  This paper is devoted to establishing the pointwise upper and lower bounds
estimates of the gradient of the solutions to a class of general elliptic
systems with H\"{o}lder continuous coefficients in a narrow region where the
upper and lower boundaries is $C^{1,\gamma},$ $0<\gamma<1$, weaker than the
previous $C^{2,\gamma}$ assumption. These estimates play a key role in the
damage analysis of composite materials. From our results, the damage may
initiate from the narrowest place.
