Following a recent paper by N. Mandache (Inverse Problems 17 (2001), pp. 1435-1444), we establish a general procedure for determining the instability character of inverse problems. We apply this procedure to many elliptic inverse problems concerning the determination of defects of various types by different kinds of boundary measurements and we show that these problems are exponentially ill-posed.

We treat the stability issue for the three dimensional inverse imaging modality called Quantitative Photoacoustic Tomography. We provide universal choices of the illuminations which enable to recover, in a Hölder stable fashion, the diffusion and absorption coefficients from the interior pressure data. With such choices of illuminations we do not need the nondegeneracy conditions commonly used in previous studies, which are difficult to be verified a-priori.

We deal with the problem of determining an inclusion within an electrical conductor from electrical boundary measurements. Under mild a priori assumptions we establish an optimal stability estimate.accounts. Unfortunately, for such a problem, the uniqueness question, not to mention stability, remains a largely open issue.Let us illustrate briefly the main steps of our arguments. We must recall that Isakov's approach to uniqueness is essentially based on two arguments a) the Runge approximation theorem, b) the use of solutions with Green's function type singularities.Also here we shall use singular solutions, and indeed we shall need an accurate study of their asymptotic behavior when the singularity gets close to the set of discontinuity ∂D of the conductivity coefficient 1+(k−1)χ D in (1.1), see Proposition 3.2. On the other hand, it seems that Runge's theorem, which is typically based on nonconstructive arguments, (Lax, [L], Kohn and Vogelius [K-V]) is not suited for stability estimates and therefore we introduced a different approach based on quantitative estimates of unique continuation, see Proposition 3.3.In Section 2 we formulate our main hypotheses and state the stability result, Theorem 2.2. In Section 3 we prove Theorem 2.2 on the basis of some auxiliary Propositions, whose proof is deferred to the following Section 4. The main resultLet us introduce our regularity and topological assumptions on the conductor Ω and on the unknown inclusion D. To this purpose we shall need the following definitions. In places, we shall denote a point x ∈ R n byDefinition 2.1. Let Ω be a bounded domain in R n . Given α, 0 < α ≤ 1, we shall say that a portion S of ∂Ω is of class C 1,α with constants r, L > 0 if, for any P ∈ S, there exists a rigid transformation of coordinates under which we have P = 0 andwhere ϕ is a C 1,α function on B r (0) ⊂ R n−1 satisfying ϕ(0) = |∇ϕ(0)| = 0 and ϕ C 1,α (Br (0)) ≤ Lr.Definition 2.2. We shall say that a portion S of ∂Ω is of Lipschitz class with constants r, L > 0 if for any P ∈ S, there exists a rigid transformation of coordinates under which we have P = 0 andwhere ϕ is a Lipschitz continuous function on B r (0) ⊂ R n−1 satisfying ϕ(0) = 0 and ϕ C 0,1 (Br (0)) ≤ Lr.Remark 2.1. We have chosen to scale all norms in a such a way that they are dimensionally equivalent to their argument. For instance, for any ϕ ∈ C 1,α (B r (0)) we set ϕ C 1,α (Br(0)) = ϕ L ∞ (Br(0)) + r ∇ϕ L ∞ (Br (0)) + r 1+α |∇ϕ| α,Br (0) .For given numbers r, M , δ, L > 0, 0 < α < 1, we shall assume

We consider an inverse problem of determining a viscosity coefficient in the Navier-Stokes equation by observation data in a neighborhood of the boundary. We prove the Lipschitz stability by the Carleman estimates in Sobolev spaces of negative order.

This paper concerns the existence of critical points for solutions to second order elliptic equations of the form ∇ · σ (x)∇u = 0 posed on a bounded domain X with prescribed boundary conditions. In spatial dimension n = 2, it is known that the number of critical points (where ∇u = 0) is related to the number of oscillations of the boundary condition independently of the (positive) coefficient σ . We show that the situation is different in dimension n ≥ 3. More precisely, we obtain that for any fixed (Dirichlet or Neumann) boundary condition for u on ∂ X , there exists an open set of smooth coefficients σ (x) such that ∇u vanishes at least at one point in X . By using estimates related to the Laplacian with mixed boundary conditions, the result is first obtained for a piecewise constant conductivity with infinite contrast, a problem of independent interest. A second step shows that the topology of the vector field ∇u on a subdomain is not modified for appropriate bounded, sufficiently high-contrast, smooth coefficients σ (x). These results find applications in the class of hybrid inverse problems, where optimal stability estimates for parameter reconstruction are obtained in the absence of critical points. Our results show that for any (finite number of) prescribed boundary conditions, there are coefficients σ (x) for which the stability of the reconstructions will inevitably degrade.