γ, we have ||∇ u σ ( x ) | − |∇ u σ + σ ′ ( x ′ ) || ≤ |∇ u σ ( x ) | − ∇ u σ ( x ′ ) | + |∇ u σ ( x ′ ) | − ∇ u σ + σ ′ ( x ′ ) |≤ C (cid:16) | x − x ′ | + k σ ′ k C ,γ (Ω) (cid:17) . That is the mapping T is continuous. We complete the proof by noting that,according to [14, Lemma 6.36, page 136], Γ × Σ is a compact subset of Γ × Σ + . (cid:3) Proof of Theorem 1.5.
Hereafter, for δ >
0, we use the notationsΩ δ = { z ∈ Ω; dist( z, Γ) ≤ δ } , Ω δ = { z ∈ Ω; dist( z, Γ) ≥ δ } . Lemma 3.1.
Under the condition min Γ f > , we have u a , q ( f ) ≥ ε for any ( a , q ) ∈ D κ , Λ , where ε = ε ( n, Ω , κ , Λ , f ) > is a constantProof. Let K be the constant in (1.4), ( a , q ) ∈ D κ , Λ and u = u a , q ( f ). If x ∈ Ω and y ∈ Γ are so that | x − y | ≤ δ , then u ( x ) ≥ u ( y ) − K | x − y | β ≥ m − Kδ β , where m = min Γ f .We get by taking δ = [ m/ (2 K )] /β (3.14) u ( x ) ≥ m/ x ∈ Ω δ . We apply Harnak’s inequality (e.g. [14, Theorem 8.21, page 199]) in order to getsup Ω δ/ u ≤ c inf Ω δ/ u, where c = c ( n, Ω , κ , Λ , f ) > m/ (2 c ) ≤ u ( x ) for any x ∈ Ω δ/ . The expected inequality follows by putting together (3.14) and (3.15). (cid:3)
As a consequence of estimate (1.4) and Lemma 3.1 we obtain, after makingstraightforward calculations, the following result.
Corollary 3.2.
Let ( a , q ) ∈ D κ , Λ , f , f ∈ C ,β (Γ) with f > . Set w = w a , q = u a , q ( f ) u a , q ( f ) , h = f f and σ = a u a , q ( f ) . Then w ∈ C ,β (Ω) is the solution of the BVP div( σ ∇ w ) = 0 in Ω , w | Γ = h. Furthermore (3.16) µ ≤ σ, k σ k C ,β (Γ) ≤ µ and k w k C ,β (Ω) ≤ M, for some positive constants µ = µ ( n, Ω , κ , Λ , β, f ) , µ = µ ( n, Ω , κ , Λ , f ) and M = M ( n, Ω , λ, Λ , β, f , f ) . YBRID INVERSE PROBLEMS 13
Let w = w a , q be as in the preceding corollary. In light of Proposition 3.1 wehave |∇ w ( y ) | ≥ η for any y ∈ Γ , for some constant η = η ( n, Ω , κ , Λ , β, f , f ) > δ >
0. Let x ∈ Ω δ and y ∈ Γ so that | x − y | ≤ δ . Then |∇ w ( x ) | ≥ |∇ w ( y ) | − M ˆ e δ ≥ η − M ˆ e δ, where ˆ e is a constant depending only on the embedding C , (Ω) ֒ → C ,β (Ω). Wethen fix δ > |∇ w ( x ) | ≥ η/ x ∈ Ω δ . We get by applying [8, Corollary 3.1] that(3.18) Cρ υ ≤ k∇ w k L ( B ( x,ρ )) for any x ∈ Ω δ and 0 < ρ < δ, where C = C ( n, Ω , κ , Λ , β, f , f , δ ) and υ = υ ( n, Ω , κ , Λ , β, f , f , δ ) are positiveconstants. Lemma 3.2. If w is as in Corollary 3.2 then (3.19) k φ k C (Ω) ≤ C k φ k − γC ,β (Ω) k φ |∇ w |k γL (Ω) , for any φ ∈ C ,β (Ω) , where C = C ( n, Ω , β, κ , Λ , f , f ) > and < γ = γ ( n, Ω , β, κ , Λ , f , f ) < areconstants.Proof. By homogeneity it is sufficient to prove (3.19) with φ ∈ C ,β (Ω) satisfying k φ k C ,β (Ω) = 1.For x ∈ Ω δ and y ∈ B ( x, ρ ), 0 < ρ < δ , we have | f ( x ) | ≤ | f ( y ) | + ρ β . In consequence | φ ( x ) | ˆ B ( x,ρ ) |∇ w ( y ) | dy ≤ ˆ B ( x,ρ ) | φ ( y ) ||∇ w ( y ) | dy + ρ β ˆ B ( x,ρ ) |∇ w ( y ) | dy. As w is non constant, we have k∇ w k L ( B ( x,ρ ) = 0 by the uniqueness of continuationproperty and hence | φ ( x ) | ≤ k φ |∇ w | k L (Ω) k∇ w k L ( B ( x,ρ )) + ρ β . This and (3.18) yield C | φ ( x ) | ≤ ρ − υ k φ |∇ w | k L (Ω) + ρ β for any x ∈ Ω δ and 0 < ρ < δ. That is we have(3.20) C k φ k L ∞ (Ω δ ) ≤ ρ − υ k φ |∇ w | k L (Ω) + ρ β for any 0 < ρ < δ. Next, assume that k φ k L ∞ (Ω δ ) = 0. Pick then x ∈ Ω δ so that | φ ( x ) | = k φ k L ∞ (Ω δ ) . For 0 < ρ < δ , we have | φ ( x ) | ≤ | φ ( x ) | + ρ β x ∈ B ( x , ρ ) ∩ Ω δ . Whence | φ ( x ) | ≤ ( η/ − | φ ( x ) ||∇ w | + ρ β x ∈ B ( x , ρ ) ∩ Ω δ implying | φ ( x ) || B ( x , ρ ) ∩ Ω δ | ≤ ( η/ − ˆ B ( x ,ρ ) ∩ Ω δ | φ ( x ) ||∇ w | + ρ α | B ( x , ρ ) ∩ Ω δ | . But since Ω has the uniform interior cone property, we have | B ( x , ρ ) ∩ Ω δ | ≥ cρ n ,for any 0 < ρ < δ/
2, where c = c (Ω) is a constant. In consequence(3.21) C k φ k L ∞ (Ω δ ) ≤ ρ − n k φ |∇ w | k L (Ω) + ρ β for any 0 < ρ < δ/ . A combination of (3.20) and (3.21) gives(3.22) C k φ k L ∞ (Ω) ≤ ρ − k k φ |∇ w | k L (Ω) + ρ β for any 0 < ρ < δ/ , where k = max( n, υ ).A very known argument consisting in minimizing the right hand side of (3.22)with respect to ρ yields the expected inequality. (cid:3) Let ( a , q ), (˜ a , ˜ q ) ∈ D κ , Λ satisfying ( a , q ) = (˜ a , ˜ q ) on Γ. With the aid of theweighted interpolation inequality (3.19), we can mimic the last part of the proofof [8, Theorem 1.1] in order to prove the following stability inequality, with for j = 1 , u j = u a , q ( f j ) and ˜ u j = u ˜ a , ˜ q ( f j ), k a − ˜ a k C ,β (Ω) + k q − ˜ q k C ,β (Ω) ≤ (cid:16) k u − ˜ u k C (Ω) + k u − ˜ u k C (Ω) (cid:17) γ , where C = C ( n, Ω , β, κ , Λ , f , f ) > < γ = γ ( n, Ω , β, κ , Λ , f , f ) < References [1] G. S. Alberti and Y. Capdeboscq, Lectures on elliptic methods for hybrid inverse problems,Société Mathématique de France, Paris, 25, pp.vii + 230, 2018. 1[2] G. Alessandrini, V. Nesi, Quantitative estimates on Jacobians for hybrid inverse problems,Bulletin SUSU MMCS 8 (3) (2015), 25-41. 5[3] G. Alessandrini, M. Di Cristo, E. Francini and S. Vessella, Stability for quantitative photoa-coustic tomography with well chosen illuminations, Ann. Mat. Pura e Appl. 196 (2) (2017),395-406. 1, 5[4] H. Ammari, Y. Capdeboscq, F. de Gournay, F. Triki and A. Rozanova-Pierrat, Microwaveimaging by elastic deformation, SIAM J. Appl. Math. 71 (6) (2011), 2112-2130. 2[5] G. Bal, K. Ren, G. Uhlmann and T. Zhou, Quantitative thermo-acoustics and related prob-lems, Inverse Prob. 27 (5), 055007, 2011. 8[6] G. Bal and J.C. Schotland, Inverse scattering and acousto-optic imaging, Phys. Rev. Letters,104, 043902, 2010. 3[7] G. Bal and G. Uhlmann, Inverse diffusion theory of photoacoustics, Inverse Prob. 26(2010):085010. 1[8] E. Bonnetier, M. Choulli and F. Triki, Stability for quantitative photoacoustic tomographyrevisited, arXiv:1905.07914. 1, 5, 11, 13, 14[9] M. Choulli, G. Hu and M. Yamamoto, Stability inequality for a semilinear elliptic inverseproblem, arXiv:2001.10940. 5, 9[10] M. Choulli and F. Triki, New stability estimates for the inverse medium problem with internaldata, SIAM J. Math. Anal. 47 (3) (2015), 1778-1799. 1[11] M. Choulli and F. Triki, Hölder stability for an inverse medium problem with internal data,Res. Math. Sci. (2019), 6:9, 15 pp. 1, 9[12] R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science andtechnology, Vol. 3, Springer, Berlin, 2000.[13] N. Garofalo and F.-H. Lin, Unique continuation for elliptic operators: a geometric-variationalapproach, Commun. Pure Appl. Math. 40 (3) (1987) 347-366. 11[14] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order,Springer, Berlin, 1998. 4, 5, 7, 8, 11, 12
YBRID INVERSE PROBLEMS 15 [15] A. Nachman, A. Tamasan, A. Timonov, Conductivity imaging with a single measurement ofboundary and interior data, Inverse Problems 23 (6) (2007), 2551-2563. 5
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