Piecewise smoothness for linear elliptic systems with piecewise smooth coefficients

TL;DR

This paper extends regularity results for linear elliptic systems with piecewise smooth coefficients, achieving higher regularity in the context of composite materials, thus solving an open problem in the field.

Contribution

It establishes piecewise $C^{m+1, ext{ extgamma}}$-regularity for elliptic systems with piecewise $C^{m, ext{ extgamma}}$ coefficients, advancing the understanding of regularity in composite material models.

Findings

Achieved higher regularity results for elliptic systems with piecewise smooth coefficients.

Solved an open problem regarding regularity in composite materials.

Extended previous regularity results from $C^{1, ext{ extgamma}}$ to $C^{m+1, ext{ extgamma}}$.

Abstract

Li and Vogelius, and Li and Nirenberg obtained piecewise $C^{1,\gamma}$-regularity for linear elliptic problems with piecewise $C^{\gamma}$-coefficients which come from composite materials. In this paper, we obtain piecewise $C^{m+1,\gamma}$-regularity for linear elliptic systems with piecewise $C^{m,\gamma}$-coefficients which also come from composite materials. This answers the open problem suggested by Li and Vogelius, and Li and Nirenberg.