71 research outputs found
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 posed on
a bounded domain with prescribed boundary conditions. In spatial dimension
, it is known that the number of critical points (where ) is
related to the number of oscillations of the boundary condition independently
of the (positive) coefficient . We show that the situation is different
in dimension . More precisely, we obtain that for any fixed (Dirichlet
or Neumann) boundary condition for on , there exists an open
set of smooth coefficients such that vanishes at least
at one point in . 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 on a
subdomain is not modified for appropriate bounded, sufficiently high-contrast,
smooth coefficients .
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 for
which the stability of the reconstructions will inevitably degrade.Comment: 26 pages, 4 figure
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
Size Estimates of Unknown Boundaries with Robin Type Condition
We deal with the problem of determining an unknown part of the boundary of an electrical conductor that is not accessible from an exter- nal observation and where a corrosion process is going on. We obtain estimates from above and below of the size of this damaged region
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