Interior Regularity Estimates in High Conductivity Homogenization and Application

TL;DR

This paper establishes sharp regularity estimates for solutions to high-contrast conductivity equations in media reinforced with thin, highly conducting fibers, with implications for imaging micro-structures.

Contribution

It provides new uniform pointwise regularity estimates near fibers in high conductivity homogenization, especially at a critical distance from the fibers, advancing understanding of micro-structure effects.

Findings

Regularity estimates are valid at a distance epsilon^{1+tau} from fibers.

Gradients of solutions are unbounded in L^p for p>2 near fibers.

Estimates depend sharply on powers of epsilon, with applications to imaging.

Abstract

In this paper, uniform pointwise regularity estimates for the solutions of conductivity equations are obtained in a unit conductivity medium reinforced by a epsilon-periodic lattice of highly conducting thin rods. The estimates are derived only at a distance epsilon^{1+tau} (for some tau>0) away from the fibres. This distance constraint is rather sharp since the gradients of the solutions are shown to be unbounded locally in L^p as soon as p>2. One key ingredient is the derivation in dimension two of regularity estimates to the solutions of the equations deduced from a Fourier series expansion with respect to the fibres direction, and weighted by the high-contrast conductivity. The dependence on powers of epsilon of these two-dimensional estimates is shown to be sharp. The initial motivation for this work comes from imaging, and enhanced resolution phenomena observed experimentally in the presence of micro-structures. We use these regularity estimates to characterize the signature of low volume fraction heterogeneities in the fibred reinforced medium assuming that the heterogeneities stay at a distance epsilon^{1+tau} away from the fibres.