Small volume expansions for elliptic equations

TL;DR

This paper develops small volume expansion formulas for elliptic equations with inclusions, extending existing theory to rapid parameter fluctuations and analyzing the influence of inclusions on boundary solutions.

Contribution

It generalizes boundary trace expansions for elliptic equations to arbitrary, possibly rapid, parameter fluctuations inside inclusions, and compares diffusion and Helmholtz cases.

Findings

Derived boundary trace expansions up to order  d for inclusions.

Constructed inclusions with influence order   d+1, lower than the typical  d.

Compared expansions for diffusion and Helmholtz equations via Liouville transformation.

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 $\nabla \cdot D^\eps \nabla u^\eps=0$ or $(-\Delta + q^\eps) u^\eps=0$ 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 $\eps^{2d}$, where $d$ is dimension and $\eps$ is the diameter of the inclusion. We construct inclusions whose leading influence is of order at most $\eps^{d+1}$ rather than the expected $\eps^d$. We also compare the expansions for the diffusion and Helmholtz equation and their relationship via the classical Liouville change of variables.