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

Gradient estimates for elliptic systems from composite materials with closely spaced stiff $C^{1,\gamma}$ inclusions