A sharp regularity result of solutions of a transmission problem
Giovanna Citti, Fausto Ferrari · Proceedings of the American Mathematical Society · 2011
In this paper we study the regularity of solutions of a transmission problem arising in studying a fiber-reinforced composite media. It is described through a divergence type equation div ( A ∇ u ) = h , \mbox {div}(A abla u) = h, in an open set D D , where h h is a bounded function and A A is a uniformly elliptic matrix, bounded and with piecewise Hölder continuous coefficients. The subdomains D i D_i where A A is of class C α C^{\alpha } have disjoint closure and are of class C 1 , α C^{1,\alpha } . Exploiting an idea contained in a paper by Li and Vogelius, we obtain the optimal regularity result for solutions, proving that they are of class C 1 , α ( D ¯ i ) C^{1,\alpha }(\bar D_i) .