Stable determination of the nonlinear term in a quasilinear elliptic equation by boundary measurements

We establish a Lipschitz stability inequality for the problem of determining the nonlinear term in a quasilinear elliptic equation by boundary measurements. We give a proof based on a linearization procedure together with special solutions constructed from the fundamental solution of the linearized problem.

div(a(u)A∇u) = 0 in Ω, where a is a scalar function and A is a matrix with variable coefficients.Assume that we can define the map Λ a : f → ∂ ν u(f ) between two well chosen spaces, where u(f ) is the solution of the BVP (1.1) when it exists.In the case where a is supposed to be unknown we ask whether Λ a determines uniquely a.This problem can be seen as a Calderón type problem for the quasilinear BVP (1.1).We are mainly interested in establishing a stability inequality for this inverse problem.
1.3.Main result.We show in Subsection 2.1 that, under assumption (a1), for each f ∈ C 2,α (Γ) the BVP (1.1) admits a unique solution u a = u a (f ) ∈ C 2,α (Ω).Furthermore, when a satisfies both (a1) and (a2) the Dirichlet-to-Neumann map Here and Henceforth • op stands for the norm of B(H The following uniqueness result is straightforward from the preceding theorem. It is worth noticing that if a 1 and a 2 are as in Corollary 1.1 then a 1 and a 2 satisfy also (a2), (a3) and (ã1) with These µ, γ and γ depends of course on a 1 and a 2 .

1.4.
Comments.There are only very few stability inequalities in the literature devoted to the determination of nonlinear terms in quasilinear and semilinear elliptic equations by boundary measurements.The semilinear case was studied in [5] by using a method based on linearization together with stability inequality for the problem of determining the potential in a Schrödinger equation by boundary measurements.The result in [5] was recently improved in [4].Both quasilinear and semilinear elliptic inverse problems were considered in [14] where a method exploiting the singularities of fundamental solutions was used to establish stability inequalities.This method was used previously in [3] to obtain a stability inequality at the boundary of the conformal factor in an inverse conductivity problem.We show in the present paper how we can modify the proof of [3, (1.2) of Theorem 1 .1]to derive the stability inequality stated in Theorem 1.1.The localization argument was inspired by that in [14].
There is a recent rich literature dealing with uniqueness issue concerning the determination of nonlinearities in elliptic equations by boundary measurements using the so-called higher order linearization method.We refer to the recent work [2] and references therein for more details.We also quote without being exhaustive the following references [1,6,7,10,11,13,15,16,17,18,19,20,21,22] on semilinear and quasilinear elliptic inverse problems.

Preliminaries
2.1.Solvability of the BVP and the Dirichlet-to-Neumann map.Suppose that a satisfies (a1) and introduce the divergence form quasilinear operator The following observation will be crucial in the sequel : As b(•, •, 0) = 0 one can easily check that Q 0 satisfies to the assumptions of [9, Theorem 15.12, page 382].Let f ∈ C 2,α (Γ).Il light of the observation above we derive that the quasilinear BVP The uniqueness of solutions of (1.1) holds from [9, Theorem 10.7, page 268] applied to Q.
Assume that a satisfies (a1) and (a2) and let Multiplying the first equation of (2.5) by v and integrating over Ω.We then obtain from Green's formula If f ∈ B m then the last identity together with (2.3) yield Here and henceforth, We endow H 1/2 (Γ) with the quotient norm For each ψ ∈ H −1/2 (Γ) we define χψ by where •, • 1/2 is the duality pairing between H 1/2 (Γ) and its dual H −1/2 (Γ).
It is not difficult to check that χψ ∈ H −1/2 (Γ), supp(χψ) ⊂ Γ 1 and the following identity holds . This identity will be very useful in the sequel.
Using that u a (f ) is the solution of the BVP (1.1), we easily check that the right hand side of the above identity is independent of v, v ∈ φ.
This identity suggests to define the Dirichlet-to-Neumann map associated to a, by the formula Using (2.6), we get 2.2.Differentiability properties.We need a gradient bound for the solution of the BVP (1.1).To this end we set Fix f ∈ B + m and a ∈ A .We apply [9, Theorem 15.9, page 380] with Next, we establish that Λ a , a ∈ A , is Fréchet differentiable in a neighborhood of the origin.For η > 0 define m , Proof.Let η > 0 to be determined later.Pick f, g ∈ B η m and set h = g − f .Let σ = a(u a (g)) and From (2.9) and Poincaré's inequality we derive then we obtain from which the expected inequality follows readily.
In the sequel η m , m > 0, will denote the constant in Lemma 2.1. where We can proceed similarly to the proof of Lemma 2.1 to derive that The expected inequality follows easily by using Lemma 2.1.
Let f ∈ B ηm m .Similarly to the calculations we carried out in the proof of Lemma 2.1, we show that the bilinear continuous form is coercive.In light of Lemma A.2, we obtain that the BVP and hence We refer to Appendix A for the exact definition of weak solutions.
Next, pick ǫ > 0 such that f Simple computations show that w is the weak solution of the BVP In particular, we have Using that and the uniform continuity of a ′ in [−̺m, ̺m], we obtain On the other hand similar estimates as above give . The last two inequalities together with (2.13) yield

In other words we proved that
Using the definition of Λ a we can then state the following result.Proposition 2.1.For each m > 0, the mapping

Proof of the main result
As we already mentioned we give a proof based on an adaptation of [3, proof of (1.2) of Theorem 1.1] combined with a localization argument borrowed from [14].
3.1.Special solutions.We construct in a general setting special solutions of a divergence form operator vanishing outside Γ 0 .To this end, let

and max
1≤i,j≤n for some constant λ > 1.
Recall that the canonical parametrix for the operator div(A∇• ) is given by Pick x 0 ∈ Γ 0 and let r 0 > 0 sufficiently small in such a way that B(x 0 , r 0 ) ∩ Γ ⋐ Γ 0 .As B(x 0 , r 0 ) \ Ω contains a cone with a vertex at x 0 , we find δ 0 > 0 and a vector ξ ∈ S n−1 such that, for each 0 < δ ≤ δ 0 , we have In the sequel Ω 0 = Ω ∪ B(x 0 , r 0 ) and u δ = u y δ , 0 < δ ≤ δ 0 , where u y δ is given by Theorem 3.1.Reducing δ 0 if necessary, we may assume that We assume that b and b * are coercive: there exists where Similarly, the BVP where C is as in (3.3).
On the other hand, using that the continuous bilinear form is coercive, we derive that the BVP where C = C(n, Ω 0 , λ) > 0.
Let z δ = zδ + u δ − H(•, y δ ).Then we have the decomposition w δ = H(•, y δ ) + z δ .Using once again Theorem 3.1 and (3.11), we obtain we obtain from Lemma (A.4) The rest of the proof is similar to that of Lemma 3.2.

We use this identity with
we obtain by taking into account (2.7) Let H j = H when A = σ j A, j = 1, 2. That is we have According to Lemmas 3.2 and 3.3, with f = g = f δ , A = σ j A and P = P j , j = 1 or j = 2, we have In the rest subsection we always need to reduce δ 0 .For simplicity convenience we keep the notation δ 0 .
and define the family of localized Dirichlet-to-Neumann maps ( Λt a ) t∈R as follows Λt