Affiliations: Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA. E-mail: [email protected] | Université de Lyon, Université Lyon 1, CNRS, UMR 5208 Institut Camille Jordan/ISTIL, Bâtiment du Doyen Jean Braconnier, 43, blvd du 11 novembre 1918, F-69622 Villeurbanne Cedex, France. E-mail: [email protected]
Abstract: This paper analyzes the influence of general, small volume, inclusions on the trace at the domain's boundary of the solution to elliptic equations of the form ∇·Dε∇uε=0 or (−Δ+qε)uε=0 with prescribed Neumann conditions. The theory is well known when the constitutive parameters in the elliptic equation assume the values of different and smooth functions in the background and inside the inclusions. We generalize the results to the case of arbitrary, and thus possibly rapid, fluctuations of the parameters inside the inclusion and obtain expansions of the trace of the solution at the domain's boundary up to an order ε2d, where d is dimension and ε is the diameter of the inclusion. We construct inclusions whose leading influence is of order at most εd+1 rather than the expected εd. We also compare the expansions for the diffusion and Helmholtz equation and their relationship via the classical Liouville change of variables.
Keywords: reconstruction of small volume inclusions, diffusion equation, Helmholtz equation, asymptotic expansions, inverse problems
