Piecewise smoothness for linear elliptic systems with piecewise smooth coefficients

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.

The results and the key idea of this paper was announced at 'International Conference on Partial Differential Equations Related to Material Science', May 6-9, 2021. One long paper is splitted into two papers 'Gradient type estimates for linear elliptic systems from composite materials' and 'Piecewise smoothness for linear elliptic systems with piecewise smooth coefficients'