Stable Determination of the Discontinuous Conductivity Coefficient of a Parabolic Equation

We deal with the problem of determining a time varying inclusion within a thermal conductor. In particular we study the continuous dependance of the inclusion from the Dirichlet-to-Neumann map. Under a priori regularity assumptions on the unknown defect we establish logarithmic stability estimates.Comment: 36 page

Stable determination of an inclusion by boundary measurements

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.Comment: 19 page

Critical Points for Elliptic Equations with Prescribed Boundary Conditions

This paper concerns the existence of critical points for solutions to second order elliptic equations of the form $\nabla\cdot \sigma(x)\nabla u=0$ 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 $\nabla u=0$) is related to the number of oscillations of the boundary condition independently of the (positive) coefficient $\sigma$. We show that the situation is different in dimension $n\geq3$. More precisely, we obtain that for any fixed (Dirichlet or Neumann) boundary condition for $u$ on $\partial X$, there exists an open set of smooth coefficients $\sigma(x)$ such that $\nabla u$ 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 $\nabla u$ on a subdomain is not modified for appropriate bounded, sufficiently high-contrast, smooth coefficients $\sigma(x)$. 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 $\sigma(x)$ for which the stability of the reconstructions will inevitably degrade.Comment: 26 pages, 4 figure

El gobierno local en el contexto de las nuevas tendencias de desarrollo. cacicazgos y redes de políticas

El objetivo de este trabajo es avanzar en la discusión acerca del papel que se le ha atribuido al gobierno local en el desarrollo ante los procesos de globalización, y para lo cual considero la experiencia mexicana. El elemento que problematiza lo anterior, es la baja capacidad política y administrativa que ostentan las autoridades locales para negociar con líderes o poderes fácticos que siguen presentes en pleno Siglo XXI. Para ello, expongo las principales teorías de desarrollo. También describo los procesos de alternancia electoral suscitados en los municipios mexicanos y sus implicaciones en las redes de políticas públicas, e involucramiento de poderes y líderes factuales en ellas. Lo anterior me permitirá discutir el papel que juega el poder informal en las acciones de desarrollo, la capacidad de los gobiernos municipales por concertar con tales cotos de poder informal a fin de cumplir con la misión que les atribuyen las teorías de desarrollo alternativas

Stable determination of an inclusion in an elastic body by boundary measurements (unabridged)

We consider the inverse problem of identifying an unknown inclusion contained in an elastic body by the Dirichlet-to-Neumann map. The body is made by linearly elastic, homogeneous and isotropic material. The Lam\'e moduli of the inclusion are constant and different from those of the surrounding material. Under mild a-priori regularity assumptions on the unknown defect, we establish a logarithmic stability estimate. For the proof, we extend the approach used for electrical and thermal conductors in a novel way. Main tools are propagation of smallness arguments based on three-spheres inequality for solutions to the Lam\'e system and refined local approximation of the fundamental solution of the Lam\'e system in presence of an inclusion.Comment: 58 pages, 4 figures. This is the extended, and revised, version of a paper submitted for publication in abridged for