On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries

We show that if $u$ is a weak solution to $\operatorname{div} (A\nabla u) = 0$ on an open set $\Omega$ containing a Lipschitz domain $D$, where $A = kI{\chi _D} + I{\chi _{\Omega /D}}(k > 0,k \ne 1)$. Then, the nontangential maximal function of the gradient of $u$ lies in ${L^2}(\partial D)$.