Eric Bonnetier

Université de Grenoble-Alpes

Asymptotic expansions of the solution to an elliptic PDE in the presence
of inclusions of small size have found succesfull applications in inverse
problems, as a means to detect inhomogeneities from boundary measurements
in a robust way.

In the course, we revisit some of this work and the applications
it led to. We also consider perturbations on the boundary, such as
replacing a Neumann boundary condition by a Dirichlet boundary condition
(or vice-versa). We derive the general form of an asymptotic expansion
between the solutions of the perturbed and unperturbed PDE’s, and show that the
convergence relies on an argument similar to compensated compactness.

We illustrate two of the applications of the resulting asymptotic formulas~:
We consider shape optimization problems, in which the subset of the boundary of
a domain, where a specific boundary condition is applied, is part of the design
variables. We also discuss the problem of how such asymptotic expansions can
be used to achieve approximate cloaking.