Publications

2016

• Uniqueness results for inverse Robin problems with bounded coefficient
  
  • Baratchart Laurent
  • Bourgeois Laurent
  • Leblond Juliette
  Journal of Functional Analysis, Elsevier, 2016. In this paper we address the uniqueness issue in the classical Robin inverse problem on a Lipschitz domain $\Omega\subset\RR^n$, with $L^\infty$ Robin coefficient, $L^2$ Neumann data and isotropic conductivity of class $W^{1,r}(\Omega)$, $r>n$. We show that uniqueness of the Robin coefficient on a subpart of the boundary given Cauchy data on the complementary part, does hold in dimension $n=2$ but needs not hold in higher dimension. We also raise on open issue on harmonic gradients which is of interest in this context. (10.1016/j.jfa.2016.01.011)
  DOI : 10.1016/j.jfa.2016.01.011
• Solvability of a Volume Integral Equation Formulation for Anisotropic Elastodynamic Scattering
  
  • Bonnet Marc
  Journal of Integral Equations and Applications, Rocky Mountain Mathematics Consortium, 2016, 28, pp.169-203. This article investigates the solvability of volume integral equations arising in elastodynamic scattering by penetrable obstacles. The elasticity tensor and mass density are allowed to be smoothly heterogeneous inside the obstacle and may be discontinuous across the background-obstacle interface, the background elastic material being homogeneous. Both materials may be anisotropic, within certainme limitations for the background medium. The volume integral equation associated with this problem is first derived, relying on known properties of the background fundamental tensor. To avoid difficulties associated with existing radiation conditions for anisotropic elastic media, we also propose a definition of the radiating character of transmission solutions. The unique solvability of the volume integral equation (and of the scattering problem) is established. For the important special case of isotropic background properties, our definition of a radiating solution is found to be equivalent to the Sommerfeld-Kupradze radiation conditions. Moreover, solvability for anisotropic elastostatics, directly related to known results on the equivalent inclusion method, is recovered as a by-product.
• Existence of guided waves due to a lineic perturbation of a 3D periodic medium
  
  • Delourme Bérangère
  • Joly Patrick
  • Vasilevskaya Elizaveta
  Applied Mathematics Letters, Elsevier, 2016. In this note, we exhibit a three dimensional structure that permits to guide waves. This structure is obtained by a geometrical perturbation of a 3D periodic domain that consists of a three dimensional grating of equi-spaced thin pipes oriented along three orthogonal directions. Homogeneous Neumann boundary conditions are imposed on the boundary of the domain. The diameter of the section of the pipes, of order ε > 0, is supposed to be small. We prove that, for ε small enough, shrinking the section of one line of the grating by a factor of √ µ (0 < µ < 1) creates guided modes that propagate along the perturbed line. Our result relies on the asymptotic analysis (with respect to ε) of the spectrum of the Laplace-Neumann operator in this structure. Indeed, as ε tends to 0, the domain tends to a periodic graph, and the spectrum of the associated limit operator can be computed explicitly. (10.1016/j.aml.2016.11.017)
  DOI : 10.1016/j.aml.2016.11.017