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In this paper we develop in detail the geometric constructions that lead to many uniqueness results for the determination of polyhedral sets, typically scatterers, by a finite minimal number of measurements. We highlight how unique continuation and a suitable reflection principle are enough to proceed with the constructions, without any other assumption on the underlying partial differential equation or the boundary condition. We also aim to keep the geometric constructions and their proofs as simple as possible. To illustrate the applicability of this theory, we show how several uniqueness results present in the literature immediately follow from our arguments. Indeed we believe that this theory may serve as a roadmap for establishing similar uniqueness results for other partial differential equations or boundary conditions., Comment: 21 pages

In this paper we consider a general class of anisotropic energies in three dimensions and give a complete characterisation of their minimisers. We show that, depending on the Fourier transform of the interaction potential, the minimiser is either the normalised characteristic function of an ellipsoid or a measure supported on a two-dimensional ellipse. In particular, it is always an ellipsoid if the transform is strictly positive, while when the Fourier transform is degenerate both cases can occur. Finally, we show an explicit example where loss of dimensionality of the minimiser does occur.

We find extremely general classes of nonsmooth open sets which guarantee Mosco convergence for corresponding Sobolev spaces and the validity of Sobolev inequalities with a uniform constant. An important feature of our results is that the conditions we impose on the open sets for Mosco convergence and for the Sobolev inequalities are of the same nature, therefore it is easy to check when both are satisfied. Our analysis is motivated, in particular, by the study of the stability of the direct acoustic scattering problem with respect to the scatterer, which we also discuss. Concerning Mosco convergence in dimension 3 or higher, our result extends all those previously known in the literature. Concerning Sobolev inequalities, our approach seems to be new and considerably simplifies the conditions previously required for the stability of acoustic direct scattering problems., Comment: 33 pages

In this paper we characterise the minimiser for a class of nonlocal perturbations of the Coulomb energy. We show that the minimiser is the normalised characteristic function of an ellipsoid, under the assumption that the perturbation kernel has the same homogeneity as the Coulomb potential, is even, smooth off the origin and sufficiently small. This result can be seen as the stability of ellipsoids as energy minimisers, since the minimiser of the Coulomb energy is the normalised characteristic function of a ball.

We consider the inverse conductivity problem with discontinuous conductivities. We show in a rigorous way, by a convergence analysis, that one can construct a completely discrete minimization problem whose solution is a good approximation of a solution to the inverse problem. The minimization problem contains a regularization term which is given by a total variation penalization and is characterized by a regularization parameter. The discretization involves at the same time the boundary measurements, by the use of the complete electrode model, the unknown conductivity and the solution to the direct problem. The electrodes are characterized by a parameter related to their size, which in turn controls the number of electrodes to be used. The discretization of the unknown and of the solution to the direct problem is characterized by another parameter related to the size of the mesh involved. In our analysis we show how to precisely choose the regularization, electrodes size and mesh size parameters with respect to the noise level in such a way that the solution to the discrete regularized problem is meaningful. In particular we obtain that the electrodes and mesh size parameters should decay polynomially with respect to the noise level., Comment: 58 pages

We deal with an inverse elastic scattering problem for the shape determination of a rigid scatterer in the time-harmonic regime. We prove a local stability estimate of log log type for the identification of a scatterer by a single far-field measurement. The needed a priori condition on the closeness of the scatterers is estimated by the universal constant appearing in the Friedrichs inequality., Comment: 32 pages, 2 figures

We prove decay estimates in the interior for solutions to elliptic equations in divergence form with Lipschitz continuous coefficients. The estimates explicitly depend on the distance from the boundary and on suitable notions of frequency of the Dirichlet boundary datum. We show that, as the frequency at the boundary grows, the square of a suitable norm of the solution in a compact subset of the domain decays in an inversely proportional manner with respect to the corresponding frequency. Under Lipschitz regularity assumptions, these estimates are essentially optimal and they have important consequences for the choice of optimal measurements for corresponding inverse boundary value problem., Comment: 42 pages

In an influential paper, Tadmor, Nezzar and Vese (Multiscale Model. Simul. (2004)) introduced a hierarchical decomposition of an image as a sum of constituents of different scales. Here we construct analogous hierarchical expansions for diffeomorphisms, in the context of image registration, with the sum replaced by composition of maps. We treat this as a special case of a general framework for multiscale decompositions, applicable to a wide range of imaging and nonlinear inverse problems. As a paradigmatic example of the latter, we consider the Calder\'on inverse conductivity problem. We prove that we can simultaneously perform a numerical reconstruction and a multiscale decomposition of the unknown conductivity, driven by the inverse problem itself. We provide novel convergence proofs which work in the general abstract settings, yet are sharp enough to settle an open problem on the hierarchical decompostion of Tadmor, Nezzar and Vese for arbitrary functions in $L^2$. We also give counterexamples that show the optimality of our general results., Comment: This version corrects some small inaccuracies from the previous version published in Advances in Mathematics 346 (2019) 1009-1066, in particular see condition 3) at the beginning of Section 2.1

The aim of this short note is to give an alternative proof, which applies to functions of bounded variation in arbitrary domains, of an inequality by Maz'ya that improves Friedrichs inequality. A remarkable feature of such a proof is that it is rather elementary, if the basic background in the theory of functions of bounded variation is assumed. Never the less, it allows to extend all the previously known versions of this fundamental inequality to a completely general version. In fact the inequality presented here is optimal in several respects. As already observed in previous proofs, the crucial step is to provide conditions under which a function of bounded variation on a bounded open set, extended to zero outside, has bounded variation on the whole space. We push such conditions to their limits. In fact, we give a sufficient and necessary condition if the open set has a boundary with $\sigma$-finite surface measure and a sufficient condition if the open set is fully arbitrary. Via a counterexample we show that such a general sufficient condition is sharp., Comment: 12 pages. To appear in Mathematische Nachrichten

We study the inverse conductivity problem with discontinuous conductivities. We consider, simultaneously, a regularisation and a discretisation for a variational approach to solve the inverse problem. We show that, under suitable choices of the regularisation and discretisation parameters, the discrete regularised solutions converge, as the noise level on the measurements goes to zero, to the looked for solution of the inverse problem., Comment: 33 pages

In this paper we characterise the equilibrium measure for a nonlocal and anisotropic weighted energy describing the interaction of positive dislocations in the plane. We prove that the minimum value of the energy is attained by a measure supported on the vertical axis and distributed according to the semi-circle law, a well-known measure which also arises as the minimiser of purely logarithmic interactions in one dimension. In this way we give a positive answer to the conjecture that positive dislocations tend to form vertical walls. This result is one of the few examples where the minimiser of a nonlocal energy is explicitly computed and the only one in the case of anisotropic kernels., Comment: 16 pages. Accepted version, to appear on Comm. Pure Appl. Math

This paper is concerned with the scattering problem for time-harmonic electromagnetic waves, due to the presence of scatterers and of inhomogeneities in the medium. We prove a sharp stability result for the solutions to the direct electromagnetic scattering problem, with respect to variations of the scatterer and of the inhomogeneity, under minimal regularity assumptions for both of them. The stability result leads to uniform bounds on solutions to the scattering problems for an extremely general class of admissible scatterers and inhomogeneities. The uniform bounds are a key step to tackle the challenging stability issue for the corresponding inverse electromagnetic scattering problem. In this paper we establish two optimal stability results of logarithmic type for the determination of polyhedral scatterers by a minimal number of electromagnetic scattering measurements. In order to prove the stability result for the direct electromagnetic scattering problem, we study two fundamental issues in the theory of Maxwell equations: Mosco convergence for H(curl) spaces and higher integrability properties of solutions to Maxwell equations in nonsmooth domains., Comment: 48 pages. To appear in JEMS

The aim of the paper is to establish optimal stability estimates for the determination of sound-hard polyhedral scatterers in $\mathbb{R}^N$, $N \geq 2$, by a minimal number of far-field measurements. This work is a significant and highly nontrivial extension of the stability estimates for the determination of sound-soft polyhedral scatterers by far-field measurements, proved by one of the authors, to the much more challenging sound-hard case. The admissible polyhedral scatterers satisfy minimal a priori assumptions of Lipschitz type and may include at the same time solid obstacles and screen-type components. In this case we obtain a stability estimate with $N$ far-field measurements. Important features of such an estimate are that we have an explicit dependence on the parameter $h$ representing the minimal size of the cells forming the boundaries of the admissible polyhedral scatterers, and that the modulus of continuity, provided the error is small enough with respect to $h$, does not depend on $h$. If we restrict to $N=2,3$ and to polyhedral obstacles, that is to polyhedra, then we obtain stability estimates with fewer measurements, namely first with $N-1$ measurements and then with a single measurement. In this case the dependence on $h$ is not explicit anymore and the modulus of continuity depends on $h$ as well., Comment: 39 pages

We investigate the continuity of boundary operators, such as the Neumann-to-Dirichlet map, with respect to the coefficient matrices of the underlying elliptic equations. We show that for nonsmooth coefficients the correct notion of convergence is the one provided by $H$-convergence (or $G$-convergence for symmetric matrices). We prove existence results for minimum problems associated to variational methods used to solve the so-called inverse conductivity problem, at least if we allow the conductivities to be anisotropic. In the case of isotropic conductivities we show that on certain occasions existence of a minimizer may fail., Comment: To appear in Inverse Problems

Photolithography is a process in the production of integrated circuits in which a mask is used to create an exposed pattern with a desired geometric shape. In the inverse problem of photolithography, a desired pattern is given and the mask that produces an exposed pattern which is close to the desired one is sought. We propose a variational approach formulation of this shape design problem and introduce a regularization strategy. The main novelty in this work is the regularization term that makes the thresholding operation involved in photolithography stable. The potential of the method is demonstrated in numerical experiments., Comment: 28 pages, 36 figures

We deal with the stability issue for the determination of outgoing time-harmonic acoustic waves from their far-field patterns. We are especially interested in keeping as explicit as possible the dependence of our stability estimates on the wavenumber of the corresponding Helmholtz equation and in understanding the high wavenumber, that is frequency, asymptotics. Applications include stability results for the determination from far-field data of solutions of direct scattering problems with sound-soft obstacles and an instability analysis for the corresponding inverse obstacle problem. The key tool consists of establishing precise estimates on the behavior of Hankel functions with large argument or order., Comment: 49 pages

We develop a very general theory on the regularized approximate invisibility cloaking for the wave scattering governed by the Helmholtz equation in any space dimensions via the approach of transformation optics., Comment: 47 pages, 12 figures

We study the stability for the direct acoustic scattering problem with sound-hard scatterers with minimal regularity assumptions on the scatterers. The main tool we use for this purpose is the convergence in the sense of Mosco. We obtain uniform decay estimates for scattered fields and we investigate how a sound-hard screen may be approximated by thin sound-hard obstacles., Comment: 23 pages

We numerically implement the variational approach for reconstruction in the inverse crack and cavity problems developed by one of the authors. The method is based on a suitably adapted free-discontinuity problem. Its main features are the use of phase-field functions to describe the defects to be reconstructed and the use of perimeter-like penalizations to regularize the ill-posed problem. The numerical implementation is based on the solution of the corresponding optimality system by a gradient method. Numerical simulations are presented to show the validity of the method., Comment: 15 pages, 12 figures

We consider the inverse problem of determining an optical mask that produces a desired circuit pattern in photolithography. We set the problem as a shape design problem in which the unknown is a two-dimensional domain. The relationship between the target shape and the unknown is modeled through diffractive optics. We develop a variational formulation that is well-posed and propose an approximation that can be shown to have convergence properties. The approximate problem can serve as a foundation to numerical methods., Comment: 28 pages, 1 figure

We treat the inverse problem of determining material losses, such as cavities, in a conducting body, by performing electrostatic measurements at the boundary. We develop a numerical approach, based on variational methods, to reconstruct the unknown material loss by a single boundary measurement of current and voltage type. The method is based on the use of phase-field functions to model the material losses and on a perimeter-like penalization to regularize the otherwise ill-posed problem.We justify the proposed approach by a convergence result, as the error on the measurement goes to zero., Comment: 28 pages

We discuss the ill-posed Cauchy problem for elliptic equations, which is pervasive in inverse boundary value problems modeled by elliptic equations. We provide essentially optimal stability results, in wide generality and under substantially minimal assumptions. As a general scheme in our arguments, we show that all such stability results can be derived by the use of a single building brick, the three-spheres inequality., Comment: 57 pages, review article

In the paper, G. Alessandrini and L. Rondi, ``Determining a sound-soft polyhedral scatterer by a single far-field measurement'', Proc. Amer. Math. Soc. 133 (2005), pp. 1685-1691, on the determination of a sound-soft polyhedral scatterer by a single far-field measurement, the proof of Proposition 3.2 is incomplete. In this corrigendum we provide a new proof of the same proposition which fills the previous gap., Comment: 3 pages

We prove that a sound-soft polyhedral scatterer is uniquely determined by the far-field pattern corresponding to an incident plane wave at one given wavenumber and one given incident direction., Comment: The proof of Proposition 3.2 is incomplete. Corrigendum is available through arXiv (math.AP/0601406)

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., Comment: 38 pages