Influence of inclusions on boundary measurements for elliptic equations

This work concerns the analysis of the influence of general, small volume, inclusions on the trace at the domain's boundary of the solution to elliptic equations of the form $ \nabla \cdot D^\varepsilon \nabla u^\varepsilon =0$ or $ (-\Delta + q^\varepsilon ) u^\varepsilon =0$ with prescribed Neumann conditions. The theory is well-known for instance when the constitutive parameters in the elliptic equation are constant in the background and inside the inclusions, and this leads in this case to high order asymptotic expansions of the trace of the solution at the domain's boundary; or, in dimension less than three, when the background is smooth but not constant, and in this configuration only first order expansions are available, that is of order $ \varepsilon ^{d}$ where $ d$ is dimension and $ \varepsilon $ is the diameter of the inclusion. We generalize the results to varying backgrounds in any dimensions and to the case of arbitrary, and thus possibly rapid, fluctuations of the parameters inside the inclusion and obtain expansions up to an order $ \varepsilon ^{2d}$. Besides, we construct inclusions whose leading influence is of order at most $ \varepsilon ^{d+1}$ rather than the expected $ \varepsilon ^d$. We also compare the expansions for the diffusion and Helmholtz equations and their relationship via the classical Liouville change of variables. See [4] for more details.

Olivier Pinaud 2010-07-30